Minors in expanding graphs

Extending several previous results we obtained nearly tight estimates on the maximum size of a clique-minor in various classes of expanding graphs. These results can be used to show that graphs without short cycles and other H-free graphs contain large clique-minors, resolving some open questions in this area

Random graphs from a weighted minor-closed class

There has been much recent interest in random graphs sampled uniformly from the n-vertex graphs in a suitable minor-closed class, such as the class of all planar graphs. Here we use combinatorial and probabilistic methods to investigate a more general model. We consider random graphs from a `well-behaved' class of graphs: examples of such classes include all minor-closed classes of graphs with 2-connected excluded minors (such as forests, series-parallel graphs and planar graphs), the class of graphs embeddable on any given surface, and the class of graphs with at most k vertex-disjoint cycles. Also, we give weights to edges and components to specify probabilities, so that our random graphs correspond to the random cluster model, appropriately conditioned. We find that earlier results extend naturally in both directions, to general well-behaved classes of graphs, and to the weighted framework, for example results concerning the probability of a random graph being connected; and we also give results on the 2-core which are new even for the uniform (unweighted) case.Comment: 46 page

Random graphs with few disjoint cycles

The classical Erd\H{o}s-P\'{o}sa theorem states that for each positive integer k there is an f(k) such that, in each graph G which does not have k+1 disjoint cycles, there is a blocker of size at most f(k); that is, a set B of at most f(k) vertices such that G-B has no cycles. We show that, amongst all such graphs on vertex set {1,..,n}, all but an exponentially small proportion have a blocker of size k. We also give further properties of a random graph sampled uniformly from this class; concerning uniqueness of the blocker, connectivity, chromatic number and clique number. A key step in the proof of the main theorem is to show that there must be a blocker as in the Erd\H{o}s-P\'{o}sa theorem with the extra `redundancy' property that B-v is still a blocker for all but at most k vertices v in B

Small Complete Minors Above the Extremal Edge Density

A fundamental result of Mader from 1972 asserts that a graph of high average degree contains a highly connected subgraph with roughly the same average degree. We prove a lemma showing that one can strengthen Mader's result by replacing the notion of high connectivity by the notion of vertex expansion. Another well known result in graph theory states that for every integer t there is a smallest real c(t) so that every n-vertex graph with c(t)n edges contains a K_t-minor. Fiorini, Joret, Theis and Wood conjectured that if an n-vertex graph G has (c(t)+\epsilon)n edges then G contains a K_t-minor of order at most C(\epsilon)log n. We use our extension of Mader's theorem to prove that such a graph G must contain a K_t-minor of order at most C(\epsilon)log n loglog n. Known constructions of graphs with high girth show that this result is tight up to the loglog n factor

Graph coloring with no large monochromatic components

For a graph G and an integer t we let mcc_t(G) be the smallest m such that there exists a coloring of the vertices of G by t colors with no monochromatic connected subgraph having more than m vertices. Let F be any nontrivial minor-closed family of graphs. We show that \mcc_2(G) = O(n^{2/3}) for any n-vertex graph G \in F. This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such F and every fixed t we show that mcc_t(G)=O(n^{2/(t+1)}). On the other hand we have examples of graphs G with no K_{t+3} minor and with mcc_t(G)=\Omega(n^{2/(2t-1)}). It is also interesting to consider graphs of bounded degrees. Haxell, Szabo, and Tardos proved \mcc_2(G) \leq 20000 for every graph G of maximum degree 5. We show that there are n-vertex 7-regular graphs G with \mcc_2(G)=\Omega(n), and more sharply, for every \epsilon>0 there exists c_\epsilon>0 and n-vertex graphs of maximum degree 7, average degree at most 6+\epsilon for all subgraphs, and with mcc_2(G)\ge c_\eps n. For 6-regular graphs it is known only that the maximum order of magnitude of \mcc_2 is between \sqrt n and n. We also offer a Ramsey-theoretic perspective of the quantity \mcc_t(G).Comment: 13 pages, 2 figure