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ЧЕБЫШЕВСКИИ СБОРНИК
Том 23. Выпуск 2.
УДК 517.518.5
DOI 10.22405/2226-8383-2022-23-2-170-178
Уникальная возможность отображения списка полных трехсторонних графиков
Ле Ван Хонг
Хонг Ван Ле — Ханойский университет природных ресурсов и окружающей среды (г. Ханой, Вьетнам).
e-mail: lxhungQhunre.edu.vn
Учитывая список Ь(у) для каждой вершины V, мы говорим, что граф С является Ь-раскрашиваемым, если существует правильная раскраска вершины в, где каждая вершина V берет свой цвет из Ь(ю). Граф является однозначно раскрашиваемым списком к, если существует присвоение списка Ь такое, что |Ь(^)| = к для каждой вершины V, и граф имеет ровно одну раскраску Ь с этими списками. Если граф С не является однозначно раскрашиваемым списком к, мы также говорим, что С обладает свойством М(к). Наименьшее целое число к, такое, что С обладит свойством М(к), называется то-числом С, обозначаемым т(С). В этой статье сначала мы охарактеризуем свойство полных трехсторонних графов, когда это однозначно ^-список раскрашиваемых графов, наконец, мы докажем, что то(К2,2,т) = то(К2,з,п) = то(К2,4,Р) = то(Кз,з,з) = 4 за каждые то > 9,п > 5,р > 4.
Ключевые слова: Раскраска вершин (раскраска), раскраска списка, однозначно раскрашиваемый список графов, полный г-частичный граф.
Библиография: 18 названий.
Для цитирования:
Ле Ван Хонг. Уникальная возможность отображения списка полных трехсторонних графиков
// Чебышевский сборник, 2022, т. 23, вып. 2, с. 170-178.
Аннотация
CHEBYSHEVSKII SBORNIK Vol. 23. No. 2.
UDC 517.518.5
DOI 10.22405/2226-8383-2022-23-2-170-178
Uniquely list colorability of complete tripartite graphs
Le Xuan Hung
Hung Xuan Le — Hanoi University for Natural Resources and Environment (Hanoi, Vietnam). e-mail: [email protected]
Abstract
Given a list L(v) for each vertex v, we say that the graph G is L-colorable if there is a proper vertex coloring of G where each vertex v takes its color from L(v). The graph is uniquely fc-list colorable if there is a list assignment L such that IL(v)l = k for every vertex v and the graph has exactly one L-coloring with these lists. If a graph G is not uniquely fc-list colorable, we also say that G has property M(k). The least integer k such that G has the property M(k) is called the rn-number of G, denoted by m(G). In this paper, first we characterize about the property of the complete tripartite graphs when it is uniquely fc-list colorable graphs, finally we shall prove that m(K2,2,m) = m(K2,3,n) = m(K2,4,P) = m(K3,3,3) = 4 for every m > 9,n > 5,p > 4.
Keywords: Vertex coloring (coloring), list coloring, uniquely list colorable graph, complete r-partite graph.
Bibliography: 18 titles. For citation:
Le Xuan Hung, 2022, "Uniquely list colorabilitv of complete tripartite graphs" , Chebyshevskii sbor-nih, vol. 23, no. 2, pp. 170-178.
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) 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 bv NG(S) (or N(S) in short). Further, for W C V(G) the set W n NG(S) is denoted by Nw(S). The subgraph of G induced by W C V(G) is denoted by G[W]. The empty and complete graphs of order n are denoted by On and Kn, respectively. Unless otherwise indicated, our graph-theoretic terminology will follow [2].
