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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 25. Выпуск 2.
УДК 519.1 DOI 10.22405/2226-8383-2024-25-2-208-221
Хроматичность полных расщепленных графов
Ле Ван Хонг
Хонг Ван Ле — Ханойский университет промышленности (г. Ханой, Вьетнам). e-mail: hungmath68@gmail.com
Аннотация
Соединение нулевого графа От и полного графа Кп, От + Кп = S(m,n), называется полным разделенным графом. В этой статье мы характеризуем хроматическую уникальность, определяем хроматический номер списка и характеризуем уникальную раскраши-ваемость списка для полного графа разделения G = S(m,n). Мы докажем, что G хроматически уникален тогда и только тогда, когда 1 lem Ie2, ch(G) = п +1 G является уникальным раскрашиваемым графом с ^списком тогда и только тогда, когда т ^ 4, п ^ 4 и т + п ^ 10 m(G) ^ 4 та каждые 1 ^ т ^ 5 и п ^ 6. Также доказано некоторое свойство графа G = S(т, п), когда от представляет собой fc-листовой раскрашиваемый граф.
Ключевые слова: хроматически уникальный, список-хроматическое число, уникально списочный раскрашиваемый граф, полный расщепляемый граф.
Библиография: 40 названий. Для цитирования:
Ле Ван Хонг. Хроматичность полных расщепленных графов // Чебышевский сборник, 2024, т. 25, вып. 2, с. 208-221.
CHEBYSHEVSKII SBORNIK Vol. 25. No. 2.
UDC 519.1 DOI 10.22405/2226-8383-2024-25-2-208-221
The chromaticity of complete split graphs
Hung Xuan Le
Hung Xuan Le — Hanoi University of Industry (Hanoi, Vietnam). e-mail: hungmath68@gmail.com
Abstract
The join of null graph Om and complete graph Kn, Om + Kn = S(m, n), is called a complete split graph. In this paper, we characterize chromatically unique, determine list-chromatic number and characterize unique list colorability of the complete split graph G = S(m,n). We shall prove that G is chromatically unique if and only if 1 < m < 2, ch(G) = n +1 G is uniquely 3-list colorable graph if and only if m > 4, n > 4 and m + n > 10 m(G) < 4 for every 1 < m < 5 and n > 6. Some the property of the graph G = S(m, n) when it is fc-list colorable graph also proved.
Keywords: chromatically unique, list- chromatic number, uniquely list colorable graph, complete split graph.
Bibliography: 40 titles.
For citation:
Le Xuan Hung, 2024. "The chromaticitv of complete split graphs" , Chebyshevskii sbornik, vol. 25, no. 2, pp. 208-221.
1. Introduction
All graphs considered in this paper are finite undirected graphs without loops or multiple edges. If G is a graph, then V(G), E(G) (or V, E in short) and G will denote its vertex-set, its edge-set and its complementary graph, 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., deg^(v)), is |Wg(v) | (resp., INw(^)|)- The subgraph of G induced by W C V(G) is denoted by G\W]. The null graphs and complete graphs of order n are denoted by On and Kn, respectively. Unless otherwise indicated, our graph-theoretic terminology will follow [2].
An acyclic graph, one not containing any cycles, is called forest. A connected forest is called a tree, a tree of order n is denoted by Tn.
Let Gl = (Vi,Ei), G2 = (V2, E2) be two graphs such that Vl n V2 = 0. Their union G = Gl UG2 has, as expected, V(G) = Vl U V2 and E(G) = El U E2. Their join defined is denoted Gl + G2 and consists of Gl U G2 and all edges joining Vl with V2.
A graph G = (V, E) is called a split graph if there exists a partition V = IU K such that G[I] and G[K] are null and complete graphs, respectively. We will denote such a graph by S(I U K,E). The join of null graph Om and complete graph Kn, Om + Kn = S(m,n), is called a complete split graph.The notion of split graphs was introduced in 1977 by Foldes and Hammer [14]. A role that split graphs play in graph theory is clarified in [14] and in [7], [9], [27], [30], [34], [35], [36]. These graphs have been paid attention also because they have connection with packing and knapsack problems [11], with the matroid theory [15], with Boolean functions [31], with the analysis of parallel processes in computer programming [18] and with the task allocation in distributed systems [19]. Many generalizations of split graphs have been made. The newest one is the notion of bisplit graphs introduced by Brandstadt et al. [6].
