DOI: 10.17516/1997-1397-2020-13-3-297-305 YAK 519.17
Colorings of the Graph K™ + Kn
Le Xuan Hung*
Hanoi University of Natural Resources and Environment
Hanoi, Vietnam
Received 04.02.2020, received in revised form 16.03.2020, accepted 13.04.2020
Abstract. In this paper, we characterize chromatically unique, determine list-chromatic number and characterize uniquely list colorability of the graph G = Km + Kn. We shall prove that G is x-unique, ch(G) = m + n, G is uniquely 3-list colorable graph if and only if 2m + n ^ 7 and m ^ 2.
Keywords: chromatic number, list-chromatic number, chromatically unique graph, uniquely list colorable graph, complete r-partite graph.
Citation: Le Xuan Hung, Colorings of the Graph Km + Kn, J. Sib. Fed. Univ. Math. Phys., 2020, 13(3), 297-305.
DOI: 10.17516/1997-1397-2020-13-3-297-305.
1. Introduction and preliminaries
All graphs considered in this paper are finite undirected graphs without loops or multiple edges. If G is a graph, then V(G) and E(G) (or V and E in short) will denote its vertex-set and its edge-set, respectively. The set of all neighbours of a subset S C V(G) is denoted by Ng(S) (or N(S) in short). Further, for W C V(G) the set W n NG(S) is denoted by Nw(S). If S = {v}, then N(S) and NW(S) are denoted shortly by N(v) and NW(v), respectively. For a vertex v e V(G), the degree of v (resp., the degree of v with respect to W), denoted by deg(v) (resp., degW(v)), is |NG(v)| (resp., \NW(v)|). The subgraph of G induced by W C V(G) is denoted by G\W]. The independent sets and complete graphs of order n are denoted by On and Kn, respectively. Unless otherwise indicated, our graph-theoretic terminology will follow [1].
A graph G = (V,, E) is called r-partite graph if V admits a partition into r classes V = = Vt U V2 U ... U Vr such that the subgraphs of G induced by Vi, i = 1,..., r, is independent set. An r-partite graph in which every two vertices from different partition classes are adjacent is called complete r-partite graph and is denoted by K\Vl,\y2\...,\Vr| . The complete r-partite graph K\v1\,\v2\,...,\^rI with \V1\ = \V2\ = ... = \Vr\ = s is denoted by K
Let Gi = (V1,E1), G2 = (V2,E2) be two graphs such that Vt n V2 = 0. Their union G = Gt U G2 has, as expected, V(G) = Vt U V2 and E(G) = Et U E2. Their join defined is denoted Gt + G2 and consists of Gt U G2 and all edges joining Vt with V2.
Let Gt = (Vi,Ei), G2 = (V2,E2) be two graphs. We call Gt and G2 isomorphic, and write Gt = G2, if there exists a bijection f : Vt ^ V2 with uv e Et if and only if f (u)f (v) e E2 for all u,v e Vi.
Let G = (V,, E) be a graph and A is a positive integer.
A A-coloring of G is a bijection f : V (G) ^ {1, 2,..., A} such that f (u) = f (v) for any adjacent vertices u, v e V(G). The smallest positive integer A such that G has a A-coloring is called the chromatic number of G and is denoted by x(G). We say that a graph G is n-chromatic if n = x(G).
*[email protected] © Siberian Federal University. All rights reserved
Let V(G) = {vi,v2,... ,vn}, two A-colorings f and g are considered different if and only if f (vk) = g(vk) for some k = 1,2,... ,n. Let P (G,A) (or simply P (G) if there is no danger of confusion) denote the number of distinct A-colorings of G. It is well-known that for any graph G, P(G, A) is a polynomial in A, called the chromatic polynomial of G. The notion of chromatic polynomials was first introduced by Birkhoff [3] in 1912 as a quantitative approach to tackle the four-color problem. Two graphs G and H are called chromatically equivalent or in short X-equivalent, and we write in notation G ~ H, if P(G,A) = P(H,A). A graph G is called chromatically unique or in short x-unique if G' = G (i.e., G' is isomorphic to G) for any graph G' such that G' ~ G. For examples, all cycles are x-unique [8]. The notion of x-unique graphs was first introduced and studied by Chao and Whitehead [4] in 1978. The readers can see the surveys [8,9] and [12] for more informations about x-unique graphs. Recently, Ngo Dac Tan and Le Xuan Hung characterized chromatically unique split graphs [12] (A graph G = (V, E) is called a split graph if there exists a partition V = I U K such that the subgraphs of G induced by I and K are independent sets and complete graphs, respectively).