A graph G = (V,E) is called r-partite graph if V admits a partition into r classes V = Vi U V2 U ... U Vr such that the subgraphs of G induced by Vi, i = 1,... ,r, is empty, if r = 2 then G is called bipartite graph, if r = 3 then G is called tripartite graph. 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\v1\,\v2|,...,|yr| • The complete r-partite graph K\v1\,\v2\,...,\vr| with \Vi\ = \ V2\ = ... = \ Vr| = s is denoted hy Kr. * *
Let Gi = (Vi,Ei), G2 = (V2, E2) ^e two graphs such that V\ n V2 = ^ union G = Gi UG2 has, as expected, V(G) = Vi U V2 and E(G) = Ei U . Their jom defined is denoted Gi + G2 and consists of Gi U G2 ^^d all edges joining Vi with V2.
Let G = (V, E) ^e a graph and A is a positive integer.
A \-coloring of G is a mapping f : V (G) ^ {1, 2,..., X} such th at f (u) = f (v) for any adjacent vertices u,v € 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 (L(v))vev be a family of sets. We call a coloring / of G with f (v) € L(v) for all v € V is a list coloring from the lists L(v). We will refer to such a coloring as an L-coloring. The graph G is called \-list-colorable, or \-choosable, if for every family (L(v))vev with \L(v)\ = X fa all v, there is a coloring of G from the lists L(v). 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). The idea of list colorings of graphs is due independently to V. G. Vizing [14] and to P. Erdos, A. L. Rubin, and H. Taylor [7].
Let G be a graph with n vertices and suppose that for each vertex v in G, there exists a list of k colors L(v), 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. If a graph G is not uniquely fc-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 fc, 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 bv m(G). The idea of uniquely colorable graph was introduced independently by Dinitz and Martin [6] and by Mahmoodian and Mahdian [10].
For example, one can easily see that the graph K\,\,2 has the property M(3) and it is U2LC, so m(Ki,it2) = 3.
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 [15]. It is a discrete optimization problem. A model for channel availability observed by the secondary users is introduced in [15]. The research of list coloring consists of two parts: the choosabilitv and the unique list colorabilitv. In [9], we characterized uniquely list color ability of the graph G = K™ + Kn.
In this paper, first we characterize about the property of the complete tripartite graphs when it is uniquely fc-list colorable graphs (Section 2), finally we shall prove that m(K2,2,m) = m(K2,3,n) = = m(K2,4,p) = m(K3,3,3) = 4 for every m ^ 9,n ^ 5,p ^ 4 (Section 3).
2. Property of the complete tripartite graphs when it is k-list colorable
We need the following Lemmas 1-6 to prove our results. Lemma 1 ([10]). Each UkLC graph is also a U(k — 1)LG graph. Lemma 2 ([10]). The graph G is UkLC if and only if k < m(G).
Lemma 3 ([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 4 ([10]). For every graph G we have m(G) ^ E(G)| + 2.
Lemma 5 ([10]). Every UkLC graph has at least 3k — 2 vertices.
Lemma 6 ([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.
Theorem 1. Let G = Km,n,p be a UkLC graph with k ^ 2. Then
(i) max[m,n,p] ^ 2;
(ii) If k ^ 3 then min{m,n,p] ^ 2; (Hi) k < m2 +"2 +p2 - (m+n+p)+4 ■ (iv) k ^
m+n+p+2 3
proof, (i) For suppose on the contrary that max{m,n,p] = 1. Then m = n = p = 1, so G is a complete graph By Lemma 3, G has the property M(2), a contradiction.
(ii) For suppose on the contrary that min{m,n,p] = 1. Without loss of generality, we may assume that min{m,n,p] = m = 1. Let V(G) = V\ u V2 u V3 is a partition of V(G) such that | Vi| = m, |V2| = n, | V31 = p,V\ = {a} and for every i = 1,2,3 the subgraphs of G induced bv Vi, is empty graph.
Since G is a U^^C ^^^re exists a list of k colors L(v) fa each vertex v, such that there
exists a unique L-coloring f for G. Set graph H = G — V1, it is not difficult to see that H is complete bipartite graph Kn,p. We assign the following lists L'(v) fa the vert ices v of H:
If f (a) € L(v) then L'(v) = L(v) \ {f (a)}.
