Crux, space constraints and subdivisions

TL;DR

This paper establishes an asymptotically optimal bound on the size of clique subdivisions in graphs, linking average degree and a new measure called crux, unifying various results and characterizing extremal cases.

Contribution

It introduces the concept of crux to measure space constraints in embedding clique subdivisions, providing a unified framework and optimal bounds for diverse graph classes.

Findings

Optimal bound on clique subdivision size based on degree and crux

Characterization of extremal graphs with tight bounds

Dichotomy in clique subdivision size in random graphs

Abstract

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 example of such a connection, 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 for 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 unions of graphs having crux size linear in $d$; (2) a unifying approach to find a clique subdivision of almost optimal size in graphs which do 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.