Coloring Graphs with Forbidden Minors

A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. My research is motivated by the famous Hadwiger\u27s Conjecture from 1943 which states that every graph with no Kt-minor is (t − 1)-colorable. This conjecture has been proved true for t ≤ 6, but remains open for all t ≥ 7. For t = 7, it is not even yet known if a graph with no K7-minor is 7-colorable. We begin by showing that every graph with no Kt-minor is (2t − 6)- colorable for t = 7, 8, 9, in the process giving a shorter and computer-free proof of the known results for t = 7, 8. We also show that this result extends to larger values of t if Mader\u27s bound for the extremal function for Kt-minors is true. Additionally, we show that any graph with no K−8 - minor is 9-colorable, and any graph with no K=8-minor is 8-colorable. The Kempe-chain method developed for our proofs of the above results may be of independent interest. We also use Mader\u27s H-Wege theorem to establish some sufficient conditions for a graph to contain a K8-minor. Another motivation for my research is a well-known conjecture of Erdos and Lovasz from 1968, the Double-Critical Graph Conjecture. A connected graph G is double-critical if for all edges xy ∈ E(G), χ(G−x−y) = χ(G)−2. Erdos and Lovasz conjectured that the only double-critical t-chromatic graph is the complete graph Kt. This conjecture has been show to be true for t ≤ 5 and remains open for t ≥ 6. It has further been shown that any non-complete, double-critical, t-chromatic graph contains Kt as a minor for t ≤ 8. We give a shorter proof of this result for t = 7, a computer-free proof for t = 8, and extend the result to show that G contains a K9-minor for all t ≥ 9. Finally, we show that the Double-Critical Graph Conjecture is true for double-critical graphs with chromatic number t ≤ 8 if such graphs are claw-free

3-Factor-criticality in double domination edge critical graphs

A vertex subset $S$ of a graph $G$ is a double dominating set of $G$ if $|N[v]\cap S|\geq 2$ for each vertex $v$ of $G$, where $N[v]$ is the set of the vertex $v$ and vertices adjacent to $v$. The double domination number of $G$, denoted by $\gamma_{\times 2}(G)$, is the cardinality of a smallest double dominating set of $G$. A graph $G$ is said to be double domination edge critical if $\gamma_{\times 2}(G+e)<\gamma_{\times 2}(G)$ for any edge $e \notin E$. A double domination edge critical graph $G$ with $\gamma_{\times 2}(G)=k$ is called $k$-$\gamma_{\times 2}(G)$-critical. A graph $G$ is $r$-factor-critical if $G-S$ has a perfect matching for each set $S$ of $r$ vertices in $G$. In this paper we show that $G$ is 3-factor-critical if $G$ is a 3-connected claw-free $4$-$\gamma_{\times 2}(G)$-critical graph of odd order with minimum degree at least 4 except a family of graphs.Comment: 14 page

Extremal Infinite Graph Theory

We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.Comment: 41 pages, 16 figure

Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures

Sufficient conditions are developed, under which the compound Poisson distribution has maximal entropy within a natural class of probability measures on the nonnegative integers. Recently, one of the authors [O. Johnson, {\em Stoch. Proc. Appl.}, 2007] used a semigroup approach to show that the Poisson has maximal entropy among all ultra-log-concave distributions with fixed mean. We show via a non-trivial extension of this semigroup approach that the natural analog of the Poisson maximum entropy property remains valid if the compound Poisson distributions under consideration are log-concave, but that it fails in general. A parallel maximum entropy result is established for the family of compound binomial measures. Sufficient conditions for compound distributions to be log-concave are discussed and applications to combinatorics are examined; new bounds are derived on the entropy of the cardinality of a random independent set in a claw-free graph, and a connection is drawn to Mason's conjecture for matroids. The present results are primarily motivated by the desire to provide an information-theoretic foundation for compound Poisson approximation and associated limit theorems, analogous to the corresponding developments for the central limit theorem and for Poisson approximation. Our results also demonstrate new links between some probabilistic methods and the combinatorial notions of log-concavity and ultra-log-concavity, and they add to the growing body of work exploring the applications of maximum entropy characterizations to problems in discrete mathematics.Comment: 30 pages. This submission supersedes arXiv:0805.4112v1. Changes in v2: Updated references, typos correcte