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3URAL MATHEMATICAL JOURNAL, Vol. 9, No. 1, 2023, pp. 29-48
DOI: 10.15826/umj.2023.1.003
AROUND THE ERDOS-GALLAI CRITERION
Vitaly A. Baransky"", Tatiana A. Senchonok"^
Ural Federal University, 51 Lenina av., Ekaterinburg, 620075, Russian Federation
"[email protected] "" [email protected]
Abstract: By an (integer) partition we mean a non-increasing sequence A = (Ax, A2,...) of non-negative integers that contains a finite number of non-zero components. A partition A is said to be graphic if there exists a graph G such that A = dpt G, where we denote by dpt G the degree partition of G composed of the degrees of its vertices, taken in non-increasing order and added with zeros. In this paper, we propose to consider another criterion for a partition to be graphic, the ht-criterion, which, in essence, is a convenient and natural reformulation of the well-known Erdos—Gallai criterion for a sequence to be graphical. The ht-criterion fits well into the general study of lattices of integer partitions and is convenient for applications. The paper shows the equivalence of the Gale—Ryser criterion on the realizability of a pair of partitions by bipartite graphs, the ht-criterion and the Erdos—Gallai criterion. New proofs of the Gale—Ryser criterion and the Erdos—Gallai criterion are given. It is also proved that for any graphical partition there exists a realization that is obtained from some splitable graph in a natural way. A number of information of an overview nature is also given on the results previously obtained by the authors which are close in subject matter to those considered in this paper.
Keywords: Integer partition, Threshold graph, Bipartite graph, Bipartite-threshold graph, Ferrers diagram.
1. Introduction
Everywhere by a graph we mean a simple graph, i.e. a graph without any loops and multiple edges.
An integer partition, or simply partition, is a non-increasing sequence A = (A1, A2,...) of nonnegative integers that contains a finite number of non-zero components (see [1]). Let sum A denote the sum of all components of the partition A and call it the weight of the partition A. It is often said that a partition of A is a partition of a non-negative integer n = sum A. The length 1(A) of a partition A is the number of its non-zero components. For convenience, the partition A will often be written as A = (A1,..., At), where t > 1(A), i.e. we will omit the zeros by starting from some zero component without forgetting that the sequence is infinite.
The theory of partitions is one of the actively developing areas of contemporary combinatorics, the foundations of which were laid by L. Euler as early as the 18th century. For some information about the achievements of this theory in the 19th and 20th centuries, see [1].
A partition A is said to be graphic if there is a graph G such that A = dpt G, where we denote by dpt G the degree partition composed by the degrees of vertices taken in non-increasing order with added zeros. In this case, the graph G is called a realization of the partition A, and A is said to be realized by the graph G. It is clear that adding or removing isolated vertices does not change the degree partition of the graph.
A finite sequence A = (A1, A2,..., An) of non-negative integers such that A1 > A2 > ■ ■ ■ > An and n is a natural number will be called an n-sequence. Such an n-sequence is called graphic if there is a simple graph G on n vertices such that deg(v1) = A1,..., deg(vn) = An, where v1,..., vn is the sequence of all its vertices; and the graph G is called a realization of the n-sequence A, and A is said to be realized by the graph G.
Obviously, an n-sequence A = (Ai, A2,..., An) is graphic if and only if the partition (Ai, A2,..., An, 0,0,...) obtained from A by adding zeros, is graphic.
It should be noted that in [4] an algorithm was constructed for generating all graphic n-sequences which does not generate any non-graphic sequences during calculations.
We call an n-sequence A = (A1, A2,..., An) proper (proper n-sequence) if
1) n - 1 > Ai > A2 > ■ ■ ■ > An;
2) the sum A of all components of the sequence A is even.
Obviously, any graphic n-sequence is proper.
The first criterion for an n-sequence to be graphic was found by Erdos and Gallai [14].
Theorem 1 [14, Erdos and Gallai]. Let A = (A1, A2,..., An) be a proper n-sequence. Then A is a graphic n-sequence if and only if it is satisfied the inequality
k n
J^Ai < k(k - 1)+ min{k,Ai} i= 1 i=k+1
for any k = 1,..., n.
It is an easy matter to prove that the condition "k = 1,..., n" can be replaced by the condition "k = 1,..., r(A)", where r(A) = max{i|Ai > i} is the rank of the n-sequence A.
The paper [17] considers all seven graphic criteria known by that time: Erdos-Gallai, Ryser, Berge, Fulkerson-Hoffman-McAndrew, Bollobas, Grunbaum, Hasselbarth. It is shown, how they deduced from each other, and a new, more elegant proof of the Erdos-Gallai criterion is given.
In this paper, we propose for consideration (in our terminology) another graphic criterion, the ht-criterion (see Theorem 2), which has the simplest and most natural form. Moreover, as will be seen below, this criterion fits well into the general study of partition lattices.
It should be noted that considerations close to the ht-criterion can be found in [15].
As can be seen below, the ht-criterion can be in essence considered as a reformulation of the Erduos-Gallai criterion which is convenient for applications.
In 2, by fairly simple reasoning, we establish the equivalence of Theorem 2 on the ht-criterion and the Erduos-Gallai Theorem 1.
§ 3 will provide a transparent proof of the Gale-Ryser theorem on the realization of two partitions by a bipartite graph, that does not use the partition graphicity criteria.
In § 4, with the Gale-Ryser theorem and without any partition graphicity criteria, we prove Theorem 2 on the ht-criterion and, therefore, obtain a new natural proof of the Erduos-Gallai theorem. From the proof of Theorem 2 we also extract Theorem 4 and Theorem 5 on the existence of a special kind of realizations for arbitrary partitions, and this result is one of the main ones in this paper.
§ 5 will give another proof of the Gale-Ryser theorem, in which the ht-criterion is used. As a result, we will show how the Gale-Ryser theorem, the ht-criterion Theorem 2 and the Erdos-Gallai Theorem 1 can be derived from each other.
At the end of paragraphs 4 and 5, we give a brief review of the previously obtained results of authors which are close in subject matter to those considered in this paper.
2. On the ht-criterion
Let us first give the necessary definitions.
