A graph G is called chromatic-choosable if its choice number is equal to its chromatic number, namely ch（G） = X（G）. Ohba＇s conjecture states that every graph G with 2X（G）＋ 1 or fewer vertices is chromatic- cho...A graph G is called chromatic-choosable if its choice number is equal to its chromatic number, namely ch（G） = X（G）. Ohba＇s conjecture states that every graph G with 2X（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. Recently, Kostochka, Stiebitz and Woodall showed that Ohba＇s conjecture holds for complete multipartite graphs with partite size at most five. But the complete multipartite graphs with no restriction on their partite size, for which Ohba＇s conjecture has been verified are nothing more than the graphs Kt＋3,2.（k-t-l）,l.t by Enotomo et al., and gt＋2,3,2.（k-t-2）,l.t for t ≤ 4 by Shen et al.. In this paper, using the concept of f-choosable （or Lo-size-choosable） of graphs, we show that Ohba＇s conjecture is also true for the graphs gt＋2,3,2.（k-t-2）,l.t when t ≥ 5. Thus, Ohba＇s conjecture is true for graphs Kt＋2,3,2,（k-t-2）,l＊t for all integers t 〉 1.展开更多
基金Supported by the National Natural Science Foundation of China(No.10871058)the project for mathematical research from the Natural Science Foundation of Hebei Province,China(No.08M004)Hebei Normal University of Science and Technology,China(ZDJS2009 and CXTD2012)
文摘A graph G is called chromatic-choosable if its choice number is equal to its chromatic number, namely ch（G） = X（G）. Ohba＇s conjecture states that every graph G with 2X（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. Recently, Kostochka, Stiebitz and Woodall showed that Ohba＇s conjecture holds for complete multipartite graphs with partite size at most five. But the complete multipartite graphs with no restriction on their partite size, for which Ohba＇s conjecture has been verified are nothing more than the graphs Kt＋3,2.（k-t-l）,l.t by Enotomo et al., and gt＋2,3,2.（k-t-2）,l.t for t ≤ 4 by Shen et al.. In this paper, using the concept of f-choosable （or Lo-size-choosable） of graphs, we show that Ohba＇s conjecture is also true for the graphs gt＋2,3,2.（k-t-2）,l.t when t ≥ 5. Thus, Ohba＇s conjecture is true for graphs Kt＋2,3,2,（k-t-2）,l＊t for all integers t 〉 1.