Let (Lv)vev be a family of sets. We call a coloring f of G with f (v) e Lv for all v e V is a list coloring from the lists Lv. We will refer to such a coloring as an L-coloring. The graph G is called A-list-colorable, or A-choosable, if for every family (Lv)veV with \Lv \ = A for all v, there is a coloring of G from the lists Lv. The smallest positive integer A such that G has a A-choosable is called the list-chromatic number, or choice number of G and is denoted by ch(G). In [7], we characterized list-chromatic number for split graphs, we have proved that if G is a split graphs then ch(G) = x(G).
Let G be a graph with n vertices and suppose that for each vertex v in G, there exists a list of k colors Lv, such that there exists a unique L-coloring for G, then G is called a uniquely k-list colorable graph or a UkLC graph for short. The idea of uniquely colorable graph was introduced independently by Dinitz and Martin [6] and by Mahmoodian and Mahdian [10] (Mahmoodian and Mahdian have obtained some results on the uniquely k-list colorable complete multipartite graphs, for example, they proved that graph G = Om + Kn is U3LC when (m,n) e {(4, 6), (5, 5), (6, 4)}).
Finding a general result for the problems raised above is a difficult task, requiring a lot of time and effort for mathematicians. There have been many interesting and insightful research results on these issues for different graph classes. However, these are still issues that have not been resolved thoroughly, so much more attention is needed. In this paper, we shall characterize chromatically unique, determine list-chromatic number and characterize uniquely list colorability of the graph G = K2m+Kn. Namely, we shall prove that G is x-unique (Section 2), ch(G) = m+n (Section 3), G is U3LC if and only if 2m + n > 7 and m > 2 (Section 4). These results contribute to solving the coloring problem for a complete multipartite graph.
2. Chromatic uniqueness
We need the following Lemmas 1-4 to prove our results. Lemma 1 ([2]). If Kn is a complete graph on n vertices then x(Kn) = n. Lemma 2. If G = Kni ,n2,...,nr is a complete r-partite graph then x(G) = r.
Lemma 3 ([11]). Let G and H be two x-equivalent graphs. Then
(i) \V(G)\ = \V(H)\;
(ii) \E(G)\ = \E(H)\;
(iii) x(G) = x(H);
(iv) G is connected if and only if H is connected;
(v) G is 2-connected if and only if H is 2-connected.
Lemma 4. Let G = (Vi U V2 U ... U Vm+n, E) be a (m + n)-partite graph with m ^ 1, n ^ 1, \Vi\ > \V2 \ > ... > \Vm+n \ and \Vi\ + \V2 \ + ... + \Vm+n\ = 2m + n. Then
.,_,. (2m + n)2 — 4m — n
\e\ ^ -~2-.
In particular,
. (2m + n)2 — 4m — n
\E\ =-2-
if and only if G is a complete (m + n)-partite graph K\Vl\,\V2\,..,\Vm+n\ with
\Vi\ = \V2\ = ... = \Vm\ = 2, \Vm+i\ = \Vm+2 \ = ... = \Vm+n\ = 1.
Proof. We prove the lemma by induction on t = m + n. For t = 2 the assertion holds, so let t > 2 and assume the assertion for smaller values of t. If \Vm+n\ > 2 then \V1\ + \V2\ + ... + \Vm+n\ ^ 2m + 2n > 2m + n, a contradiction. So, \Vm+n\ = 1. If \Vm\ > 3 then \V1\ + \V2\ + ... + \Vm+n\ ^ 3m + n > 2m + n, a contradiction. Therefore, \Vm\ < 2. Now we consider separately two cases. Case 1: There exists i e {1, 2,..., m} such that \Vi\ = 2.