We denote by IPL the set of all partitions of all natural numbers with added zero partition, and by IPL(m) for a non-negative integer m we denote the set of all its partitions. On the sets IPL and IPL(m), consider the dominance relation < [13], by setting A < ^ if
A1 + A2 +-----+ Ai < + ^2 +-----+ №
for any i = 1, 2, . . . , i. e. the prefix partial sums of the partition A do not exceed the corresponding prefix partial sums of the partition
The partition can be conveniently depicted as a Ferrers diagram, which can be thought of as a set of square boxes of the same size (see an example in Fig. 1, which shows the partition (6, 5,4,4,3,2,1,1) of the number 26, the length of this partition is 8). We will use Cartesian notation for Ferrers diagrams aligned to the bottom-left corner of the 1st quadrant. Components correspond to columns and decrease in size from left to right. The coordinates for boxes resemble the standard Cartesian coordinates for the Euclidean plane.
Figure 1. The Ferrers diagram of the partition (6, 5, 4, 4, 3, 2, 1, 1).
Let us define two types of elementary transformations (see [2-5]) of the partition A = (A1, A2,..., An), where n = 1(A) + 1.
Let there be natural numbers i, j € {1,..., n} such that i < j < 1(A) + 1 and
1) Ai - 1 > Ai+1,
2) Aj-1 > Aj + 1,
3) Ai = Aj + 5, where 5 > 2.
We will say that the partition n = (A1,..., Ai — 1,..., Aj + 1,..., An) is obtained from the partition A = (A1,...,Ai,..., Aj,..., An) by an elementary transformation of the first type (or through box movement). It should be noted that n differs from A on exactly two components with numbers i and j.
For the Ferrers diagram, such a transformation means moving the top box from the i-th column to the right to the top of the j-th column. The conditions 1), 2) and 3) guarantee that after such a move, a partition will again be obtained. It should be noted that a box can also be thrown to the zero component with the number 1(A) + 1.
The fact that n is obtained from A by moving a box will be briefly written in the form A ^ n It should be noted that an elementary transformation of the first type preserves the weight of the partition, while the length of the partition can be preserved or lifted by 1.
We now define elementary transformations of the second type for the partition A = (A1, A2,...).
Let Ai — 1 > Ai+1, where i < 1(A). A transformation that replaces A by
n = (A1,..., Ai-1, Ai - 1, Ai+1,...) will be called an elementary transformation of the second type (or a box removal).
As in the previous case, we will briefly write A ^ n, i.e., the notation A ^ n means that n obtained from A by an elementary transformation of the first or second type. It should be noted that box removal reduces the weight of the partition by exactly 1, while the length of the partition can be preserved or lowered by 1.
On the set IPL and on sets of the form IPL(m), we define the relation < by setting n < A if n can be obtained from A by sequentially applying a finite number (possibly a zero one) of elementary transformations of the stated types.
Of course, in the case of IPL(m) we are forced to use only elementary transformations of the first type, which do not change the weight of the partitions. It was proved in [3] and [5] that the relation < on each of the considered sets coincides with the dominance relation <1, and each of these sets is a lattice.
It is essential to note that the use of elementary transformations is often more convenient than the use of inequalities appearing in the definition of the dominance relation.
It should be noted that the IPL lattice is a disjoint union of lattices IPL(m), where m ranges over non-negative integers corresponding to some natural transitive system of embeddings [5].
Let A = (A1, A2,...) be a partition. We determine the rank r(A) of the partition A by setting r(A) = max{i|Ai > i}. Obviously, the rank r = r(A) of a partition A is equal to the number of boxes on the main diagonal of the Ferrers diagram of this partition. The maximum square made up of boxes and symmetrical about the main diagonal is called the Durfey square of the partition A (see Fig. 2).
Fig. 2 shows the partition A = (6, 5,4,4,3,2,1,1). Here r(A) = 4 and the Durfey square consists of 16 = 4 ■ 4 boxes. Any row and any column of a Durfey square consists of r = r(A) boxes.
For each partition A, we will consider an conjugate partition A* whose components are equal to the number of boxes in the corresponding rows of the Ferrers diagram of the partition A. It is clear that the Ferrers diagram of the partition A* can be obtained from the Ferrers diagram of the partition A by mirror symmetry with respect to the main diagonal. For Fig. 2, A* = (8, 6, 5, 4, 2, 1) is satisfied. Of course, the equality r(A*) = r(A) is true.
It should be noted (see [3]) that for any m € N the mapping A ^ A* is an involutive anti-automorphism of the lattice IPL(m) such that (A*)* = A and the condition y1 < y2 implies the condition y* > y2.
Let £, n € IPL(m) and f be an elementary transformation of the first type £ ^ n, transforming £ into n. It is plain to see (see Fig. 3) that the inverse transformation f * to the transformation f is an elementary transformation of the first type n* £*, transforming n* into £ *. Ferrers diagrams stated in Fig. 3 can also be considered as Ferrers diagrams of the corresponding conjugate partitions, only then they need to be considered lying "on their side", i.e., the components should be read in rows.
Similarly, if f is an elementary transformation of the second type £ ^ n (box removal) that transforms £ into n, then the inverse transformation f * of f is a box insertion that transforms n*
Figure 2. The main diagonal in the Durfey square.
Figure 3. The elementary transformathion of the first type and the inverse transformation.
into £*, i. e., n —r
£ * (it is convenient to use the same symbol — to indicate box insertion).
We now define the head and tail of the partition A = (Ai, A2,...), the rank of which is equal to r.
As the head hd (A), we take the partition that is obtained from the partition A by reducing the first r components by the same number r—1 and zeroing all components with numbers r+1, r+2,... (for an example, see the diagram in Fig. 4).
As the tail tl (A) we take a partition for which the Ferrers diagram of the conjugate partition is obtained from the Ferrers diagram of the partition A by deleting the first r columns, i. e. the Ferrers diagram of the partition tl (A)* is located to the right of the Durfey square (see Fig. 4).
tl(A)*
Figure 4. The head and the tail of the partition (6, 5, 4, 4, 3, 2, 1, 1).
The arrows in Fig. 4 indicate the directions in which the components of the partitions hd(A) and tl(A) are read. It is clear that the upper row of the Durfey square enters the Ferrers diagram of the partition hd(A), the partition hd(A) is "read" by column from left to right, and the length of the partition hd(A) is equal to r. The partition tl(A) is "read" by row from bottom to top and the length of the partition tl(A)* is equal to 1(A) — r(A), and the length of the partition tl(A) is equal to tl(A)* — the value the first component of the partition tl(A)*, hence l(tl(A)) < r(A).
