Packing graphs of bounded codegree

TL;DR

This paper investigates graph packing under bounded codegree conditions, proving new sufficient conditions for packing when one graph excludes certain bipartite subgraphs and has high maximum degree.

Contribution

It establishes new packing criteria for graphs with bounded codegree, extending the Bollobás-Eldridge-Catlin conjecture under specific structural constraints.

Findings

Proves packing when one graph excludes $K_{2,t}$ and has high maximum degree.

Provides improved conditions if the second graph excludes $K_{1,1,s}$.

Extends classical packing conjecture to graphs with bounded codegree.

Abstract

Two graphs $G_1$ and $G_2$ on $n$ vertices are said to 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 $(\Delta_1(G)+1) (\Delta_2(G)+1) \le n+1$, then $G_1$ and $G_2$ pack. We consider the validity of this assertion under the additional assumption that $G_1$ or $G_2$ has bounded codegree. In particular, we prove for all $t \ge 2$ that, if $G_1$ contains no copy of the complete bipartite graph $K_{2,t}$ and $\Delta_1 > 17 t \cdot \Delta_2$, then $(\Delta_1(G)+1) (\Delta_2(G)+1) \le n+1$ implies that $G_1$ and $G_2$ pack. We also provide a mild improvement if moreover $G_2$ contains no copy of the complete tripartite graph $K_{1,1,s}$, $s\ge 1$.