Set G' = G — Vi. It is clear that G' is a (m + n — 1)-partite graph
(Vi u V2 u ... u Vi_i u Vi+i u ... u Vm+n, E').
By the induction hypothesis,
(2( m — 1) + n) 2 — 4( m — 1) — n \E \ ^-2-.
We have
\E\ < \E'\ + \Vi\(\Vi\ + ... + \Vi-i\ + \Vi+i\ + ... + \Vm+n\) <
(2(m — 1) + n)2 2
(2m + n) 2 — 4m —
(2(m 1) + n)2 4(m 1) n < —- 2-----+ 2(2m + n — 2)
It is not difficult to see that
(2m + n) 2 — 4m — \E\ = -o-
if and only if G is a complete (m + n)-partite graph K\Vl\,\V2\,..,\Vm+n\ with
\Vi \ = \ V2 \ = ... = \Vm\ = 2, \Vm+i\ = \Vm+2\ = ... = \Vm+n\ = 1.
Case 2: \Vi\ = 2 for every i = 1, 2,... ,m.
In this case, \Vi\ > 3. Let h e {1,2,... ,m} such that \Vh\ = 1 and \Vh-i\ > 3. Let Gi = = KP1,P2,..,Pm+n be a complete (m+n)-partite graph such that ph = \Vh\ + 1 = 2, ph-i = \Vh-i\ — 1 and pi = \Vi\ for every i e {1, 2,... ,m + n}\{h — 1,h}. By Case 1,
\E(Gi)\ ^ (^m + n) — 4m — n.
We have
\E(Gi)\ = ]T pipj =
2
= 53 PiPj + PiPh-l +
i,je{l,...,m+n}\{h-l,h} ie{l,...,m+n}\{h-l,h}
+ E PiPh + Ph-lPh =
ie{l,...,m+n}\{h-l,h}
= E \Vi\\Vj \ + E m^-^-1) +
i,je{l,...,m+n}\{h-l,h} ie{l,...,m+n}\{h-l,h}
+ E Vi^ + i) + (\Vh-i\- i)(\Vh\ + 1) =
ie{l,...,m+n}\{h-l,h}
= E \Vi\\Vj \ + |Vh-l|-|Vh|- 1 >
l^i<j^m+n
> \E \ + 1.
It follows that
. (2m + n)2 — 4m — n \E\ < --
2 □
Now we characterize chromatically unique for the graph G = K2p + Kn. Theorem 5. The graph G = K2p + Kn is x-unique.
Proof. It is clear that G is a complete (m + n)-partite graph KPl,P2,..,Pm+n with
Pi = P2 = ... = Pm = 2, Pm+l = Pm+2 = ... = Pm+n = 1.
Let G' = (V', E') is a graph such that G' ~ G. Since Lemma 2 and (iii) of Lemma 3 we have
x(G') = x(G) = m + n.
Let G' has a coloring f using m + n colors 1, 2,... ,m + n. Set
Vi = {u e V' \ f (u) = i}.
for every i = 1, 2,..., m+n. It follows that G' is a (m+n)-partite graph (V{UV2U.. .UVm+n, E'). By (i) and (ii) of Lemma 3 we have
\V(G')\ = \V(G)| = 2m + n, \E(G')\ = \E(G)\ = {2m + n)* ~ 4m ~ n. Without loss of generality we may
\Vl\ > \VJt\ > ... > \Vm+n\.
By Lemma 4, we have
|Vi'| = V \ = ... = \Vm\ =2, \Vm+i \ = \Vm+2\ = ... = \Vm+n\' = 1. It follows that G' = G. Thus G is x-unique. □
3. List-chromatic number
We need the following Lemmas 6-8 to prove our results. Lemma 6 ([5]). If G is a graph then ch(G) > x(G).
Lemma 7 ([5]). If Gi is a subgraph of G2 then ch(Gi) < ch(G2).
We determine list-chromatic number for complete graphs. Lemma 8. If Kn is a complete graph on n vertices then ch(Kn) = n.
Now we determine list-chromatic number for the graph G = K2. Theorem 9. List-chromatic number of G = K2 is
ch(G) = r.
