Chariya Uiyyasathian

Graphs with large clique-chromatic numbers

The clique-chromatic number of a graph G, χc(G), is the least number of colors on V (G) without a monocolored maximal clique of size at least two. If G is triangle-free, χc(G) = χ(G); we then consider only graphs with a triangle. Unlike the chromatic number, the clique-chromatic number of a graph is not necessary to be at least those of its subgraphs. Thus, for any family of graphs ℱ, the boundedness of {χc(G)|G ∈ℱ} has been investigated. Many families of graphs are proved to have a bounded set of clique-chromatic numbers. In literature, only few families of graphs are shown to have an unbounded set of clique-chromatic numbers, for instance, the family of line graphs. This paper gives another family of graphs with such an unbounded set. These graphs are obtained by the well-known Mycielski’s construction with a certain property of the initial graph.

More Results on Clique-chromatic Numbers of Graphs with No Long Path

The clique-chromatic number of a graph is the least number of colors on the vertices of the graph so that no maximal clique of size at least two is monochromatic. In 2003, Gravier, Hoang, and Maffray have shown that, for any graph \(F\), the class of \(F\)-free graphs has a bounded clique-chromatic number if and only if \(F\) is a vertex-disjoint union of paths, and they give an upper bound for all such cases. In this paper, their bounds for \(F=P_2+kP_1\) and \(F=P_3+kP_1\) with \(k \ge 3\) are significantly reduced to \(k+1\) and \(k+2\) respectively, and sharp bounds are given for some subclasses.

On Non 3-Choosable Bipartite Graphs

In 2003, Fitzpatrick and MacGillivray proved that every complete bipartite graph with fourteen vertices except K 7,7 is 3-choosable and there is the unique 3-list assignment L up to renaming the colors such that K 7,7 is not L-colorable. We present our strategies which can be applied to obtain another proof of their result. These strategies are invented to claim a stronger result that every complete bipartite graph with fifteen vertices except K 7,8 is 3-choosable. We also show all 3-list assignments L such that K 7,8 is not L-colorable.