A Proof of a Conjecture of Ohba
We prove a conjecture of Ohba which says that every graph $G$ on at most $2χ(G)+1$ vertices satisfies $χ_\ell(G)=χ(G)$.
math.CO↗
arXiv subjects
Bruce A. Reed
Publications and source records attributed to Bruce A. Reed.
Claw-free graphs, skeletal graphs, and a stronger conjecture on $ω$, $Δ$, and $χ$
The second author's $ω$, $Δ$, $χ$ conjecture proposes that every graph satisties $χ\leq \lceil \frac 12 (Δ+1+ω)\rceil$. In this paper we prove that the conjecture holds for all claw-free graphs. Our approach uses the structure theorem of Chudnovsky and Seymour. Along the way we discuss a stronger local conjecture, and prove that it holds for claw-free graphs with a three-colourable complement. To prove our results we introduce a very useful $χ$-preserving reduction on homogeneous pairs of cliques, and thus restrict our view to so-called "skeletal" graphs.
cs.DM↗
A short proof that $χ$ can be bounded $ε$ away from $Δ+1$ towards $ω$
In 1998 the second author proved that there is an $ε>0$ such that every graph satisfies $χ\leq \lceil (1-ε)(Δ+1)+εω\rceil$. The first author recently proved that any graph satisfying $ω> \frac 23(Δ+1)$ contains a stable set intersecting every maximum clique. In this note we exploit the latter result to give a much shorter, simpler proof of the former. We include, as a certificate of simplicity, an appendix that proves all intermediate results with the exception of Hall's Theorem, Brooks' Theorem, the Lovász Local Lemma, and Talagrand's Inequality.
cs.DM↗
Polynomial treewidth forces a large grid-like-minor
Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an $\ell\times\ell$ grid minor is exponential in $\ell$. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A \emph{grid-like-minor of order} $\ell$ in a graph $G$ is a set of paths in $G$ whose intersection graph is bipartite and contains a $K_{\ell}$-minor. For example, the rows and columns of the $\ell\times\ell$ grid are a grid-like-minor of order $\ell+1$. We prove that polynomial treewidth forces a large grid-like-minor. In particular, every graph with treewidth at least $c\ell^4\sqrt{\log\ell}$ has a grid-like-minor of order $\ell$. As an application of this result, we prove that the cartesian product $G\square K_2$ contains a $K_{\ell}$-minor whenever $G$ has treewidth at least $c\ell^4\sqrt{\log\ell}$.
math.CO↗