Proof. By Lemma 2 and Lemma 6, we have ch(G) ^ r. Now we prove ch(G) ^ r by induction on r. For r = 1 the assertion holds, so let r > 1 and assume the assertion for smaller values of r.
Let V (G) = Vi U V2 U ... U Vr is a partition of V (G) such that for every i = 1,...,r, \Vi \ =2 and the subgraphs of G induced by Vi, is independent set. Set
Vi = {vii ,vi2 }
for every i = 1,...,r. Let LVjj be the lists of colors of vij such that \LVij \ = r for every
i = 1, 2,... ,r; j = 1,2. Now we consider separately two cases. Case 1: There exists i e {1, 2,... ,r} such that LVi1 n LVi2 = 0.
Without loss of generality we may assume that LV11 n LV12 = 0 and a e LV11 n
LLvi2. set
G' = G — Vi. It is clear that G' is a graph KZ,-1. Again set
LVj C LVi3 \ {a}
such that \L'Vij \ = r — 1 for every i = 2, 3,... ,r; j = 1, 2.
By the induction hypothesis, there exists (r — 1)-choosable g of G' with the lists of colors L for every i = 2, 3,... ,r; j = 1, 2.
Let f be the coloring of G such that
f (vij) = g(vij) for every i = 2, 3,...,r; j = 1, 2 f (vij) = a for every j = 1, 2. Then f is a r-choosable for G, ie., ch(G) ^ r.
Case 2: LVi1 n LVi2 = 0 for every i = 1, 2,... ,r. Let b e Lv11 . Set G' = G — Vi = and
/
LVij C LVij \{b}
such that \L'Vij \ = r — 1 for every i = 2, 3,... ,r; j = 1, 2.
By the induction hypothesis, there exists (r — 1)-choosable g of G' with the lists of colors L'V.. for every i = 2,3,... ,r; j = 1,2. Since \LV11 U LV12 \ = 2r and \V(G'\ = 2(r — 1), it follows that
\(LV11 U Lv12) \ g(V(G'))\ > 2.
We again divide this case into two subcases. Subcase 2.1: ((Lvh U Lv12 ) \ g(V(G'))) n Lv12 = 0.
Let c e ((LV11 U LV12) \ g(V(G'))) n LV12. Let f be the coloring of G such that f (vij) = g(vij) for every i = 2, 3,...,r; j = 1, 2, f (vii) = b, f (vi2) = c. Then f is a r-choosable for G, ie., ch(G) ^ r. Subcase 2.2: ((Lv11 U Lv12) \ g(V(G'))) n Lv12 = 0.
By \(Lv11 U Lv12) \ g(V(G'))\ > 2, there exists d e (Lv11 U Lv12) \ g(V(G')), d = b. It is clear that b,d e LV11. Since \LV12 \ = r and \g(V(G'))\ < 2(r — 1), there exists i e {2,3,..., r} such
that g(vii), g(vi2) e LVl2. Without loss of generality we may assume that g(v2l),g(v22) e LVl2. Let e e (LV21 U LV22) \ g(V(G')). First assume that e e LV21. If e = b then coloring f of G such that
f (vij) = g(vij) for every i = 3,4,...,r; j = 1, 2, f(v22) = g(v22), f(v2i) = e, f(vii) = b, f(vi2) = g(v2i). is a r-choosable for G. If e = b then coloring f of G such that f (vij) = g(vij) for every i = 3,4,...,r; j = 1, 2, f(v22) = g(v22), f(v2i) = e, f(vii) = d, f (vi2) = g(v2i). is a r-choosable for G. By symmetry, we can show that ch(G) < r if e e LV22. □
Theorem 10. List-chromatic number of G = K2l + Kn is
ch(G) = m + n.
