Partition universality for graphs of bounded degeneracy and degree

TL;DR

This paper establishes near-optimal bounds on the number of edges needed in a graph to ensure that any edge coloring contains a monochromatic subgraph of a certain degeneracy or degree, using probabilistic methods.

Contribution

It provides asymptotically tight bounds for universality in edge colorings for graphs with bounded degeneracy and degree, improving previous bounds and employing random graph models.

Findings

Derived optimal edge bounds for degeneracy-universal graphs.

Improved bounds for degree-universal graphs with Δ ≥ 4.

Showed random graphs likely contain the universal subgraphs.

Abstract

We prove asymptotically optimal bounds on the number of edges a graph $G$ must have in order that any $r$-colouring of $E(G)$ has a colour class which contains every $D$-degenerate graph on $n$ vertices with bounded maximum degree. We also improve the upper bounds on the number of edges $G$ must have in order that any $r$-colouring of $E(G)$ has a colour class which contains every $n$-vertex graph with maximum degree $\Delta$, for each $\Delta\ge 4$. In both cases, we show that a binomial random graph with $Cn$ vertices and a suitable edge probability is likely to provide the desired $G$.