Some problems in list coloring bipartite graphs

Abstract

A proper coloring of a graph is an assignment of colors to the vertices so that no two adjacent vertices have the same color. A list coloring is a generalization of this concept, where each vertex is assigned a list of colors and a proper coloring is found so that the color assigned to each vertex must come from its list. The choice number, ch(G) of a graph G is then the smallest integer k such that G can be list colored given any lists of size k assigned to the vertices. We focus our research on a number of interesting questions about the choice number of bipartite graphs.^ First, what happens to the choice number of the complete bipartite graph \$K\sb{m,n}\$ when edges are removed? We prove that if \$n=m\sp{m}\$ then removing an edge lowers the choice number of \$K\sb{m,n}.\$ When \$n\not= m\sp{m}\$ we prove that in certain cases we can remove several edges of \$K\sb{m,n}\$ without lowering the choice number. We also investigate several examples to illustrate that these certain cases seem to be in the majority.^ Second, we introduce the concept of defective list colorings. That is, in a proper coloring the graph induced by each color class (say, all the red vertices) is an independent set. What if we relax this condition and allow each color class to induce a graph of maximum degree d? This is called a defective list coloring with defect d. Our main question then is: does the defective choice number, \$ch\sb{d}(K\sb{m,n})\$ of complete bipartite graphs behave like the choice number? We prove that \$ch\sb{d}(K\sb{m,n})\$ behaves asymptotically like \$ch(K\sb{m,n}).\$ Also, as is the case for \$ch(K\sb{m,n}),\$ we find formulas for \$ch\sb{d}(K\sb{m,n})\$ when n is much larger than m.^ Third, we investigate the defective choice number of various classes of planar graphs. We say a graph G is (k, d)-choosable if \$ch\sb{d}(G)\le k.\$ We prove that all planar graphs are (4,2)-choosable, all outerplanar graphs are (2,2)-choosable, all triangle-free outerplanar graphs are (2,1)-choosable and there is no d such that all planar bipartite graphs are (2,d)-choosable. ^

Mathematics

Recommended Citation

Thomas Clinton Hull, "Some problems in list coloring bipartite graphs" (1997). Dissertations and Master's Theses (Campus Access). Paper AAI9805236.
http://digitalcommons.uri.edu/dissertations/AAI9805236

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