Chromatic number, clique subdivisions, and the conjectures of Haj\'os and Erd\H{o}s-Fajtlowicz

For a graph $G$, let $\chi(G)$ denote its chromatic number and $\sigma(G)$ denote the order of the largest clique subdivision in $G$. Let H(n) be the maximum of $\chi(G)/\sigma(G)$ over all $n$-vertex graphs $G$. A famous conjecture of Haj\'os from 1961 states that $\sigma(G) \geq \chi(G)$ for every graph $G$. That is, $H(n) \leq 1$ for all positive integers $n$. This conjecture was disproved by Catlin in 1979. Erd\H{o}s and Fajtlowicz further showed by considering a random graph that $H(n) \geq cn^{1/2}/\log n$ for some absolute constant $c>0$. In 1981 they conjectured that this bound is tight up to a constant factor in that there is some absolute constant $C$ such that $\chi(G)/\sigma(G) \leq Cn^{1/2}/\log n$ for all $n$-vertex graphs $G$. In this paper we prove the Erd\H{o}s-Fajtlowicz conjecture. The main ingredient in our proof, which might be of independent interest, is an estimate on the order of the largest clique subdivision which one can find in every graph on $n$ vertices with independence number $\alpha$.Comment: 14 page

Crux, space constraints and subdivisions

For a given graph $H$, its subdivisions carry the same topological structure. The existence of $H$-subdivisions within a graph $G$ has deep connections with topological, structural and extremal properties of $G$. One prominent examples of such connections, due to Bollob\'{a}s and Thomason and independently Koml\'os and Szemer\'edi, asserts that the average degree of $G$ being $d$ ensures a $K_{\Omega(\sqrt{d})}$-subdivision in $G$. Although this square-root bound is best possible, various results showed that much larger clique subdivisions can be found in a graph from many natural classes. We investigate the connection between crux, a notion capturing the essential order of a graph, and the existence of large clique subdivisions. This reveals the unifying cause underpinning all those improvements for various classes of graphs studied. Roughly speaking, when embedding subdivisions, natural space constraints arise; and such space constraints can be measured via crux. Our main result gives an asymptotically optimal bound on the size of a largest clique subdivision in a generic graph $G$, which is determined by both its average degree and its crux size. As corollaries, we obtain (1) a characterisation of extremal graphs for which the square-root bound above is tight: they are essentially disjoint union of graphs each of which has the crux size linear in $d$; (2) a unifying approach to find a clique subdivision of almost optimal size in graphs which does not contain a fixed bipartite graph as a subgraph; (3) and that the clique subdivision size in random graphs $G(n,p)$ witnesses a dichotomy: when $p = \omega(n^{-1/2})$, the barrier is the space, while when $p=o( n^{-1/2})$, the bottleneck is the density.Comment: 37 pages, 2 figure

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

Combinatorics and Probability

The main theme of this workshop was the use of probabilistic methods in combinatorics and theoretical computer science. Although these methods have been around for decades, they are being refined all the time: they are getting more and more sophisticated and powerful. Another theme was the study of random combinatorial structures, either for their own sake, or to tackle extremal questions. Both themes were richly represented at the workshop, with many recent exciting results presented by the lecturers

Topological cliques of random graphs

Given a graph G, denote by tcl(G) the largest integer r for which G contains a TKr, a toplogical complete r-graph. We show that for every ε{lunate} \u3e 0 almost every graph G of order n satisfies (2-ε)n 1 2 ≤ tlc(G)≤(2+ε) 1 2. © 1981