Let G = (V, E) be a graph and A is a positive integer.
A \-coloring of G is a bijection f : V(G) ^ (1,2,..., X} 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 %(G). We say that a graph G is n-chromatic iin = x(G)-
Let V (G) = (vl ,v2,... ,vn}, two A-colorings / and g are considered different if and only if f (vk) = 9(vk) for som e k = 1, 2,... ,n. Let P (G, A) (or simp lv 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, X) is a polynomial in A, called the chromatic polynomial of G. The notion of chromatic polynomials was first introduced by Birkhoff [4] 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-muivalent, and we write in notation G ~ if P(G, X) = P(H, X). A graph G is called chromatically unique or in short x~uniQue if G' = G (i.e., G' is ^^^^orphic to G) to ^ot graph G' such th at G' ~ G. For examples, all cycles are %-unique [25]. The notion of %-unique graphs was first introduced and studied by Chao and Whitehead [10] in 1978. The readers can see the surveys [22], [25], [26] and [36] for more informations about %-unique graphs.
Let (Lv)V£v 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 \-list-colorable, or X-choosable, if for every family (Lv)v^v with | = A to 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).
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 UfcLC graph for short. The idea of uniquely colorable graph was introduced independently by Dinitz and Martin [13] and by Mahmoodian and Mahdian [29] (Mahmoodian and Mahdian have obtained some results on the uniquely fc-list colorable complete multipartite graphs). There have been many interesting and insightful research results on these issues for different graph classes (see [16], [20], [21], [23], [24], [29]). 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 unique list colorabilitv of the complete split graph G = S(m,n). Namely, we shall prove that G is chromatically unique if and only if 1 < m < 2 (Section 2), ch(G) = n + 1 (Section 3), G is uniquely 3-list colorable graph if and only if m ^ 4, n ^ 4 and m + n ^ 10 m(G) < 4 for every 1 < m < ^d n ^ 6 (Section 5), some the property of the graph G when it is fc-list colorable graph also proved (Section 4).
2. Chromatically unique
For a graph G and a positive integer k, a partition [A\, A2,..., A^} of V(G) is called a fc-independent partition in G if each Ai is a non-empty independent set of G. Let a(G, k) denote the number of fc-independent partitions in G. Hence, P(G,\) = ^ a(G,k)(X)k where
1<k<n
(X)k = X(X - 1)... (A - k + 1).
The polynomial a(G, x) = ^ a(G, k)xk is called the ^^^^^^^^mial of G. For convenience,
1<fc<ra
simply denote a(G, x) hy ff(G^d G = H by G = H. The following lemmas will be used to prove our main results.
Lemma 7 ([25]). If Kn is a complete graph on n vertices then x(Kn) = n and G is x-Mrngwe.
Lemma 8 ([32]). Let G and H be two x-e<luiva^en^ graphs. Then
(i)\V (G)| = (H )|; (ti) \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 9 ([32]). (i) All trees of the same order are x-equivalent;
(ii) A tree Tn is x-unique if and only if 1 < n < 3.
Lemma 10 ([8]). Let G and H be two disjoint graphs. Then
a(G + H,x) = a(G, x)a(H, x).
Lemma 11 ([28]). Let G and H be two graphs. Then P(G,\) = P(H,\) if and only if a(G, x) = a(H, x).
Lemma 12 ([36]). LetG = S (I UK, E) be a split graph with\K \ = nandk = max{deg(u) \ u e I}. Then
(i) G is n-chromatic if and only if k < n;
(ii) G is (n + 1)-chromatic if and only if k = n.
Lemma 13 ([21]). The graph G = KV? + Kn is x-unique.
Now we characterize %-unique complete split graphs.
