Constructing dense graphs with sublinear Hadwiger number

Mader asked to explicitly construct dense graphs for which the size of the largest clique minor is sublinear in the number of vertices. Such graphs exist as a random graph almost surely has this property. This question and variants were popularized by Thomason over several articles. We answer these questions by showing how to explicitly construct such graphs using blow-ups of small graphs with this property. This leads to the study of a fractional variant of the clique minor number, which may be of independent interest.Comment: 10 page

Extremal density for sparse minors and subdivisions

We prove an asymptotically tight bound on the extremal density guaranteeing subdivisions of bounded-degree bipartite graphs with a mild separability condition. As corollaries, we answer several questions of Reed and Wood on embedding sparse minors. Among others, $\bullet$ $(1+o(1))t^2$ average degree is sufficient to force the $t\times t$ grid as a topological minor; $\bullet$ $(3/2+o(1))t$ average degree forces every $t$-vertex planar graph as a minor, and the constant $3/2$ is optimal, furthermore, surprisingly, the value is the same for $t$-vertex graphs embeddable on any fixed surface; $\bullet$ a universal bound of $(2+o(1))t$ on average degree forcing every $t$-vertex graph in any nontrivial minor-closed family as a minor, and the constant 2 is best possible by considering graphs with given treewidth.Comment: 33 pages, 6 figure

Extremal density for sparse minors and subdivisions

We prove an asymptotically tight bound on the extremal density guaranteeing subdivisions of bounded-degree bipartite graphs with a mild separability condition. As corollaries, we answer several questions of Reed and Wood on embedding sparse minors. Among others, ∙ (1+o(1))t2 average degree is sufficient to force the t×t grid as a topological minor; ∙ (3/2+o(1))t average degree forces every t-vertex planar graph as a minor, and the constant 3/2 is optimal, furthermore, surprisingly, the value is the same for t-vertex graphs embeddable on any fixed surface; ∙ a universal bound of (2+o(1))t on average degree forcing every t-vertex graph in any nontrivial minor-closed family as a minor, and the constant 2 is best possible by considering graphs with given treewidth

Spanning trees of smallest maximum degree in subdivisions of graphs

\newcommand{\subdG}[1][G]{#1^\star} Given a graph $G$ and a positive integer $k$, we study the question whether $G^\star$ has a spanning tree of maximum degree at most $k$ where $G^\star$ is the graph that is obtained from $G$ by subdividing every edge once. Using matroid intersection, we obtain a polynomial algorithm for this problem and a characterization of its positive instances. We use this characterization to show that $G^\star$ has a spanning tree of bounded maximum degree if $G$ is contained in some particular graph class. We study the class of 3-connected graphs which are embeddable in a fixed surface and the class of $(p-1)$-connected $K_p$-minor-free graphs for a fixed integer $p$. We also give tightness examples for most of these classes

Graph Theory

This was a workshop on graph theory, with a comprehensive approach. Highlights included the emerging theories of sparse graph limits and of infinite matroids, new techniques for colouring graphs on surfaces, and extensions of graph minor theory to directed graphs and to immersion

Extremal connectivity for topological cliques in bipartite graphs

AbstractLet d(s) be the smallest number such that every graph of average degree >d(s) contains a subdivision of Ks. So far, the best known asymptotic bounds for d(s) are (1+o(1))9s2/64⩽d(s)⩽(1+o(1))s2/2. As observed by Łuczak, the lower bound is obtained by considering bipartite random graphs. Since with high probability the connectivity of these random graphs is about the same as their average degree, a connectivity of (1+o(1))9s2/64 is necessary to guarantee a subdivided Ks. Our main result shows that for bipartite graphs this gives the correct asymptotics. We also prove that in the non-bipartite case a connectivity of (1+o(1))s2/4 suffices to force a subdivision of Ks. Moreover, we slightly improve the constant in the upper bound for d(s) from 1/2 (which is due to Komlós and Szemerédi) to 10/23

Graph Theory

Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures

Structural sparsity of complex networks: bounded expansion in random models and real-world graphs

This research establishes that many real-world networks exhibit bounded expansion, a strong notion of structural sparsity, and demonstrates that it can be leveraged to design efficient algorithms for network analysis. Specifically, we give a new linear-time fpt algorithm for motif counting and linear time algorithms to compute localized variants of several centrality measures. To establish structural sparsity in real-world networks, we analyze several common network models regarding their structural sparsity. We show that, with high probability, (1) graphs sampled with a prescribed sparse degree sequence; (2) perturbed bounded-degree graphs; (3) stochastic block models with small probabilities; result in graphs of bounded expansion. In contrast, we show that the Kleinberg and the Barabási–Albert model have unbounded expansion. We support our findings with empirical measurements on a corpus of real-world networks