  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 ($\ge 940060$).
