Additive Non-Approximability of Chromatic Number in Proper Minor-Closed Classes

Robin Thomas asked whether for every proper minor-closed class G, there exists a polynomial-time algorithm approximating the chromatic number of graphs from G up to a constant additive error independent on the class G. We show this is not the case: unless P=NP, for every integer k >= 1, there is no polynomial-time algorithm to color a K_{4k+1}-minor-free graph G using at most chi(G)+k-1 colors. More generally, for every k >= 1 and 1 <=beta <=4/3, there is no polynomial-time algorithm to color a K_{4k+1}-minor-free graph G using less than beta chi(G)+(4-3 beta)k colors. As far as we know, this is the first non-trivial non-approximability result regarding the chromatic number in proper minor-closed classes. We also give somewhat weaker non-approximability bound for K_{4k+1}-minor-free graphs with no cliques of size 4. On the positive side, we present an additive approximation algorithm whose error depends on the apex number of the forbidden minor, and an algorithm with additive error 6 under the additional assumption that the graph has no 4-cycles

Number of cliques in graphs with a forbidden subdivision

We prove that for all positive integers $t$, every $n$-vertex graph with no $K_t$-subdivision has at most $2^{50t}n$ cliques. We also prove that asymptotically, such graphs contain at most $2^{(5+o(1))t}n$ cliques, where $o(1)$ tends to zero as $t$ tends to infinity. This strongly answers a question of D. Wood asking if the number of cliques in $n$-vertex graphs with no $K_t$-minor is at most $2^{ct}n$ for some constant $c$.Comment: 10 pages; to appear in SIAM J. Discrete Mat

Hadwiger's conjecture for graphs with forbidden holes

Given a graph $G$, the Hadwiger number of $G$, denoted by $h(G)$, is the largest integer $k$ such that $G$ contains the complete graph $K_k$ as a minor. A hole in $G$ is an induced cycle of length at least four. Hadwiger's Conjecture from 1943 states that for every graph $G$, $h(G)\ge \chi(G)$, where $\chi(G)$ denotes the chromatic number of $G$. In this paper we establish more evidence for Hadwiger's conjecture by showing that if a graph $G$ with independence number $\alpha(G)\ge3$ has no hole of length between $4$ and $2\alpha(G)-1$, then $h(G)\ge\chi(G)$. We also prove that if a graph $G$ with independence number $\alpha(G)\ge2$ has no hole of length between $4$ and $2\alpha(G)$, then $G$ contains an odd clique minor of size $\chi(G)$, that is, such a graph $G$ satisfies the odd Hadwiger's conjecture

Forbidden minor characterizations for low-rank optimal solutions to semidefinite programs over the elliptope

We study a new geometric graph parameter \egd(G), defined as the smallest integer $r\ge 1$ for which any partial symmetric matrix which is completable to a correlation matrix and whose entries are specified at the positions of the edges of $G$, can be completed to a matrix in the convex hull of correlation matrices of \rank at most $r$. This graph parameter is motivated by its relevance to the problem of finding low rank solutions to semidefinite programs over the elliptope, and also by its relevance to the bounded rank Grothendieck constant. Indeed, \egd(G)\le r if and only if the rank-$r$ Grothendieck constant of $G$ is equal to 1. We show that the parameter \egd(G) is minor monotone, we identify several classes of forbidden minors for \egd(G)\le r and we give the full characterization for the case $r=2$. We also show an upper bound for \egd(G) in terms of a new tree-width-like parameter \sla(G), defined as the smallest $r$ for which $G$ is a minor of the strong product of a tree and $K_r$. We show that, for any 2-connected graph $G\ne K_{3,3}$ on at least 6 nodes, \egd(G)\le 2 if and only if \sla(G)\le 2.Comment: 33 pages, 8 Figures. In its second version, the paper has been modified to accommodate the suggestions of the referees. Furthermore, the title has been changed since we feel that the new title reflects more accurately the content and the main results of the pape

Ramsey numbers of cubes versus cliques

The cube graph Q_n is the skeleton of the n-dimensional cube. It is an n-regular graph on 2^n vertices. The Ramsey number r(Q_n, K_s) is the minimum N such that every graph of order N contains the cube graph Q_n or an independent set of order s. Burr and Erdos in 1983 asked whether the simple lower bound r(Q_n, K_s) >= (s-1)(2^n - 1)+1 is tight for s fixed and n sufficiently large. We make progress on this problem, obtaining the first upper bound which is within a constant factor of the lower bound.Comment: 26 page