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

Two graphs $G_1$ and $G_2$ on $n$ vertices are said to $\mathit{\text{pack}}$ 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 $\mathrm{\Delta }(G_1)$ or $\mathrm{\Delta }(G_2)$ is large enough ($\ge 940060$).