Proof. It is clear that G = K2m + Kn is a complete (m + n)-partite graph. By Lemma 2 and Lemma 6, we have ch(G) ^ m + n. Now we prove ch(G) ^ m + n. It is not difficult to see that G is a subgraph of K2m+n. By Lemma 7 and Theorem 9, ch(G) ^ m + n. Thus, ch(G) = m + n. □
4. Uniquely list colorability
If a graph G is not uniquely k-list colorable, we also say that G has property M(k). So G has the property M(k) if and only if for any collection of lists assigned to its vertices, each of size k, either there is no list coloring for G or there exist at least two list colorings. The least integer k such that G has the property M(k) is called the m-number of G, denoted by m(G). This conception was originally introduced by Mahmoodian and Mahdian in [10].
We need the following Lemmas 11-16 to prove our results.
Lemma 11 ([10]). A connected graph G has the property M(2) if and only if every block of G is either a cycle, a complete graph, or a complete bipartite graph.
Lemma 12 ([10]). For every graph G we have m(G) < \E(G)\ +2.
Lemma 13 ([10]). Every UkLC graph has at least 3k — 2 vertices.
Lemma 14. If 2m + n = 7 and m > 2 then G = Km + Kn is U3LC.
Proof. It is clear that G = Km + Kn is a complete (m + n)-partite graph. Let V(G) = Vi U V2 U ... U Vm+n is a partition of V (G) such that \Vi\ = \V2\ = ... = \Vm\ = 2, \Vm+i\ = \Vm+2\ = ... = \Vm+n\ = 1 and for every i = 1,... ,m the subgraphs of G induced by Vi, is independent set. Set Vi = {uii,ui2} for every i = 1,... ,m and Vm+i = {vi} for every i = 1,... ,n. Now we consider separately two cases.
Case 1: m = 2 and n = 3.
We assign the following lists for the vertices of this graph:
LUii = {1, 2, 3}, L„i2 = {1, 4, 5}, Lu21 = {1, 2, 3}, LU22 = {2, 4, 5},
Lvi = {1, 2, 5}, LV2 = {1, 2, 4}, Lv3 = {1, 2, 3}. A unique coloring f exists from the assigned lists:
f(u11) = 1, f(u12) = 1, f(u21) = 2, f(u22) = 2,
f (vi) = 5, f (V2)=4, f(vs) = 3.
Case 2: m = 3 and n = 1.
We assign the following lists for the vertices of this graph:
LU11 = {1,4, 5}, LU12 = {2, 4, 5}, LU21 = {1, 2, 3}, LU22 = {3,4, 5},
LU31 = {1, 2,4}, Lu32 = {3,4, 5}, Lvi = {3, 4, 5}. A unique coloring f exists from the assigned lists:
f(uii) = 1, f(ui2) = 2, f(u2i) = 3, f(U22) = 3,
f(usi) = 4, f(us2) = 4, f(vi) = 5.
□
Lemma 15. If m = 2 and n ^ 3 then G = Km + Kn is U3LC.
Proof. We prove G is U3LC by induction on n. If n = 3 then by Lemma 14, G is U3LC. So let n > 3 and assume the assertion for smaller values of n.
Let V (G) = Vi U V2 U... U Vn+2 is a partition of V (G) such that |Vi| = |V2| = 2, |Vs| = |V4| = = ... = = 1 and for every i = 1, 2 the subgraphs of G induced by Vi, is independent set.
Set Vi = {uii, ui2} for every i = 1,2 and Vi+2 = {vi} for every i = 1,... ,n. Set G' = G — vn. By the induction hypothesis, for each vertex v in G', there exists a list of 3 colors L'v, such that there exists a unique f for G .
We assign the following lists for the vertices of G:
T = T' T = T' T = T' T = T'
LU11 LU11 , LU12 LU12 , LU21 LU21 , LU22 LU22 ,
Lvi = L'v1, ...,LVn-1 = LVn-1, Lvn = {f(v1),f(v2),t},
with t <t LU U LU U LU U LU U lv U ... U L'v .
y- U11 U12 U21 U22 v1 v n— 1
A unique coloring f of G exists from the assigned lists: f (v)= f'(v) if v £ V(G'), f (vn) = t.
□
Lemma 16. If m = 3 and n > 1 then G = Km + Kn is U3LC.
Proof. We prove G is U3LC by induction on n. If n = 1 then by Lemma 14, G is U3LC. So let n > 1 and assume the assertion for smaller values of n.
