Partitions of graphs into cographs

AbstractCographs form the minimal family of graphs containing K1 that is closed with respect to complementation and disjoint union. We discuss vertex partitions of graphs into the smallest number of cographs. We introduce a new parameter, calling the minimum order of such a partition the c-chromatic number of the graph. We begin by axiomatizing several well-known graphical parameters as motivation for this function. We present several bounds on c-chromatic number in terms of well-known expressions. We show that if a graph is triangle-free, then its chromatic number is bounded between the c-chromatic number and twice this number. We show that both bounds are sharp for graphs with arbitrarily high girth. This provides an alternative proof to a result by Broere and Mynhardt; namely, there exist triangle-free graphs with arbitrarily large c-chromatic numbers. We show that any planar graph with girth at least 11 has a c-chromatic number at most two. We close with several remarks on computational complexity. In particular, we show that computing the c-chromatic number is NP-complete for planar graphs

Induced Subgraphs and Subtournaments

Proceedings of Graph Theory@Georgia Tech, a conference honoring the 50th Birthday of Robin Thomas, May 7-11, 2012 in the Clough Undergraduate Learning Commons.Let H be a graph and let I be the set of all graphs that do not contain H as a minor. Then H is planar if and only if there is an upper bound on the treewidth of the members of I; and there are many other similar theorems that relate properties of H to structural properties of the members of I. Let us call them "structure theorems". What about structure theorems for other containment relations instead of minor containment? For induced subgraph containment, there are virtually no such theorems known, but for subtournament containment there are some quite pretty structure theorems, mostly excluding tournaments that are almost transitive. A transitive tournament is the tournament analogue of both a stable set and a clique, so what happens if we exclude as induced subgraphs TWO graphs instead of one; one almost a stable set, and the other almost a clique? This turns out to be a better problem with some nice answers. Partly joint work with Maria Chudnovsky, Sasha Fradkin, Ilhee Kim, Gaku Liu, Sergei Norin, Bruce Reed.NSF, NSA, ONR, IMA, Colleges of Sciences, Computing and Engineerin

Digraph Coloring Games and Game-Perfectness

In this thesis the game chromatic number of a digraph is introduced as a game-theoretic variant of the dichromatic number. This notion generalizes the well-known game chromatic number of a graph. An extended model also takes into account relaxed colorings and asymmetric move sequences. Game-perfectness is defined as a game-theoretic variant of perfectness of a graph, and is generalized to digraphs. We examine upper and lower bounds for the game chromatic number of several classes of digraphs. In the last part of the thesis, we characterize game-perfect digraphs with small clique number, and prove general results concerning game-perfectness. Some results are verified with the help of a computer program that is discussed in the appendix