If f (a) € L(v) then L'(v) = L(v) \ {6}, where b € L(v) and b = f (v). It is clear that \L'(v)\ = k — 1 ^ 2 for every v € V(H). By Lemma 3, H has the property M(2). So bv Lemma 1, H has the property M(k — 1). It follows that with lists L'(v), there exists at least two list colorings for the vertices v of So it is not difficult to see that with lists L(v), there exists at least two list colorings for the vertices v of G, a contradiction.
(iii) It is not difficult to see that \E(G)\ = m +n +p 2~^m+n+p). By Lemma 4, we have
m(G) < \E(G)\ +2 = m2 + n2 + p2 —(m + n + P) + 4.
Bv Lemma 2, we have k < m +n +p -(m+n+p}+4.
(iv) Assertion (iii) follows immediately from Lemma 5.
Let G = Km,n,Pbe a UfcLC graph with V(G) = Vi u V2 u V3, G[Vi] = Om, G[V2] = On, G[V3] = = Op, 2 < m < n < p,k ^ 3. Set
Vi = {Ui,U2,...,Um},V2 = {Vi,V2,...,Vn},V3 = {Wi,W2,...,Wp}.
Suppose that, for the given fc-list assignment L: LUi = {a.i,i, ai,2,..., ai,k} for every i = 1,...,m, Lvi = {bi,i,bi,2,.. .,bi,k} for every i = 1,...,n, LWi = {ci,i,Ci,2,..., Ci,k} for every i = 1,...,p, there is a unique fc-list color f:
f (ui) = aiti for every i = 1,... ,m, f (vi) = biti for every i = 1,... ,n, f (wj) = Citi for every i = 1,... ,p.
Theorem 2. (i) ai,i = bjti for every i = 1,... ,m,j = 1,... ,n;
(ii) a,iti = Cj,i for eve ry i = 1,... ,m, j = 1,... ,p;
(iii) biti = Cj,i for eve ry i = 1,... ,n, j = 1,... ,p;
(iv) ai,i € Wj,2, aj,3,..., aj,k} for eve ry i,j = 1, 2,..., m;
(v) bi,i € {bj,2, bj,3,..., bj,k} for eve ry i,j = 1, 2,..., n;
(vi) Ci,i € {cj,2, Cjt3,.. .,Cj,k} for eve ry i,j = 1, 2,... ,p.
Proof, (i) Since G = Km,n,p is a complete tripartite graph, u,, is adjacent to Vj for every i = 1,.. .,m,j = 1,..., n. So it is not difficult to see that ai,i = f (uj) = f (vj) = bjti for every i = 1,... ,m,j = 1,... ,n.
(ii) Similar proofs (i).
(iii) Similar proofs (i).
(iv) 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,i € {aj0,2,aj0,3,... ,aj0,k }• It is clear that ai0,i = ajo;i- Let f' be the coloring of G such that
(a) f'(Ujo) = aio
(b) f'(ui) = ai,i for every i € {1,..., m}, i = jo;
(c) f'(vi) = biti for every i = 1,... ,n;
(d) f'(wj) = Ci,i for every i = 1,... ,p.
Then f ^ a ^^^ coloring for G rnd f = f, a contradiction.
(v) Similar proofs (iii).
(vi) Similar proofs (iii).
Set f (v) = L(v) \ {f (v)} for every v € V(G).
Theorem 3. (i) If (Vi)l > k — 2 for every i = 1,2,3;
(ii) Uv^Vi f (v) C f (Vj U Vt) for eve ry i,j,t £ {1, 2, 3} and i,j,t are doubles a distinction; (tit) UV£v(G)W) C f (V(G)); _
(iv) There exists v £ Vi U Vj such th at f (v) C f (Vt) for eve ry i,j,t £ {1, 2, 3} an d i,j,t are doubles a distinction.
Proof, (i) For suppose on the contrary that |/(V1 )| = t ^ k — 2,. Set H = G — V1: it is not difficult to see that H is complete bipartite graph Kn,p. We assign the following lists L'(v) for the vertices v of H:
If f (Vi) C L(v) then L'(v) = L(v) \ f (Vi).