For n-sequences, the concepts of rank, head,, and tail are introduced in exactly the same way. In order to consider the ht-criterion for partitions to be graphic, we present two auxiliary lemmas.
Lemma 1. Let A = (A1, A2,..., An) be an n-sequence. Then for any k = 1,..., r = r(A), the condition
k n
J^Ai < k(k — 1)+ min{k,Ai} (2.1)
¿=i ¿=k+i
is equivalent to the condition
k k £hd(A)i tl(A)i,
¿=i ¿=i
where hd (A) and tl(A) are the i-components, respectively, of the head and tail of the partition A for any i = 1,..., k.
fa
hd(A) = (3, 2,1,1) tl(A) = (4, 2,1)
hd(A)
Proof. Note first that for k = 1,..., r the sum
n
y^ min{k, Ai}
i=k+1
is equal to the number of boxes of the Ferrers diagram of the sequence A located in the strip standing at the intersection of rows with numbers from 1 to k and columns with numbers from k + 1 to n (see the shaded area in Fig. 5).
hd(A)
r — 1
k r — k
Figure 5. The Ferrers diagram of the sequence.
Let us rewrite inequality (2.1) in the equivalent form
kn
Ai — k(k — 1) — k(r — k) < min{k, Ai} — k(r — k),
i=1 i=k+1
after transformations, the resulting inequality is equivalent to the inequality
kn
y^ Ai — k(r — 1) < ^ min{k,Ai} — k(r — k).
i=1 i=k+1
It is plain to see that
k k n k
J^Ai — k(r — 1) = J^hd(A)i and ^ min{k, Ai} — k(r — k) = J^tl(A)i
i=1 i=1 i=k+1 i=1
Therefore, inequality (2.1) is equivalent to the inequality
¿hd(A)i tl(A)i.
i=1
i=1
□
Lemma 2. Let A = (A1, A2,..., An) be an n-sequence and hd (A) < tl (A). Then A1 < n — 1.
Proof. Let r = r(A). The condition hd (A) < tl (A) implies hd (A)1 < tl (A)1, so
A1 — (r — 1) < 1(A) — r.
Therefore,
A1 < 1(A) — 1 < n — 1.
□
Since l(hd (A)) = r(A) and l(tl (A)) < r(A) for any partition A, due to Lemmas 1 and 2 the statement of the Erdos-Gallay theorem is equivalent to the following statement.
Theorem 2. Let A = (A1, A2,...) be an arbitrary nonzero partition of even weight. Then A is a graphic partition if and only if
The criterion for a partition to be graphic specified in Theorem 2 will be called the ht-criterion.
Our next main goal is to prove Theorem 2 without using the Erdos-Gallai theorem and other criteria for graphic partitions. To do this, we first give a direct, transparent proof of the well-known Gale-Ryser theorem on the representation of a pair of partitions by a bipartite graph, in which we will not use any of the criteria for the graphicity of partitions.
For a bipartite graph H = (V1,E, V2), where V and V2 are its parts, we denote by dptH(V1) and dptH(V2) the degree partitions corresponding to its parts, i.e. partitions composed of the degrees of the vertices of the corresponding parts in non-increasing order and added with zeros.
Theorem 3 [16, Gale D., Ryser H.J. (1957)]. Let a = (a1, a2,...) and ft = (ft1,ft2,...) be nonzero partitions. Then there is a bipartite graph H = (V1,E, V2) such that dptH(V1) = a and dptH(V2) = ft if and only if sum(a) = sum(ft) and a < ft*.
We need the following
Lemma 3. Let G = (V, E) be a graph, V = {v1, v2,..., vn} and A = (A1, A2,...) is a degree partition corresponding to the graph G such that Ai = deg(vi) for any i = 1,... , 1(A) and Aj = 0 for any j > 1(A). Then hd (A) < tl(A).
Proof. By virtue of Lemma 1, it suffices to check the validity of inequality (2.1) for any k = 1,..., r = r(A). For k = 1,..., r, we estimate the sum
Let us first estimate the part of this sum contributed by edges from G({v1, v2,..., vk}). Obviously, degG(vi) < k — 1 for any i = 1,... ,k. Therefore, this part of the sum does not exceed k(k — 1).
Let us now estimate the contribution to the sum made by edges of the form Vjvi, where 1 < j < k and k + 1 < i < n. For a given i such that k + 1 < i < n, the number of such edges does not exceed k and does not exceed Ai, i.e., does not exceed min{k, Ai}.
hd (A) < tl (A).
3. About the Gale—Ryser theorem
k
Consequently, inequality (2.1) is satisfied for any k = 1,..., r = r(A), and therefore, by virtue of Lemma 1, the inequality hd (A) < tl (A) also holds. □
A graph G is said to be splitable if the set of its vertices can be represented as a disjoint union of a clique V1 and a coclique V2 (i.e., V1 n V2 = 0, V1 generates a complete subgraph K(V1), and V2 generates a zero subgraph O(V2) with an empty set of edges). For such a graph G, the set of all edges can be represented as a disjoint union of the set of all edges of the complete subgraph K(V1) and the set E of all its edges connecting vertices from V with vertices from V2. Therefore, it is convenient to represent a splitable graph G in the form G = (K(V1 ),E,O(V2)). We will simply write G = (K(V1), E, V2).
The following lemma proves the necessity of the conditions in the Gale-Ryser theorem.
Lemma 4. Let H = (V1, E, V2) be an arbitrary bipartite graph and dpt (V1) = a, dpt (V2) = P be the degree partitions of its parts. Then
1) sum(a) = sum(^) = m, where m = |E|;
2) a < P*.
(It should be noted that the condition a < P* is equivalent to the condition P < a*, since the transformation 7 — 7* is an involutive antiautomorphism of the lattice IPL(m).)
Proof. 1) It is obvious.
2) Without loss of generality, we will assume that H does not have any isolated vertices. Let V1 = {u1,u2,...,up} and V2 = {v1, v2,..., }, where a» = deg(-Uj) for any i = 1,...,p and Pj = deg(vj) for any j = 1,..., q.
Let us embed the graph H into a splitable graph H+ = (K(V1),E, V2) by adding to H all possible edges connecting pairs of different vertices from V1. In the graph H+, the set V is a clique and the set V2 is a coclique. Then
a1 + (p — 1) > a2 + (p — 1) > ■ ■ ■ > ap + (p — 1) > p > P1 > P2 > ■ ■ ■ > Pq,
i. e.
dpt (H+) = (a1 + (p — 1), a2 + (p — 1),..., ap + (p — 1), P1, P2,..., Pq).
