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

ON CHOOSABILITY OF COMPLETE MULTIPARTITE GRAPHS K_{4,3*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 K_{4,3*t,2*(k-2t-2),1*(t+1)} for all integers t ≥ 1 and k ≥ 2t+2, that is, ch(K_{4,3*t,2*(k-2t-2),1*(t+1)}) = k, which extends the results ch(K_{4,3,2*(k-4),1*2}) = k given by Shen et al. (Discrete Math. 308 (2008) 136-143), and ch(K_{4,3*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

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