If there exists A C f (Vi) such th at A n L(v) = 0, then
L'(v) = L(v) \ {di,d2,.. .,dt-\A\,ei,Z2,.. .,e\A\},
where
di,d2,..., dt-\A\ £ L(v) \ A,ei,e2,..., e\A\ £ L(v) and f(v) £ {ei,e2,...,e\A\}.
It is clear that IL'(v)I = k — t ^ 2 for every v £ V(H). By Lemma 3, H has the property M(2). So bv Lemma 1, H has the property M (k — t). 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 L(v), there exist at least two list colorings for the vertices v of G, a contradiction. Thus, |/| > k — 2.
By the same method of proof as above, we can also prove that If(V2)| > k — 2 Mid If(Vs)| > k — 2.
(ii) For suppose on the contrary that f (v) C f (V2 U V3). Then there exists io,jo such that aio,jo £ f (V2 U V3) with 1 ^ i0 ^ m, 2 ^ j0 ^ ^^et f ^e ^te raloring of G such that
(a) f'(Ui0) = ai0,j0;
(b) f'(ui) = ai,i for every i £ {1,..., m}, i =
(c) f'(vi) = bi,i for every i = 1,... ,n;
(d) f '(wi) = Ci,i for every i = 1,... ,p.
Then f 'is a fc-list coloring for G and f = f, a contradiction. Thus,
UveVl m c f (V2 U V3).
Bv the same method of proof as above, we can also prove that Uey2 f (v) C f (Vi U V3) and u^ev3 m c f (Vi U V2). _
(iii) For suppose on the contrary that U^ey(G)f (v) C f (V(&)■ Without loss of generality, we may assume that there exists io,jo such th at ai0,j0 £ f (V (G)) with 1 ^ i0 ^ m, 2 ^ j0 ^ k.
Let f' ^e ^te coloring of G such that
(a) f'(Ui0) = ai0,j0;
(b) f'(ui) = ai,i for every i £ {1,..., m}, i =
(c) f'(vi) = bi,i for every i = 1,... ,n;
(d) f '(wi) = Ci,i for every i = 1,... ,p.
Then f 'is a fc-list coloring for G and f = f, a contradiction.
(iv) For suppose on the contrary that f (v) C f (Vi) for every v £ V2 U V3, then |/(v) \ f | ^ 1 for every v £ V2 U V^^o lL(v) \ f (V^)| ^ 2 for every v £ V2 U V3. Set graph
H = G — Vi = G[V2 U V3] = Kn,p.
Let L' (v) C L(v) \ f (V-\) such th at (v)| = 2 for every v £ V2 U V3. By Lemma 3, H has the property M(2), it follows that with lists V(v), there exist at least two list colorings for the vertices v for every v £ V2 U V3. ^o it is not difficult to see that with lists L(v), there exist at least two list colorings for the vertices v of G, a contradiction. Thus, there exists v £ V2 U V3 such that W) C f (Vi).
Bv the same method of proof as above, we can also prove that there exists v £ Vi U V3 such that f (v) C f (V2) ^^d ^tee exists v £ Vi U V2 such th at f (v) C f (V3).
3. On property M(4) of some complete tripartite graphs
Set the complete tripartite graph G = Km,n,p. Let V(G) = ViUV2UV3 is a partition of V(G) such that Vi = {ui,u2,..., um}, V2 = {vi,v2,..., vn}, V3 = {wi,w2,..., wp} and fa every i = 1,2,3 the subgraphs of G induced by Vi, is empty graph.
Lemma 7. m(K2,2,p) = 3 if 1 4 p 4 2.
Proof. By Lemma 3, G is U2LC. Suppose that G is U3LC. Bv Lemma 5, IV(G)| ^ 7, a contradiction. So m(G) = 3.
Lemma 8 ([10]). m(K2,2,3) = m(K2A3) = 3.
Lemma 9 ([17]). m(K2,2,P) = 3 if 4 4 p 4 8.
