The Bollob\'as-Eldridge-Catlin conjecture for even girth at least $10$

TL;DR

This paper investigates the Bollobás-Eldridge-Catlin conjecture for graph packing, proving it under conditions that exclude small cycles and require large maximum degree, advancing understanding of graph packing constraints.

Contribution

It extends the conjecture's validity to graphs with large maximum degree and no small even cycles, a significant step in graph packing theory.

Findings

Proves the conjecture for graphs with large maximum degree and no small even cycles.

Establishes new bounds on maximum degree for graph packing.

Provides conditions under which two graphs can be packed without small cycles.

Abstract

Two graphs $G_1$ and $G_2$ on $n$ vertices are said to \textit{pack} if there exist injective mappings of their vertex sets into $[n]$ such that the images of their edge sets are disjoint. A longstanding conjecture due to Bollob\'as and Eldridge and, independently, Catlin, asserts that, if $(\Delta(G_1)+1) (\Delta(G_2)+1) \le n+1$, then $G_1$ and $G_2$ pack. We consider the validity of this assertion under the additional assumptions that neither $G_1$ nor $G_2$ contain a $4$-, $6$- or $8$-cycle, and that $\Delta(G_1)$ or $\Delta(G_2)$ is large enough ($\ge 940060$).