Let dpt (H+) = A. Then
r(A) = p, hd(A) = (a1,a2,...,ap) = a, tl *(A) = (P1 ,P2,...,Pq )= P. By virtue of Lemma 3, we have hd (A) < tl (A). Since tl (A) = P*, we obtain a < P*. □
To prove the sufficiency of the Gale-Ryser theorem conditions, we need additional definitions and two lemmas.
Let (x, v, y) be a triple of different vertices of the graph G = (V, E) such that xv € E and vy € E. We call such a triple 1) lifting if deg(x) < deg(y), 2) lowering if deg(x) > 2 + deg(y), and 3) preserving if deg(x) = 1 +deg(y).
A transformation ^ of a graph G such that <^(G) = G — xv + vy, i.e., first the edge xv is removed from G and then the edge vy is added (see Fig. 6), is called the rotation of the edge (in the graph G around vertex v) corresponding to the triple (x,v,y).
The rotation of an edge in the graph <^(G) corresponding to the triple (y,v,x) is called the reverse rotation of an edge for the rotation
x<
•V
x•
(v W
Figure 6. The rotation of an edge.
V
The rotation of an edge in G corresponding to a triple (x, v, y) is called 1) lifting if the triple (x,v,y) is lifting, 2) lowering if the triple (x,v,y) is lowering, and 3) preserving if the triple (x, v,y) is preserving.
It should be noted that the cases when deg(x) = 1 or deg(y) = 0 will be considered admissible, i. e. after the rotation of an edge, an isolated vertex may appear or the rotation of an edge will occur in the graph G with the addition of a new isolated vertex.
It should be noted that the rotation of an edge in the graph G is lowering if and only if the reverse rotation of the edge is lifting.
If the graph G2 obtained from the graph Gi by rotating an edge, then we write Gi — G2.
Let dpt (G) be the degree partition corresponding to the graph G and ^ be the rotation of the edge in the graph G corresponding to the triple (x,v,y), where xv € E and vy / E. Then the following assertions are correct.
1. If is a lifting rotation of an edge, then dpt (G) < dpt (<^(G)), moreover, dpt (G) is obtained from dpt (<^(G)) with one elementary transformation of the first type, and G is obtained from <^(G) with the reverse (lowering) rotation of an edge.
2. If is the lowering rotation of an edge, then dpt (G) > dpt (<^(G)), moreover, dpt (<^(G)) is obtained from dpt (G) with one elementary transformation of the first type, and G is obtained from <^(G) with the reverse (lifting) rotation of an edge.
3. If is the preserving rotation of an edge, then dpt (G) = dpt (<^(G)), and G is obtained from <^(G) with the reverse (preserving) rotation of an edge.
Let (x, v, y) be a triple of distinct vertices of a bipartite graph H = (V1, E, V2) such that xv € E and vy / E. If x,y € V1 and v € V2, then we call the triple V1-triple, but if x,y € V2 and v € V1, then the triple will be called a V2-triple. We will say that V1-triples correspond to V1-rotations of edges, and V2-triples correspond to V2-rotations of edges.
Lemma 5. 1. Let H1 = (V1,E1,V2) and H2 = (Vl,E2,V2) be bipartite graphs, and the graph H2 is obtained from the graph H1 by the lowering V1-rotation of the edge H1 — H2. Then dptH2 (V1) is obtained from dptHl (V1) with an elementary transformation of the first type, i. e., dptHl (V1) — dpt^2 (V1), and dpt^2 (V2) = dptHl (V2).
2. Let H1 = (V1,E1, V2) be a bipartite graph and the partition i be obtained from the partition dptHl (V1) with an elementary transformation of the first type, i. e., dptH2 (V1) — Then there exists a bipartite graph H2 = (V1,E2, V2) that is obtained from the graph H1 by means of a lowering V1-rotation of an edge H1 — H2 and for which i = dptH2 (V2) and dptH2 (V2) = dptHi (V2).
Proof. Assertion 1 is obvious. Let us prove the assertion 2. Let dptHl(V1) = (A1,..., Aj,..., Aj,..., At), where Aj > 2 + Aj, 1 < i < j < t and ¡i = (A1,..., Aj — 1,..., Aj + 1,...,At). Let for vertices x,y € V1, degHl(x) = Aj and degHl(y) = Aj. Since Aj > Aj, there is a vertex v € V2 such that xv € E1 and vy / E1. Let ^ be a lowering V1-rotation of an
edge corresponding to the triple (x,v,y) in the graph H1. Then ^ = dptH2 (V1) in the graph H2 = ^(H1). The equality dptH2 (V2) = dptHl (V2) is obvious, since the lowering ^-rotation of an edge does not change the degrees of vertices in V2. □
The following lemma guarantees the sufficiency of the Gale-Ryser theorem conditions.
Lemma 6. Let a = (a1,a2,...) and P = (P1,P2,...) be nonzero partitions such that sum (a) =sum(P) and a < P*. Then there is a bipartite graph H = (V1,E, V2) such that dpt (V1) = a and dpt (V2) = P.
Proof. Let 1(a) = p and l(P) = q. Take two sets V1 and V2 such that |V1| = |V2| = m, where m = sum (a) = sum(P). It is clear that p, q < m. Let V1 = {u1,u2,... ,um} and V2 = {v1,v2,... ,vm|.
First, we construct a bipartite graph Ho = (V1,Eo, V2). To do this, it suffices to specify the neighborhoods of its vertices v1, v2,..., vm. Let
nho (v1) = {U1,U2, . . . },nHO (v2) = {U1,U2, . . . , }, . . . , Nffo (vq) = {U1,U2, . . . , }
and nHo(vj) = 0 if i > q (for such i, Pj = 0). Neighborhoods of vertices v1,v2,...,vm form a system of nested subsets in V and uniquely define the graph H0.
Let us consider an m x m bipartite adjacency matrix A of the bipartite graph H0, where the columns of the matrix A correspond to the vertices v1, v2,..., vm and are numbered from left to right, and the rows correspond to the vertices u1,u2,...,um and are numbered from bottom to top. In matrix A, boxes containing 1's are concentrated in the lower left corner and form a Ferrers diagram for P = (P1, P2,...), and by reading 1's row by row, we get a Ferrers diagram for P*, i. e. dptHo (V1) = P* and dptHo (V2) = P.