Lemma 10. m(K2,2,p) = 3 if 1 4 p 4 8.
PROOF. It follows from Lemma 7, Lemma 8 and Lemma 9.
Lemma 11 ([18]). The graph K2,3,4 has the property M(3).
Lemma 12. m(K2,3,4) = 3.
PROOF. It follows from Lemma 3 and Lemma 11.
Theorem 4. G = Km,n,p is U3LC if one of the following conditions occurs.
(i) m ^ 2,n ^ 2 and p ^ 9;
(ii) m ^ 2,n ^ 3 and p ^ 5;
(iii) m ^ 2,n ^ 4 and p ^ 4;
(iv) m,n,p ^ 3.
PROOF, (i) We assign the following lists for the vertices of G: L(u\) = {1, 2, 6} L(u2) = L(u3) = = ... = L(um) = {3, 4, 5};
L(vi) = {1, 3, 6} L(V2) = L(v3) = ... = L(vn) = {2, 4, 6};
L(wi) = {1, 4, 5} L(w2) = {1, 3, 6} L(w3) = {1, 4, 6} L(w4) = {1, 5, 6} L(w5) = {2, 3, 4}, L(w6) = {2, 3, 5} L(wj) = {2, 3, 6} L(w8) = {2, 4, 6} L(wg) = L(ww) = ... = L(wp) = {2, 5, 6}.
A unique coloring / of G exists from the assigned lists: f (u\)=6, f (u2) = f (u3) =... = f (um) = 5;
f(Vi) = 3, f(V2) = f(V3) = ... = f(vn) = 4;
f(wi) = f(W2) = f(W3) = f(W4) = 1, f(w5) = f(we) = ... = f(wp) = 2.
(ii) We assign the following lists for the vertices of G: L(ui) = {1, 3, 6} L(u2) = L(u3) = = ... = L(um) = {2, 4, 5};
L(vi) = {1, 2, 3} L(V2) = L(v3) = ... = L(vn) = {2, 4, 5};
L(wi) = {1, 3, 5} L(w2) = {1, 4, 5} L(W3) = {1, 4, 6} L(Wi) = {2, 3, 4} L(w5) = L(w6) = = ... = L(wp) = {2, 5, 6}.
A unique coloring / of G exists from the assigned lists: f (ui)=6,f (u2) = f (u3) =... = f (um) = 5;
f(Vi)=3, f(V2) = f(V3) = ... = f(vn)=4;
f (Wi) = f (W2) = f (W3) = 1 f (W4) = f (W5) = ... = f (Wp) = 2.
(iii) We assign the following lists for the vertices of G: L(ui) = {1, 3, 5} L(u2) = L(u3) = ... = = L(um) = {2, 4, 6};
L(vi) = {1, 2, 3} L(V2) = {1, 3, 5} L(V3) = {1, 2, 4}, L(v^) = L(v5) = ... = L(vn) = {2, 4, 6};
L(wi) = {1, 4, 5} L(w2) = {1, 3, 6} L(w3) = {2, 3, 4}, L(w4) = L(w5) = ... = L(wp)=2, 5, 6.
A unique coloring / of G exists from the assigned lists: f (ui) = 5, f (u2) = f (u3) = ... = = f (um~) = 6
f (vi) = f (V2) = 3 f (V3) = f (V4) = .. = f (vn) = 4;
f (wi) = f (W2) = 1 f (W3) = f (w4) = ... = f (wp) = 2.
(iv) We assign the following lists for the vertices of G: L(ui) = {1, 4, 6} L(u2) = {2, 3, 6}, L(u3) = L(ua) = ... = L(um) = {2, 4, 5};
L(vi) = {2, 3, 6} L(v2) = {1, 2, 4} L(v3) = L(v4) = ... = L(vn) = {4, 5, 6};
L(wi) = {2, 3, 5} L(w2) = {2, 4, 6} L(w3) = L(w4) = ... = L(wp) = {3, 4, 6}.
