Discussiones Mathematicae Graph Theory 30(2) (2010) 237-244
doi: 10.7151/dmgt.1489

[BIBTex] [PDF] [PS]

ON CHOOSABILITY OF COMPLETE MULTIPARTITE GRAPHS K4,3*t,2*(k-2t-2),1*(t+1)

Guo-Ping Zheng,  Yu-Fa Shen,  Zuo-Li Chen,  Jin-Feng Lv

School of Mathematics and Information Science and Technology
Hebei Normal University of Science and Technology
Qinhuangdao 066004, P.R. China
and
Center for Mathematics of Hebei Province
Hebei Normal University
Shijiazhuang 050016, P.R. China
e-mail: zhengguoping9199@126.com

Abstract

A graph G is said to be chromatic-choosable if ch(G) = χ(G). Ohba has conjectured that every graph G with 2χ(G)+1 or fewer vertices is chromatic-choosable. It is clear that Ohba's conjecture is true if and only if it is true for complete multipartite graphs. In this paper we show that Ohba's conjecture is true for complete multipartite graphs K4,3*t,2*(k-2t-2),1*(t+1) for all integers t ≥ 1 and k ≥ 2t+2, that is, ch(K4,3*t,2*(k-2t-2),1*(t+1)) = k, which extends the results ch(K4,3,2*(k-4),1*2) = k given by Shen et al. (Discrete Math. 308 (2008) 136-143), and ch(K4,3*2,2*(k-6),1*3) = k given by He et al. (Discrete Math. 308 (2008) 5871-5877).

Keywords: list coloring, complete multipartite graphs, chromatic-choosable graphs, Ohba's conjecture.

2010 Mathematics Subject Classifications: 05C15.

References

[1]H. Enotomo, K. Ohba, K. Ota and J. Sakamoto, Choice number of some complete multipartite graphs, Discrete Math. 244 (2002) 55-66, doi: 10.1016/S0012-365X(01)00059-0.
[2]P. Erdös, A.L. Rubin and H. Taylor, Choosability in graphs, Congr. Numer. 26 (1979) 125-157.
[3]S. Gravier and F. Maffray, Graphs whose choice number is equal to their chromatic number, J. Graph Theory 27 (1998) 87-97, doi: 10.1002/(SICI)1097-0118(199802)27:2<87::AID-JGT4>3.0.CO;2-B.
[4]W. He, L. Zhang, Daniel W. Cranston, Y. Shen and G. Zheng, Choice number of complete multipartite graphs K3*3,2*(k-5),1*2 and K4,3*2,2*(k-6),1*3, Discrete Math. 308 (2008) 5871-5877, doi: 10.1016/j.disc.2007.10.046.
[5]H.A. Kierstead, On the choosability of complete multipartite graphs with part size three, Discrete Math. 211 (2000) 255-259, doi: 10.1016/S0012-365X(99)00157-0.
[6]K. Ohba, On chromatic choosable graphs, J. Graph Theory 40 (2002) 130-135, doi: 10.1002/jgt.10033.
[7]K. Ohba, Choice number of complete multipartite graphs with part size at most three, Ars Combinatoria 72 (2004) 133-139.
[8]B. Reed and B. Sudakov, List colouring when the chromatic number is close to the order of the graph, Combinatorica 25 (2005) 117-123, doi: 10.1007/s00493-005-0010-x.
[9]Y. Shen, W. He, G. Zheng and Y. Li, Ohba's conjecture is ture for graph with independence number at most three, Applied Mathematics Letters 22 (2009) 938-942, doi: 10.1016/j.aml.2009.01.001.
[10]Y. Shen, W. He, G. Zheng, Y. Wang and L. Zhang, On choosability of some complete multipartite graphs and Ohba's conjecture, Discrete Math. 308 (2008) 136-143, doi: 10.1016/j.disc.2007.03.059.
[11]Y. Shen, G. Zheng and W. He, Chromatic choosability of a class of complete multipartite graphs, J. Mathematical Research and Exposition 27 (2007) 264-272.
[12]Zs. Tuza, Graph colorings with local constrains-A survey, Discuss. Math. Graph Theory 17 (1997) 161-228, doi: 10.7151/dmgt.1049.
[13]V.G. Vizing, Coloring the vertices of a graph in prescribed colors (in Russian), Diskret. Anal. 29 (1976) 3-10.
[14]D.R. Woodall, List colourings of graphs, in: J.W.P. Hirschfeld (ed.), Surveys in Combinatorics, 2001, London Math. Soc. Lecture Note Series, vol. 288 (Cambridge University Press, Cambridge, UK, 2001) 269-301.

Received 22 January 2009
Revised 22 June 2009
Accepted 22 June 2009