Example 1. Let P = (3,2,1,1,0,...). Then P* = (4,2,1,0,...) and m = 7. Then the matrix A has the following form:
U" 0 0 0 0 0 0 0
«6 0 0 0 0 0 0 0
V>5 0 0 0 0 0 0 0
«4 0 0 0 0 0 0 0
V>3 1 0 0 0 0 0 0
U'2 1 1 0 0 0 0 0
U\ 1 1 1 1 0 0 0
Vi V-2 v-3 v4 V5 V6 v7
Since P* > a and the partitions P* and a have the same weight m, in IPL(m) there is a sequence of elementary transformations of the first type such that
P* = £(0)-£(1) - ••• -£(t) = a.
According to this sequence, by applying t times the assertion 2 of Lemma 5, we obtain, with lowering V1-rotations of edges, a sequence of bipartite graphs:
Ho = (V1, Eo, V2) - H1 = (V1, E1, V2) -----► Ht = (V1, Et, V2)
such that dptHi (V1) = £(i) and dptHi (V2) = P for any i = 0,1,..., t. The graph Ht = (V1, Et, V2) is the sought one, since dptHt (V1) = £(t) = a and dptHt (V2) = P. □
Gale-Ryser theorem proceeds from Lemmas 4 and 6.
4. Proof of Theorem 2 using the Gale—Ryser theorem
Now our goal is to prove Theorem 2 without using the Erdos-Gallai theorem and other criteria for partitions to be graphic. In addition, along the way, we will prove one of the main results of the paper, Theorem 5, on the existence for any nonzero partition A of a realization that is obtained from some splitable graph with a certain sequence of lowering rotations of edges.
For this we need two auxiliary lemmas.
Lemma 7. 1. Let H1 = (V1,E1) and H2 = (V2,E2) be graphs and let the graph H2 be obtained from the graph H1 with the lowering rotation of an edge H1 — H2. Then dpt (H2) is obtained from dpt(H1) with an elementary transformation of the first type dpt (H1) — dpt(H2).
2. Let H1 = (V1, E1) be a graph and let the partition i be obtained from the partition dpt (H1) with an elementary transformation of the first type dpt (H1) — Then there exists a graph H2 = (V2,E2) which is obtained from the graph H1 by means of a lowering rotation of the edge H1 — H2 and for which i = dpt (H2) is satisfied.
Proof. Assertion 1 is obvious. Assertion 2 is proved similarly to assertion 2 of Lemma 5. □
Lemma 8. For any partition A of even weight, the number C = sum tl (A) — sum hd (A) is even.
P r o o f. Since
sumA = sum hd (A) + r (r — 1) + sum tl (A),
where r = r(A), sumtl(A) +sumhd(A) is even. It follows that the number sumtl(A) — sumhd(A) is also even. □
The necessity of the condition of Theorem 2 is proved in Lemma 3.
Let us now give a proof of the sufficiency of the condition of Theorem 2, in which the Erdos-Gallai criterion and other criteria for the graphicity of partitions are not used, but the Gale-Ryser theorem is used.
Let A = (A1, A2,...) be an arbitrary nonzero partition of even weight, hd(A) < tl (A). Our goal is to prove the existence of a realization for the partition A and to reveal a special kind of the realization that we obtain.
Let r = r(A) be the rank of the partition A. It should be noted that l(hd (A)) = r and l(tl (A)) = (tl *(A))1 < r, where (tl *(A))1 is the first component of the partition tl *(A).
Since hd (A) < tl (A), there is a sequence of elementary transformations from tl (A) to hd(A), and both types of elementary transformations are admissible. Let us apply the algorithm [8] for constructing the shortest sequence of this type. For this, we take the component wise difference of the partitions
tl (A) — hd (A) = (tl (A)1 — hd (A)b tl (A)2 — hd (A)2,..., tl (A)r — hd (A)r, 0,...).
Example 2. Assuming that A = (8, 7, 7, 7,6,6, 5,3,3,2,2). Then
r(A) = 6, hd (A) = (3,2,2,2,1,1), tl *(A) = (5,3,3,2,2), C = sumtl *(A) — sum hd (A) = 15 — 11 = 4, tl (A) = (5, 5,3,1,1).
hd(A)
Figure 7. The head and the tail of the partition.
It should be noted that the conditions of Theorem 2 are satisfied since sum A = 56 and
hd (A) = (3,2,2,2,1,1) < (5, 5,3,1,1) = tl (A),
since the prefix sums of the sequence (3,2,2,2,1,1) do not exceed the corresponding prefix sums of the sequence (5, 5,3,1,1,0). Take the component-wise difference of the partitions tl (A) and hd (A):
tl (A) = (5, 5, 3, 1, 1, 0)
_hd (A) = (3, 2, 2, 2, 1, 1)
tl (A) - hd (A) = (+2 +3 +1 -1 0 -1)
The partition tl (A) over the partition hd(A) in components with numbers 1, 2, and 3 has hills (see [8]) of heights 2, 3, and 1, respectively, and in components with numbers 4 and 6, it has pits (see [8]), each of which has depth 1. It should be noted that the sum of the heights of all hills is C plus the sum of the depths of all pits [8]. A pit is called admissible if adding 1 to it does not change the non-increasing order for the resulting partition (preserves the stepped form of the Ferrers diagram). Here the 6-pit (in the component with number 6), like the 4-pit, is admissible for the partition tl (A). According to [8], if there is a pit, then there should be an admissible pit.
The algorithm for constructing some shortest sequence of elementary transformations from tl (A) to hd (A) [8] consists in sequentially moving a box into an admissible pit from the hill closest to it on the left or in removal the upper box from the last hill; be removal exactly C boxes. The admissible pits in the partition to be transformed will be chosen from right to left. The length of such a sequence is equal to the sum of the heights of all the hills of the partition tl (A) over the partition hd (A). Let us build such a sequence in our example. First we remove two boxes, then we fill two pits, and finally we remove two more boxes.
tl (A) = (5, 5, 3, 1, 1, 0)4; (5, 5, 2, 1, 1, 0)3 ;
+2, +3, +1, -1, 0, -1 +2, +3, 0, -1, 0, -1
(5, 4, 2, 1, 1, o)2 ; (5, 3, 2, l, 1, 1)2; (5, 2, 2, 2, 1, 1)2
+2, +2, 0, -1, 0, -1 +2, +1, 0, -l, 0, 0 +2, 0, 0, 0, 0, 0
(4, 2, 2, 2, 1, 1) 1; ; (3, 2, 2, 2, 1, 1)0 = hd (A)
+1, o, 0, 0, 0, 0 0, 0, 0, 0, 0, 0
In this context, underlining at each step shows the choice of a hill for the subsequent elementary transformation of the second type or the choice of a hill and an admissible pit for the subsequent elementary transformation of the first type. In addition, at the top right of the current partition, we state the number of boxes that still need to be removed.