A unique coloring /of G exists from the assigned lists: f (u\) = 1, f (u2) = f (u3) =... = f (um) = 2;
f(Vi) = 3, f(V2) = f(v3) = ... = f(vn) = 4;
f(Wi) = 5 f(W2) = f(W3) = ... = f(WP) = 6.
Corollary, (i) G = K2,2yP is U3LC if and only Hp ^ 9;
(ii) G = K2,3,p is U3LC if and only if p ^ 5;
(iii) G = K2,4,p is U^^C if and only if p ^ 4;
(iv) G = K3,3,p is U^^C if and onlv if p ^ 3.
proof, (i) It follows from Lemma 10 and (i) of Theorem 3.
(ii) It follows from (ii) of Theorem 1, Lemma 8 and Lemma 12.
(iii) It follows from (ii) of Theorem 1, Lemma 10 and Lemma 12.
(iv) It follows from (ii) of Theorem 1 and Lemma 8.
Lemma 13. The graph G = K2,n,p has the property M(4).
proof. For suppose on the contrary that G is U4LC. Then fa each vertex v in G, there exists a list of 4 colors L(v), such that there exists a unique L-coloring for G. By (i) of Theorem 3 we have 2 = |Vi| ^ |/(Vi)l > 4 - 2 = 2, contradiction. Thus, G = K2,n,P has the propertv M(4).
The join of Om mvd Kn, Om + Kn = S(m,n), is called a complete split graph.
Lemma 14 ([10]). For every n ^ 2, we have m(S(3,n)) = 3.
Theorem 5. (i) m(K2y2,p) = 4 if and only if p ^ 9;
(ii) m(K2,3,p) = 4 if and only if p ^ 5;
(iii) m(K2,4,p) = 4 if and onl у if p ^ 4;
(iv) m(K3,3,3) = 4.
PROOF, (i) It follows from (i) of Theorem 4 and Lemma 13.
(ii) It follows from (ii) of Theorem 4 and Lemma 13.
(iii) It follows from (iii) of Theorem 4 and Lemma 13.
(iv) For suppose on the contrary that G = K3,3,3 is U4LC. Then fa each vertex v in G, there exists a list of 4 colors L(v), such that there exists a unique L-coloring for G. By (i) of Theorem 3, i/(Vi)l, If (V2)l > 4-2 = 2, it follows that |/(Vi)l = |/Ш1 = 3. So f (щ) = f (и/) and f (vz) = f (Vj) for every i, j = 1,2,3,i = j. Set graph G' = (V', E'^th V' = V(G),
E' = E(G) u {rnuj Ii,j = 1,2,... ,m; i = j} u {vivj Ii,j = 1, 2,...,n; i = j}.
It is clear that G' is complete split graph S(3, 6). By Lemma 14, G' has the property M(3). Bv Lemma 1, G' has the property M(4) да with lists L(v), 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 L(v), there exist at least two list colorings for the vertices v of G, a contradiction. Thus, G has the property M(4). Bv (iv) of Theorem 4, we have m(K3,3,3) = 4.
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8. M. Ghebleh and E.S. Mahmoodian, On uniquely list colorable graphs, Ars Combin. № 59 (2001) 307-318.
9. Le Xuan Hung, Colorings of the graph K™ + Kn, Journal of Siberian Federal University. Mathematics & Physics, to appear.
10. M. Mahdian and E.S. Mahmoodian, A characterization of uniquely 2-list colorable graphs, Ars Combin. № 51 (1999) 295-305.
11. R.C. Read, An introduction to chromatic polynomials, J. Combin. Theory № 4 (1968) 52-71.
12. Ngo Dac Tan and Le Xuan Hung, On colorings of split graphs, Acta Mathematica Vietnammica, Vol. 31, № 3, 2006, pp. 195 - 204.
13. V.G. Vizing, On an estimate of the chromatic class of a p-graph, Discret. Analiz. № 3 (1964), pp. 23-30. (In Russian)
14. V. G. Vizing. Coloring the vertices of a graph in prescribed colors. In Diskret. Analiz, № 29 in Metodv Diskret. Anal, v Teorii Kodov i Shem, pp. 3-10, 1976.
