Packing graphs of bounded codegree

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$(\Delta_1(G)+1) (\Delta_2(G)+1) \le n+1$impliesthat$G_1$and$G_2$pack\mathrm{.}Wealsoprovideamildimprovementifmoreover$G_2$containsnocopyofthecompletetripartitegraph$K_{1,1,s}$,$s\ge 1$.