Since the partition A has an even weight, by virtue of Lemma 8 the number C is even. Assuming that s = 0.5C. With the component-by-component difference of the partitions tl (A) and hd (A), by using the considered algorithm [8], we construct the shortest sequence of elementary transformations from tl (A) to hd(A), and at the beginning we remove s boxes and at the end we remove s more boxes:
tl (A) = n(0) - n(i) ------ n(s) = t = t(0) ------ T(t) = £ (4 2)
= £(0) — £(1) — ■ ■ ■ — £(s) = hd (A).
Since l(tl (A)) < r and l(hd (A)) = r, in components with numbers greater than r in the difference tl (A) — hd (A) all components are equal to 0, i. e. among them there are no hills or pits. Obviously, l(£) < r is true, since in (4.2) when the pits are filled and the hills are removed, the lengths of the partitions cannot become larger than r. Hence, due to the equality l(hd (A)) = r and the fact that in the sequence of transformations
£ = £(0) — £(1) —----- £(s) = hd (A)
only boxes are removed, we get the equality l(£) = r, i. e. for £ = (£1, £2,..., £r) £r > 1.
Let us also consider a sequence of inverse transformations in reverse order from t* to tl * (A), which are box insertions:
t* = n*s) — n*s-i) —----- n*0) = tl *(A).
Since in the sequence of transformations from t * to tl * (A) only block insertions occur and in the partition tl (A) all components do not exceed r, in this sequence all components of all partitions do not exceed r and, in particular, (t *)1 < r.
Let us now take a pair of partitions: a = £ and ft = t *. Then
sum a = sum t = sum t * = sum ft,
since the transition from t to £ in (4.2) does not remove the boxes, and by virtue of (4.2) a = £ < t = ft * also holds. Therefore, by virtue of the Gale-Ryser theorem, there is a bipartite graph H = (V1, R, V2) such that dptH(V1) = a and dptH(V2) = ft. We add V1 to a complete subgraph by adding 1/2 ■ r(r — 1) edges. We obtain a splitable graph H + = (K(V1), R, V2). Since r > (t *)1 = ft1 and the partition £ = (£1,£2,... , £r) satisfies £r > 1, we have
£1 + (r — 1) > ■ > £r + (r — 1) > r > ft1 > ft2 > ...
Therefore, the degree partition corresponding to the graph H+ has the form:
dpt(H+) = (£1 + (r — 1),..., £r + (r — 1), ft1, ft2,..., fti(^), 0,...).
Let CT(0) = dpt(H+). Since ft = t *,
a(0) = dpt(H+) = (£1 + (r — 1),... £r + (r — 1), (t *)1, (t *)2,..., (t *)i(r»), 0,...).
It is clear that r(a(0)) = r, hd (ct(0)) = £ = £(0) and tl *(o"(0)) = t* = n*s). It should be noted that for s = 0, due to (4.2) we have
hd (^(0)) = £ = hd (A), tl >(0)) = t * = tl *(A)
and r(dpt(H+)) = r, so dpt(H+) = A, i. e. the splitable graph H + is a realization of the partition A.
Thus, in the case s = 0, the existence of a splitable realization for A has been proved.
In what follows, we will assume that s > 0.
Starting from the partition CT(0) = dpt(H+), we sequentially perform s elementary transformations of the first type in the partitions.
At step 1), we remove a box from the head of the partition CT(0) by removing the box £(0) ^ and insert this box into the partition n*s) by inserting the box n*s) ^ n*s-1).
As a result, we get a partition CT(1) such that r(a(1)) = r, hd (^(1)) = £(1) and tl*(a(1)) = n*s-1), and n(1) is obtained from CT(0) with an elementary transformation of the first type CT(0) ^ CT(1) (see
Fig. 8).
At step 2) remove a box from the head of the current partition CT(1) by removing the box C(1) ^ £(2) and insert this box into the partition n*s-1) by inserting the box n*s-1) n*s-2). As a result, we get a partition CT(2) such that r(a(2)) = r, hd (ct(2)) = £(2) and tl *(o"(2)) = n*s-2), and CT(2) is obtained from CT(1) with an elementary transformation of the first type CT(1) ^ CT(2).
We perform such steps s times.
At step s) we obtain a partition a(s) such that hd (o"(s)) = £(s) = hd (A), tl *(o"(s)) = n*0) = tl * (A) and r(a(s)) = r(A). Therefore a(s) = A and
dpt(H+) = 0"(0) ^ ^(1) ^ ^(2) ^ ■ ■ ■ ^ &(s) = A-
Now, starting from the graph H+, we apply Lemma 7 s times to this sequence, and we obtain graph G such that dpt(G) = A.
Thus, in the case s > 0, the sought realization of the partition A is obtained from the splitable graph H+ with the s lowering rotations of edges.
The theorem has been proved. □
It is plain to see that this proof also shows the validity of the two assertions as follows.
Theorem 4. Let A be a graphic partition. Then A has a realization that is a splitable graph if and only if sumhd (A) = sumtl (A).
Theorem 5. Let A be a graphic partition and
s = ^[sumtl*(A) — sumhd (A)].
Then the partition A has a realization G that is obtained from some splitable graph H by s successive lowering rotations and, conversely, H is obtained from G by s successive lifting rotations of edges.
a)
1 4
2
2 c)
3
K\
1
3
22
Figure 9. All realizations of the partition.
d)
34
22
Example 3. Assuming that A = (4,3,2,2,2,1). Here sumA = 14, r = 2, hd(A) = (3,2) and tl (A)* = (2,2,2,1), therefore tl (A) = (4,3), sumtl(A) — sumhd(A) = 7 — 5 = 2 and s = 1. To get hd (A) from tl (A) we need to remove two boxes, hence, hd (A) < tl (A).
Up to isomorphism and isolated vertices, there are 4 realization of the partition A (see Fig. 9). According to the proof of Theorem 2, we have
tl (A) = (4,3) — (4,2) = t = £ — (3,2) = hd (A).