15. WT. Wang and X. Liu, List-coloring based channel allocation for open-spectrum wireless networks, in Proceedings of the IEEE International Conference on Vehicular Technology (VTC '05), 2005, pp. 690 - 694.
16. R.J. WTilson, Introduction to graph theory, Longman group ltd, London, (1975).
17. Yancai Zhao and Erfang Shan, On characterization of uniquely 3-list colorable complete multipartite graphs, Discussiones Mathematicae Graph Theory, № 30, 2010, pp. 105-114.
18. Y.Q. Zhao, WT.J. He, Y.F. Shen and Y.N. Wang, Note on characterization of uniquely 3-list colorable complete multipartite graphs, in: Discrete Geometry, Combinatorics and Graph Theory, LNCS 4381 (Springer, Berlin, 2007, pp. 278-287.
REFERENCES
1. M. Behzad, 1965, "Graphs and thei chromatic number", Doctoral Thesis (Michigan State University).
2. М. Behzad and G. Chartrand, 1971, "Introduction to the theory of graphs", Allyn and Bacon, Boston.
3. M. Behzad, G. Chartrand and J. Cooper, 1967, "The coloring numbers of complete graphs", J. London Math. Soc. № 42, pp. 226 - 228.
4. J.A. Bondv and U.S.R. Murtv, 1976, "Graph theory with applications", MacMillan.
5. R. Diestel, 2000, "Graph Theory", Springer - Verlag New York.
6. J.H. Dinitz and W.J. Martin, 1995, "The stipulation polynomial of a uniquely list colorable graph", Austran. J. Combin. № 11, pp. 105-115.
7. P. Erdos, A. L. Rubin, and H. Taylor, 1979, "Choosabilitv in graphs". In Proceedings of west coast conference on combinatorics, graph theory, and commuting, number 26 in Congr. Numer., pp. 125-157, Areata, CA.
8. M. Ghebleh and E.S. Mahmoodian, 2001, "On uniquely list colorable graphs", Ars Combin. № 59, pp. 307-318.
9. Le Xuan Hung, "Colorings of the graph К™ + Kn", Journal of Siberian Federal University. Mathematics & Physics, to appear.
10. M. Mahdian and E.S. Mahmoodian, 1999, "A characterization of uniquely 2-list colorable graphs", Ars Combin. № 51, pp. 295-305.
11. R.C. Read, 1968, "An introduction to chromatic polynomials", J. Combin. Theory, № 4, pp. 52-71.
12. Ngo Dac Tan and Le Xuan Hung, 2006, "On colorings of split graphs", Acta Mathematica Vietnammica, Volume 31, № 3, pp. 195 - 204.
13. V.G. Vizing, 1964, "On an estimate of the chromatic class of a p-graph", Discret. Analiz. № 3, pp. 23-30. (In Russian).
14. V. G. Vizing ,1976, "Coloring the vertices of a graph in prescribed colors". In Diskret. Analiz, number 29 in Metodv Diskret. Anal, v Teorii Kodov i Shem, pp. 3-10.
15. W. Wang and X. Liu, 2005, "List-coloring based channel allocation for open-spectrum wireless networks
, in Proceedings of the IEEE International Conference on Vehicular Technology (VTC '05), pp. 690 - 694.
16. R.J. WTilson, 1975, Introduction to graph theory, Longman group ltd, London.
17. Yancai Zhao and Erfang Shan, 2010, "On characterization of uniquely 3-list colorable complete multipartite graphs", Discussiones Mathematicae Graph Theory, № 30, pp. 105-114.
18. Y.Q. Zhao, WT.J. He, Y.F. Shen and Y.N. Wang, 2007, "Note on characterization of uniquely 3-list colorable complete multipartite graphs", in: Discrete Geometry, Combinatorics and Graph Theory, LNCS 4381 (Springer, Berlin, pp. 278-287.
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