Theorem 1. G = S(m, n) is chromatically unique if and only if 1 ^ m ^ 2.
Proof. Let V (G) = I U K is a partition of V (G) such th at G[I] = Om, G[K] = Kn, G = G[I] + G[K], I = (ul,u2,..., um} and K = (vl,v2,..., vn}.
Firrst we prove the necessity. Suppose that G = S(m, n) is %-unique. For suppose on the contrary that m ^ 3. If n = 1 then G = Tm+l, where Tm+l is a tree of order m + 1. By (ii) of Lemma 9, G is not %-unique because m + 1 ^ 4, a contradiction. So n ^ 2. Set G' = (I' U K', E') with
I' = (Ul,U2, . . . ,Um},K' = (Vl,V2,...,Vn} and E' = El U E2 U E3 with
El = (VlUl,U2U3, . . .,Um-lUm},
E2 = (mvj | i = 1, 2,... ,m,j = 2,..., n},
E3 = (viVj | i = j; i,j = 1, 2,..., n}. It is not difficult to see that
G = G[(V2,V3,.. .,vn}\ + G[(vl,Ul,U2,... ,um}] = Kn-l + Tm+l,
G' = G'[(v2,v3,.. .,vn}\ + G'[(vl,Ul,U2,... ,um}] = Kn-l + T^+l,
where Tm+l and T'm+l ^re trees of order m + 1. ^v (i) of Lemma 9, P(Tm+l, X) = P(T^+l, X). By Lemma 10 and Lemma 11, it follows that P(G, X) = P(G', X) , ie., G ~ G'. It is clear that
l(u e V(G) | degG(u) = A(G) = m + n - 1}| = I(vl,v2,... ,vn}| = n,
l(u e V(G') | degG/(u) = A(G') = m + n - 1}| = I(v2,v3,... ,vn}l = n - 1.
So, G ^ G' and G is not %-unique, a contradiction. Thus, 1 ^ m ^ 2.
Now we prove the sufficiency. Suppose that 1 ^ m ^ ^^f m = 1 then G is a complete graph Kn+l. ^v Lemma 7, G is ^^^^^e. If m = 2 then G = K2> + Kn. ^v Lemma 13, G is ^^^^ue. □
3. List-chromatic number
We need the following Lemmas 14, 15 to prove our results. Lemma 14 ([12]). For a graph G, ch(G) ^ x(G).
We determine list-chromatic number for complete graphs.
Lemma 15. If Kn is a complete graph on n vertices then ch(Kn) = n.
proof. By Lemma 7 and Lemma 14, ch(Kn) ^ n. Set V(Kn) = {v\,v2,..., vn} and LVi is a list of colors of Vi such th at lLVi | = n for every i = 1,2,... ,n. Let f ^e a coloring of Kn such that
f (Vl) e LV1 ,f (V2) e LV2 \ (f (Vl)},..., f (vn) e LVn \ (f (vi), f (V2),..., f (vn-l}.
Then ^s ^^^^^^ble for Kn, i.e., ch(Kn) ^ n. Thus, ch(Kn) = n. □ Now we determine list-chromatic number for complete split graphs.
Theorem 2. List-chromatic number of G = S(m, n) is
ch(G) = n + 1.
PROOF. By (ii) of Lemma 12 and Lemma 14, ch(G) ^ n + 1. Let V(G) = I U K is a partition of V (G) such th at G[I] = Om, G[K] = Kn, I = {u1 ,u2,..., um} and K = {v1, v2,..., vn}. Let
Ly,i, LU2,..., LUm, LV1, LV2,..., LVn
be the lists of colors of
U1,U2, . . . ,Um,Vi,V2, ...,Vn,
respectively, such that
\Lu-i \ = \Lu2 \ = ... = \Lum \ = \LV1 \ = \LV2 \ = ... = \LVn \ = n + 1.