Therefore, a = £ = (4,2) and ft = t * = (2,2,1,1). We sequentially construct a bipartite graph H, a splitable graph H +, two vertices of which generate a clique and six vertices generate a coclique, then we perform one lowering rotation of an edge, we obtain the sought realization G of the partition A = (4,3,2,2,2,1) (see Fig. 10). Graph G is shown in Fig. 9a.
5 3
2 2 11
2211 Figure 10. The graph G obtaining.
2221
2
2
3
2
1
2
It is easy to check that in this example, each realization of the partition A = (4, 3, 2, 2, 2, 1) can be obtained from a suitable splitable graph with one lowering rotation of an edge.
At the end of this section, we present a brief review of related results previously obtained by the authors.
It is worth reminding that any graphic partition has an even weight. The set of all graphic partitions of fixed weight 2m is an order ideal of the lattice IPL(2m), i. e. it is closed under smaller partitions. A graphic partition A of weight 2m will be called a maximal graphic partition if it is maximal in the set of all graphic partitions of the lattice IPL(2m).
The graph G is called a threshold one (see [16]) if its set of vertices can be represented as a disjoint union of the clique V1 and coclique V2 (i. e. V1U V2 = 0, V1 generates the complete subgraph K(V1) and V2 is the zero subgraph O(V2) in G), and the set of neighbourhoods in G of vertices from V2 forms a chain of subsets of the set V1 with respect to the set-theoretic inclusion. It should be noted that the cases V = 0 or V2 = 0 are allowed, i. e. the complete and zero graphs are threshold. For the threshold graph G, the set of all edges can be represented as a disjoint union of the set of all edges of the complete subgraph K(V1) and the set E of all its edges connecting vertices from V1 with vertices from V2. Thus, the threshold graph can be represented as G = (K(V1),E, O(V2)). We will simply write G = (K(V1),E, V2). A bipartite subgraph H = (V1,E, V2) will be called its sandwich subgraph. In the trivial cases when V1 = 0, or V2 = 0, or V2 consists of isolated vertices, the sandwich subgraph H is an empty subgraph in G.
The following statements are true, which were proved in [6, 7].
1. An arbitrary partition A is a maximal graphic partition if and only if hd (A) = tl (A).
2. A graph is threshold if and only if it does not contain any lifting triples of vertices.
3. The degree partition corresponding to the graph G is the maximum graphic partition if and only if the graph G is threshold.
4. Any graph can be reduced to a threshold form with a finite sequence of lifting rotations of edges.
5. For an arbitrary graphic partition A, all of its realizations H, and only they are obtained from the threshold realizations G of maximal graphic partitions ^ such that ^ > A and sum(^) = sum(A) with the finite sequences of lowering edge rotations from G to H.
Assume that ^ and A are two arbitrary non-zero partitions and ^ > A. The height A) of a partition ^ over a partition A is the number of transformations in the shortest sequence of elementary transformations transforming ^ into A.
For a given graphic partition A, a maximal graphic partition ^ such that ^ > A and sum = sum (A) is said to be closest in height to a partition A if it has the minimum possible height over A among all such partitions.
The following assertion was proved in [9].
Assume that A be an arbitrary graphic partition and ^ be the maximum graphic partition closest to it in height. Then
1) either r(^) = r(A) — 1 and l(tl(A)) < r(A), or r(^) = r(A);
2) height^, A) = height(tl(A),hd(A)) -i[sum(tl(A)) -sum(hd(A))] = ^ £[=i |tl(A), - hd(A),|, where r = r(A).
An algorithm was found in [9] that constructs some maximum graphic partition ^ closest to A in height such that r(^) = r(A). In the case when l(tl(^)) < r(^), we also found an algorithm that constructs some maximum graphic partition ^ closest to A in height such that r(^) = r(A) — 1.
Assuming that A be an arbitrary non-zero graphic partition of weight 2m and there is maximal graphic partitions ^ such that ^ > A and r(^) = r, where r is some natural number. Then the set of heads of all such maximal graphic partitions ^ creates an interval in the lattice IPL(m — 1/2 ■ r(r — 1)). This result was obtained by our PhD-student V.V. Zuev (Ural Federal University). The full version of this result will be published in the article being prepared.
5. Proof of the Gale—Ryser theorem with the ht-criterion
Let us now give another rather simple proof of the Gale-Ryser theorem, in which the ht-criterion is used.
The necessity of the condition of the theorem is satisfied by virtue of Lemma 4.
Let us prove the sufficiency of the condition of the theorem. Let a = (ai,a2,... ) and ft = (ft1 , ft2 ,ft) be non-zero partitions such that sum a = sum ft and a < ft *. Assume that m = sum a = sum ft, i. e. a, ft € IPL(m), p = 1(a) and q = l(ft).
Since ap + (p — 1) > p > ft1, the sequence as follows
A = (ai + (p — 1),..., ap + (p — 1),fti,ft2,...,ftq, 0,... )
is a partition. Obviously, r(A) = p and
hd (A) = (0:1,0:2,...) = a, tl *(A) = (P1 ,P2,...) = P,
thus, tl (A) = P*.
It should be noted that
sum A = sum hd (A) + p(p — 1) + sum tl (A) = sum a + p(p — 1) + sum P = 2m + p(p — 1)
is even number and hd(A) = a < P * = tl (A). Therefore, by virtue of the ht-criterion, there is a graph H realizing the partition A, i. e. a graph H such that dpt(H) = A. It is clear that 1(A) = p+q. Without loss of generality, we assume that H has no isolated vertices and = {v1,v2,..., vp+q}.
Assume that V1 = {v1, v2,..., vp}, where degH vj = aj + (p — 1) for any i = 1,2,... ,p, and V2 = {vp+1, vp+2,..., vp+q}, where degH vp+i = Pj for any j = 1,2,..., q.
Let us remove all edges in H that connect pairs of different vertices from V1. We obtain a graph G. For each i = 1,2,..., p, the degree aj of the vertex vj will decrease by no more than p — 1 when passing from H to G, so degG vj = aj + ¿^ where ¿j will hold in G, where ¿j is a non-negative integer. Moreover, ¿j = 0 is satisfied if, when passing from H to G, the degree of the vertex vj decreases by p — 1.
Case 1. Assume that the set V = {v1, v2,..., vp} is not a clique in H.
