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## Discussiones Mathematicae Graph Theory

2010 | 30 | 1 | 105-114
Tytuł artykułu

### On characterization of uniquely 3-list colorable complete multipartite graphs

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
For each vertex v of a graph G, if there exists a list of k colors, L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k-list colorable graph. Ghebleh and Mahmoodian characterized uniquely 3-list colorable complete multipartite graphs except for nine graphs: $K_{2,2,r}$ r ∈ {4,5,6,7,8}, $K_{2,3,4}$, $K_{1*4,4}$, $K_{1*4,5}$, $K_{1*5,4}$. Also, they conjectured that the nine graphs are not U3LC graphs. After that, except for $K_{2,2,r}$ r ∈ {4,5,6,7,8}, the others have been proved not to be U3LC graphs. In this paper we first prove that $K_{2,2,8}$ is not U3LC graph, and thus as a direct corollary, $K_{2,2,r}$ (r = 4,5,6,7,8) are not U3LC graphs, and then the uniquely 3-list colorable complete multipartite graphs are characterized completely.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
105-114
Opis fizyczny
Daty
wydano
2010
otrzymano
2009-02-26
poprawiono
2009-04-02
zaakceptowano
2009-04-14
Twórcy
autor
• Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China
• Department of Science, Bengbu University, Anhui 233030, P.R. China
autor
• Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China
Bibliografia
• [1] N. Alon, Restricted colorings of graphs, in: K. Walker, editor, Surveys in Combinatorics, Number 187 in London Math. Soc. LNS, pp. 1-33, 1993.
• [2] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (American Elsevier Publishing Co., INC., New York, 1976).
• [3] J.H. Dinitz and W.J. Martin, The stipulation polynomial of a uniquely list colorable graph, Austral. J. Combin. 11 (1995) 105-115.
• [4] P. Erdös, A.L. Rubin and H. Taylor, Choosability in graphs, in: Proceedings of West Coast Conference on Combinatorics, Graph Theory and Computing, number 26 in Congr. Number., pp. 125-157, Arcata, CA, September 1979.
• [5] Y.G. Ganjali, M. Ghebleh, H. Hajiabohassan, M. Mirzadeh and B.S. Sadjad, Uniquely 2-list colorable graphs, Discrete Appl. Math. 119 (2002) 217-225, doi: 10.1016/S0166-218X(00)00335-8.
• [6] M. Ghebleh and E.S. Mahmoodian, On uniquely list colorable graphs, Ars Combin. 59 (2001) 307-318.
• [7] W.J. He, Y.N. Wang, Y.F. Shen and X. Ma, On property M(3) of some complete multipartite graphs, Australasian Journal of Combinatorics, to appear.
• [8] M. Mahdian and E.S. Mahmoodian, A characterization of uniquely 2-list colorable graphs, Ars Combin. 51 (1999) 295-305.
• [9] E.S. Mahmoodian and M. Mahdian, On the uniquely list colorable graphs, in: Proceedings of the 28th Annual Iranian Mathematics Conference, Part 1, number 377 in Tabriz Univ. Ser., Tabriz, 1997.
• [10] Y.F. Shen and Y.N. Wang, On uniquely list colorable complete multipartite graphs, Ars Combin. 88 (2008) 367-377.
• [11] V.G. Vizing, Coloring the vertices of a graph in prescribed colors, (in Russian) Discret. Anal. 29 (1976) 3-10.
• [12] Y.Q. Zhao, W.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) 278-287, doi: 10.1007/978-3-540-70666-3₃0.
Typ dokumentu
Bibliografia
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