By Lemma 15, there exists (n + 1)-choosable g of G[K U {«1}] = Kn+1 with the lists of colors LV1, LV2,..., LVn, LU1 .Let f ^e ^te raloring of G such that f (Vi) = g(vi) for every i = 1, 2,... ,n, f (U1) = g(m),
f (Ui) e Lui \ g(N (Ui)) for every i = 2,3,...,m. Then ^s (n + for G, i.e., ch(G) < n + 1. Thus ch(G) = n + 1 □
4. Property of S (m, n) when it is k-list colorable
If a graph G is not uniquely fc-list colorable, we also say that G has property M(&). 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 fc 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 [29].
Lemma 16 ([29]). Each UkLC graph is also a U(k - 1)LC graph.
Lemma 17 ([29]). The graph G is UkLC if and only if k < m(G).
Lemma 18 ([29]). 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 19 ([29]). For every graph G we have m(G) ^ IE(G)| +2.
Lemma 20 ([29]). Every UkLC graph has at least 3k — 2 vertices.
For example, one can easily see that the graph S(2, 2) has the property M(3) and it is U2LC, so m(S(2, 2)) = 3.
Proposition 1. Let G = S(m,n) be a UkLC graph with k ^ 2. Then
(i) m ^ 2;
(ii) k < m2-™+4 ; (Hi) k < [ \.
PROOF, (i) If m = 1 then G is a complete graph Kn+\. Lemma 18, G has the property M(2), a contradiction.
(ii) It is not difficult to see that IE(G)| = . By Lemma 19, we have
m(G) < IE(G)| + 2 = — m + 4
By Lemma 17, we have k < m2 2>a+4 ■
□
Let G = S (m, n) be a UfcLC graph with F (G)= IUK,G[I]= Om,G[K]= Kn,m ^ 2,n ^ 1,k ^ 3.
Set
I = (Ul,U2, . . .,Um},K = (Vl,V2, . . . , Vn}.
Suppose that, for the given fc-list assignment L: LUi = (ai,l, ai,2,..., a,i,k} for every i = 1,...,m, LVi = (bi,l,bi,2,..., bi,k} for every i = 1,...,n, there is a unique fc-list color f:
f (ui) = ai,l for every i = 1,... ,m, f (vi) = fe^ for every i = 1,... ,n.
Theorem 3. (i) bitl = bj^, where 1 ^ i,j ^ n and i = j;
(ii) ai,l = bjl, where 1 ^ i ^ m, 1 ^ j ^ n;
(Hi) aitl e (aj,2, a3,3,..., ai,k}, where i,j = 1, 2,... ,m.
Proof, (i) Since G[K] = Kn, it is not difficult to see that bi,l = f (vi) = f (vj) = bj,l, where 1 ^ i,j ^ n and i = j.
(ii) Since G[K U (ui}] = Kn+l for every i = 1,..., m, it is not difficult to see that
ai,l = f (Ui) = f (Vj) = bj,u
where 1 ^ i ^ m, 1 ^ j ^ n.
(ui) If i = j, then it is obvious that the conclusion is true. If i = j, then we suppose that there exists i0,jo such th at i0,jo = 1,...,m; i0 = jo and ai0,l e (aj0,2,aj0,3,... ,a,j0,k }• It is clear that Uiol = %>,:l- Let f' be the coloring of G such that
(a) f'(Ujo) = aio,i;
(b) f'(ui) = ai,i for every i e (1,..., m}, i = jo;
(c) f'(vi) = bi,: for every i = 1,... ,n.
Then f ^ a fc-list coloring for G and f = f, a ^^^todiction. □ Set J(v) = Lv \ (f (v)} for every v e V(G) = IU K.
Theorem 4. (i) 2 ^ If(I)l; (ii) If (I )| ^ m — 2, where m ^ 4; (m) UveIf&) C f (K);
(w) UveV(G)f (v) C f (V(G)); _
(v) There exists i e (1,... ,n} such th at f (vi) C f (I).