Let V(G) = Vi U V2 U Vs U ... U Vn+s is a partition of V(G) such that |Vi | = |V>| = |Vs| = 2, = V | = ... = V^n+^i| = 1 and for every i = 1, 2, 3 the subgraphs of G induced by Vi, is independent set. Set Vi = {uii,ui2} for every i = 1,2,3 and Vi+s = {vi} for every i = 1,... ,n. Set G' = G — vn. By the induction hypothesis, for each vertex v in G', there exists a list of 3 colors Lv, such that there exists a unique f' for G'. We assign the following lists for the vertices of G:
T = T' T = T' T = T' T = T' T = T' T = T'
LU11 = LU11 , LU12 = LU12 , LU21 = LU21 , LU22 = LU22 , LU31 = LU31 , LU32 = LU32
Lv1 = L'v1, ..., Lvn—1 = L'v—, Lvn = {f'(vi),f'(v2),t},
with t £ LU11 U LU12 U LU21 U LU22 U LU31 U LU32 U L'v1 U ... U L'vn — 1 .
A unique coloring f of G exists from the assigned lists: f (v)= f'(v) if v £ V(G'), f (vn) = t.
□
Theorem 17. G = Km + Kn is U3LC if and only if 2m + n ^ 7 and m ^ 2.
Proof. Firrst we prove the necessity. Suppose that G = Km + Kn is U3LC. By Lemma 13, |V(G)| =2m + n > 7. If m = 1 then \E(G)\ = 1, by Lemma 12, m(G) < \E(G)| + 2 = 3, a contradiction. Therefore, m ^ 2.
Now we prove the sufficiency. We prove G is U3LC by induction on m. If m = 2 then by Lemma 15, G is U3LC. If m = 3 then by Lemma 16, G is U3LC. So let m > 3 and assume the assertion for smaller values of m.
Let V (G) = Vi U V2 U V3 U ... U Vm+n is a partition of V (G) such that \Vi\ = \V2\ = ... = = \Vm\ = 2, = |Vm+2| = ... = \Vm+n \ = 1 and for every i = 1,2,... ,m the subgraphs of
G induced by Vi, is independent set. Set Vi = {uii,ui2} for every i = 1,... ,m and G' = G — Vm. By the induction hypothesis, for each vertex v in G', there exists a list of 3 colors L'V, such that there exists a unique f' for G'.
We assign the following lists for the vertices of G: LUm1 = LUm2 = {f'(uii),f'(u2i),t} with t ef'(G'), Lv = LV if v e V(G').
A unique coloring f of G exists from the assigned lists: f (umi) = f (um2) = t, f (v) = f '(v) if v e V(G'). □
5. Conclusion
The coloring problem, including the list coloring problem, has always been much researched in graph theory because it has many applications in computer science. The list coloring model can be used in the channel assignment. The fixed channel allocation scheme leads to low channel utilization across the whole channel. It requires a more effective channel assignment and management policy, which allows unused parts of channel to become available temporarily for other usages so that the scarcity of the channel can be largely mitigated [13]. It is a discrete optimization problem. A model for channel availability observed by the secondary users is introduced in [13]. The research of list coloring consists of two parts: the choosability and the unique list colorability.
The main results of the paper have identified the list-chromatic number (Theorem 10), characterized chromatically unique (Theorem 5) and characterized uniquely list colorability (Theorem 17) of the graph G = Km + Kn. The desire in the future will achieve deeper results on the issues raised in this article.
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Раскраски графа K™ + Kn
Ле Хуан Хунг
Ханойский университет природных ресурсов и окружающей среды
Ханой, Вьетнам
Аннотация. В этой статье мы характеризуем хроматически уникальное хроматическое число в списке и однозначно характеризуем окрашиваемость графа списка К™ + Кп. Мы докажем, что О X единственно, еЬ(О) = т + п, О является однозначным трехцветным графом раскраски тогда и только тогда, когда 2т + п ^ 7 и т ^ 2.
Ключевые слова: хроматическое число, хроматический номер списка, хроматически уникальный граф, однозначный список раскрашиваемого графа, полный г-раздельный граф.