Then there are vertices vj from the set V such that ¿j > 0. Therefore,
p p p
y^ degG vj = ^ aj + ^ ¿j > sum a = m.
j=1 j=1 j=1
Therefore, in the graph G there are more than m edges going from V1 to V2. Since sum p is greater than or equal to the number of such edges, we get the sum P > m, which contradicts sum P = m.
Therefore, this case is impossible.
Case 2. Let the set V1 = {v1, v2,..., vp} is a clique in H.
Then degG vj = aj for any i = 1,2,... ,p. Since sum a = m, the number of edges of the graph G going from V1 to V2 is equal to m. By virtue of the equality sum p = m, it follows that there are no edges in the graph G that connect pairs of different vertices from V2. Therefore, V1 and V2 are two cocliques in G.
Since dptG(V1) = a and dptG(V2) = P, the graph G is the sought bipartite graph with parts V1 and V2.
The theorem has been proved. □
At the end of this section, we present a brief review of the results previously obtained by the authors and similar in subject matter to the Gale-Ryser theorem.
We first give the necessary definitions.
We say that a bipartite graph H = (V1,E, V2) contains a bipartite 4-pseudocycle x1,x2,x3,x4,x1, if x1,x3 € V2; x2,x4 € V1; x1x2 € E; x2x3 / E; X3X4 € E; x4x1 / E.
We call the bipartite graph H = (V1,E, V2) a bipartite-threshold graph [10] if it does not have any lifting triples of both the first and second parts, i. e. such lifting triples (x, v, y), that x, y € V1, v € V2, or x, y € V2, v € V1.
It should be reminded [7] that a graph is a threshold one if and only if it does not contain any lifting triples of vertices. Therefore, bipartite-threshold graphs are analogues of threshold graphs for the class of bipartite graphs.
In [10], the properties of bipartite-threshold graphs were studied. The following assertion is true [10].
Let H = (V1,E, V2) be a bipartite graph. Then the following conditions are equivalent
1. H is a sandwich subgraph of the threshold graph G = (K(V1), E, V2).
2. H is a sandwich subgraph of the threshold graph G = (K(V2), E, V1).
3. The neighborhoods in H of the vertices of each of the parts V and V2 are nested, i.e. they form chains with respect to the set-theoretical inclusion.
4. Neighborhoods in H of the vertices of the part V1 are nested, i.e. it forms a chain with respect to the set-theoretical inclusion.
5. Neighborhoods in H of the vertices of V2 are nested, i. e. it forms a chain with respect to the set-theoretical inclusion.
6. H is a bipartite-threshold graph, i.e. it does not contain lifting V1-triples and lifting V2-triples.
7. H does not contain lifting V1-triples.
8. H does not contain lifting V2-triples.
9. dptH(V2) =dptH(V1)*.
10. dptH(V1) =dptH(V2)*.
11. H has no bipartite 4-pseudocycles.
Assume that a and ft be non-zero partitions such that sum a = sum ft = m and a < ft . A bipartite graph H = (V1,E, V2) such that dptH(V1) = a and dptH(V2) = ft will be called a realization of a pair of partitions (a, ft). The class of all such bipartite graphs is denoted by BG (a, ft) (the family of bipartite graphs corresponding to the pair (a, ft)).
For an arbitrary partition 7, we denote by btg(Y, 7*) a bipartite threshold graph with parts V1 and V2 without any isolated vertices such that dptG(V1) = 7 and dptG(V2) = 7*. It should be noted that this graph is unique up to isomorphism (see [12]).
Any bipartite graph H = (V, E, V2) from the family of graphs BG (a, ft) can be reduced with finite sequences of bipartite lifting rotations of edges to bipartite threshold graphs, each of which, up to isomorphism and isolated vertices, has the form btg(Y, 7*) for a suitable partition 7, and the graph H = (V1,E, V2) is obtained from such graphs btg(Y, 7*) with reverse sequences of bipartite lowering edge rotations.
We denote by BTG| (a, ft) the family of all bipartite threshold graphs that can be obtained from the graphs of the family BG (a, ft) with bipartite lifting rotations (the family of bipartite threshold graphs over the pair (a, ft)).
Let a bipartite graph H = (V, E2, V2) be obtained from a bipartite graph G = (V1, E1, V2) with a finite sequence of bipartite lifting edge rotations. The least number of bipartite lifting rotations of edges in the sequence taking G to H is denoted by updist (G, H) and is called the upper distance from G to H.
The following two theorems are valid [12].
1. The family of bipartite threshold graphs BTG^ (a, ft) up to isomorphisms and isolated vertices consists of graphs of the form btg (7,7 *), where a < 7 < ft * (compare with the Zuev theorem about the interval of heads of maximal graphic partitions over the given partition given at the end of Section 4).
2. Let the bipartite threshold graph H = (V1 ,E2, V2) = btg(Y, 7*) € BTGf (a, P) be obtained from the graph G = (V1, E1, V2) € BG(a, P) with a finite sequence of bipartite lifting rotations of edges. Then
updist (G, H) > height (P *, a) = height (a *, P).
This estimate is achieved on the graphs btg (P *,P) and btg(a, a *), i.e. for 7 = P * and for 7 = a.
It is clear that any bipartite graph is reduced by successive rotations of edges, each of which corresponds to a lifting triple of only the first part, to a bipartite-threshold graph.
Assume that the bipartite graph H2 = (V1,E2, V2) can be obtained from the graph H1 = (V1,E1, V2) with a finite sequence of ^-rotations of the edges. Let V -dist(H1,H2) denote the smallest number of V1-rotations of edges in the sequence that maps H1 to H2 and call it as the VVdistance from H1 to H2. In [11], with the Hungarian algorithm, a polynomial algorithm was constructed that transforms an arbitrary bipartite graph H = (V1, E, V2) into a bipartite-threshold graph G with a finite sequence of the smallest possible length consisting of ^-rotations of edges, i. e. the length equal to V1-dist (H, G).
In conclusion we make the following important remark. Let A be an arbitrary nonzero partition. It corresponds to two partitions a = hd (A) and P = tl (A) . According to the ht-criterion, a partition A is graphic if and only if its sum is even and a < P . It is clear that the ht-criterion is essentially an analog of the Gale-Reiser criterion when passing from the class of bipartite graphs to the class of all graphs. There are many facts indicating a deep analogy between the properties of degree partitions in the class of all bipartite graphs and in the class of all graphs.
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