Proof, (i) For suppose on the contrary that If(/)| = 1, then al:l = a2,: = ... = am,: = a. Set H = G — I, it is not difficult to see that H is a complete graph Kn. We assign the following lists L^ for the vert ices v of H:
If a e Lv then L'v = Lv \ (a},
If a e Lv then L'v = Lv \ (b}, where b e Lv and b = f (v). It is clear that IL'V| = k — 1 ^ 2 for every v e V(H). By Lemma 18, H has the property M(2). So by Lemma 16, H has the property M(k — 1). It follows that with lists L'v, there exist at least two list colorings for the vertices v of So it is not difficult to see that with lists Lv, there exist at least two list colorings for the vertices v of G, a contradiction.
(ii) For suppose on the contrary that If (I )| ^ m — 1. We consider separately two cases. Case 1: If(I)l = m — 1.
Without loss of generality, we may assume that a1,1 = a2,1 and ai,1 = aj,1 for every i,j e {2,..., m}, i = j. Set graph G' = (V', E'), with
V' = IU K,E' = (E (G) U {UIU]\i, j = 1,2,...,m; i = j}) \ {U1U2}.
It is clear that G' is complete split graph S(2, m + n — 2) with V(G') = I' U K', where
I' = {U1,U2},K' = {U3,U4, . . . ,Um,V1,V2, ...,Vn}
Since a1i1 = 0,21, ^ is not difficult we have got a contradiction. Case 2: \f (I)\ = m.
In this case, ai,1 = aj, 1 for every i,j e {1, 2,..., m}, i = j. Set graph G" = (V'', E"), with
V'' = IU K, E" = E(G) U {muj\i, j = 1,2,... ,m; i = j}.
It is clear that G" is a complete graph Km+n. By Lemma 18, G" has the propertv M(2), so with lists Lv, there exist at least two list colorings for the vertices v of G". Since V(G) = V(G"), it is not difficult to see that with lists Lv, there exist at least two list colorings for the vertices v of G, a contradiction.
(iii) For suppose on the contrary that Uveif (v) £ f (K). there exists io,jo such that aio,jo e f (K) with 1 ^ i0 ^ m, 2 ^ j0 ^ ^^et f be the coloring of G such that
(a) f'(Ui0) = ai0,j0;
(b) f'(ui) = a,i, 1 for every i e {1,..., m}, i =
(c) f'(vi) = bi, 1 for every i = 1,... ,n.
Then f ^ a ^^^ coloring for G and f = f, a contradiction.
(iv) For suppose on the contrary that Uveiukf (v) £ f (V(G). We consider separately two cases. Case 1: There exists i0, jo such that a,i0tj0 e f (V(G)) with 1 ^ i0 ^ m, 2 ^ j0 ^ k.
Let f' be the coloring of G such that
(a) f'(Ui0) = ai0,j0;
(b) f'(ui) = a,i, 1 for every i e {1,..., m}, i = i0\
(c) f (vi) = bi, 1 for every i = 1,... ,n.
Then f is a k-hst coloring for G and f = f, a contradiction.
Case 2: There exists i0,j0 such th at bi0,j0 e f (V (G)) with 1 ^ i0 ^ n, 2 ^ j0 ^ k. Let f" be the coloring of G such that
(a) f'(ui) = a,i, 1 for every i e {1,..., m}-,
(b) f"(vio) = bio,jo;
(c) f'(vi) = bi, 1 for every i e {1,..., n}, i = i0.
Then f" is a coloring for G and f" = f, a contradiction.
(v) For suppose on the contrary that f (vi) £ f (I) for every i e {1,..., n^^en \ f (vi)\ f (/)\ ^ 1 for every i e {1,...,n^^o \LVi \ f (/)\ ^ 2 for every i e {1,... ,n}. Set graph
H = G — I = G[K ] = Kn.
Let L'v. £ LVi \ f (I) such th at \L'V. \ = 2 for every i e {1,..., n}. ^v Lemma 18, H has the property M(2), it follows that with lists L'v., ^^^re exist at least two list colorings for the vertices Vi for every i e {1,..., n}. ^o it is not difficult to see that with lists Lv, there exist at least two list colorings for the vertices v of G, a contradiction. □
5. Uniquely 3-list colorable complete split graphs
We need the following Lemmas 21-29 to prove our results.
Lemma 21. (i) m(S(1,n)) = 2 for every n ^ 1; (ii) m(S (r, 1)) = 2 for eve ry r ^ 1; (Hi) m(S (2,n)) = 3 for eve ry n ^ 2.
Proof, (i) It is clear that 5(1, n) is a complete graph for every n ^ ^^^^tama 18, m(S(1,n)) = 2 for every n ^ 1.
(ii) It is clear that S(r, 1) is a complete bipartite graph for every r ^ 1, by Lemma 18, m(S(r, 1)) = 2 for every r ^ 1.
(iii) By Lemma 18, G = S(2, n) is U2LC to every n ^ 2.
It is not difficult to see that IE(G)| = 1. By Lemma 19, m(S(2,n)) ^ 3 for every n ^ 2. Thus, m(S(2, n)) = 3 for every n ^ 2. □
Lemma 22 ([16]). m(S (3,n)) = 3 for eve ry n ^ 2;
Lemma 23 ([16]). For every r ^ 2, m(S(r, 3)) = 3.
Lemma 24 ([17]). Graphs S(5,4) and S(4, 4) have property M(3).
Lemma 25 ([33]). The graph S (4, 5) has prope rty M (3).
Lemma 26. G = S (4, n) has the property M (4) for eve ry n ^ 2;
Proof. Let G = S (4, n) is a complete split graph with V (G) = I UK, G[I] = 04, G[K] = Kn,n ^ 2. Set
I = (m,U2,U3,U4},K = (V1,V2, . . . ,Vn}.
For suppose on the contrary that graph G = S(4, n) is U4LC. So there exists a list of 4 colors Lv to each vertex v e V(G), such that there exists a unique L-coloring f for G. By (i) and (ii) of Theorem 4, If(I)l = 2.
Let f (I) = (a, b}. Set graph H = G — I, it is not difficult to see that H is a complete graph Kn. We assign the following lists L'v to the vert ices v of H:
(a) If a, b e Lv then L'v = Lv \ (a, b},
(b) If a e Lv ,b e Lv then L'v = Lv \ (a, c}, wher e c e Lv and c = f (v),
(c) If a e Lv, b e Lv then L'v = Lv \ (b, c}, where c e Lv and c = f (v),
(d) If a, b e Lv then L'v = Lv \ (c, d}, where c,d e Lv, c = d and c,d = f (v).
It is clear that IL'V| = 2 for every v e V(H). ^v Lemma 18, H has the property M(2). It follows that with lists L'v, there exist at least two list colorings for the vertices v of H. So it is not difficult
to see that with lists Lv, ^^^re exist at least two list colorings for the vertices v of G, a contradiction. □
Lemma 27 ([39]). (i) For every n ^ 2, S(5, n) has the property M(4); (ii) If n ^ 5 then m(S(5,n)) = 4.
Lemma 28 ([38]). For every m ^ 1,k ^ 2, S(m, 2k — 3) has the property M(k). Lemma 29 ([38]). For every n ^ 1,fc ^ 2, S(2k — 3,n) has the property M(k). Now we prove our results.
Theorem 5. The graph G = S(m,n) is uniquely 3-list, colorable graph if and only if m ^ 4, n ^ 4 and m + n ^ 10.
Proof. Firrst we prove the necessity. Suppose that G = S(m, n) is U3LC. If m < 4 or n < 4 then bv Lemma 28 and Lemma 29, it is not difficult to see that G has the property M(3), a contradiction. Therefore, m ^ ^d n ^ 4. ft follows that m + n ^ 8. If m + n = 8 then m = ^d n = 4, bv Lemma 24, G has property M(3), a contradiction. If m + n = 9 then (m,n) e {(4, 5), (5, 4)}, bv Lemma 24 and Lemma 25, G has property M(3), a contradiction. Thus, m + n ^ 10.
Now we prove the sufficiency. Suppose that m ^ 4, n ^ 4 and m + n ^ 10. Let V (G) = I U K,G[I] = Om,G[K] = Kn, I = {m ,U2 ,...,um},K = {vuv2,... ,vn}. We prove G is U3LC by induction on m + n.lim + n = 10, then we consider separately three cases.
(i) m = 4 and n = 6.
We assign the following lists for the vertices of G:
Lui = {1, 3, 4}, Lu2 = {1, 7, 8}, Lu3 = {2, 5, 6}, LUi = {2, 7, 8};
LVi = {1, 2, 3},LV2 = {1, 2, 4},LV3 = {1, 2, 5},LV4 = {1, 2, 6},LV6 = {1, 2, 7},LV6 = {1, 2, 8}. A unique coloring / of G exists from the assigned lists: f (m) = 1, f (U2) = 1, f (us) = 2, f (U4) = 2; f (V1) = 3, f (V2) = 4, f (vs) = 5, f (V4) = 6, f (v5) = 7, f (v6) = 8.
(ii) m = 5 and n = 5.
We assign the following lists for the vertices of G:
Lui = {1, 4, 5}, Lu2 = {1, 3, 6}, LU3 = {2, 3, 7}, LUi = {2, 4, 5}, LUh = {2, 6, 7} LVi = {1, 2, 3},LV2 = {1, 2, 4},LV3 = {1, 2, 5},LV4 = {1, 2, 6},LV6 = {1, 2, 7}. A unique coloring / of G exists from the assigned lists: f (U1) = 1, f (U2) = 1, f (us) = 2, f (U4) = 2, f (u5) = 2; f (V1) = 3, f (V2) = 4, f (vs) = 5, f (V4) = 6, f (v5) = 7. (Hi) m = 6 and n = 4.
We assign the following lists for the vertices of G:
Lui = {1, 3, 5}, Lu2 = {1, 4, 5}, Lu3 = {2, 3, 6}, LUi = {2, 3, 4}, LUh = {2, 4, 6}, Lu& = {2, 5, 6}; L-vi = {1, 2, 3},LV2 = {1, 2, 4},LV3 = {1, 2, 5},LV4 = {1, 2, 6}. A unique coloring / of G exists from the assigned lists:
f (U1) = 1, f (U2) = 1, f (us) = 1, f (U4) = 2, f (u5) = 2, f (ue) = 2; f (V1) = 3, f (V2) = 4, f (vs) = 5, f (V4) = 6.
Now let m + n > 10 and assume the assertion for smaller values of m+n. We consider separately two cases.
Case 1: m ^ 5.
Set G' = G — um = S(m — 1, n). 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:
Lum = L'Um_i, Lv = L; if v e V(G>). A unique coloring / of G exists from the assigned lists: f (um) = f (um-1), f (v) = f (v) if v e V(G'). Case 2: n ^ 5.
Set G' = G — vn = S(m,n — 1^. 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:
LVn = {f (vn-1), f (vn-2), t}wt h t e f (G'), Lv = L'vii V e V (G'). A unique coloring / of G exists from the assigned lists: f (vn)= t,f (v) = f (v) if v e V(G'). □
Corollary 1. m(S(4,n)) = 4 for every n ^ 6.
□
Theorem 6. m(S(r,n)) ^ 4 for every 1 ^ r ^ 5 and n ^ 6.
PROOF. It follows from Lemma 21 to Lemma 27. □
6. Conclusion
The coloring problem and the list coloring problem are interesting topics in graph theory. Coloring graphs found application in many practical problems, for example, coding theory or security. Clearly, to estimate the chromatic as well as the chromatic uniqueness is very important. So far there have been many research results on this topic for different graph layers. However, the problem has not been generally solved, and further research is needed. This article contributes to enriching the research results on the problem of listing colors.
The main results of the paper have identified the characterized chromatically unique (Theorem 1), list-chromatic number (Theorem 2) and characterized unique list colorabilitv (Theorem 5 and Theorem 6) of complete split graph G = S(m, n). Some the property of the graph G = S(m,n) when it is fc-list colorable graph also proved (Theorem 3 and Theorem 4).The desire in the future will achieve deeper results on the issues raised in this article.
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