Toward \.Zak's conjecture on graph packing

Abstract

Two graphs $G_{1} = (V_{1}, E_{1})$ and $G_{2} = (V_{2}, E_{2})$, each of order $n$, pack if there exists a bijection $f$ from $V_{1}$ onto $V_{2}$ such that $uv \in E_{1}$ implies $f(u)f(v) \notin E_{2}$. In 2014, \.{Z}ak proved that if $\Delta (G_{1}), \Delta (G_{2}) \leq n-2$ and $|E_{1}| + |E_{2}| + \max \{ \Delta (G_{1}), \Delta (G_{2}) \} \leq 3n - 96n^{3/4} - 65$, then $G_{1}$ and $G_{2}$ pack. In the same paper, he conjectured that if $\Delta (G_{1}), \Delta (G_{2}) \leq n-2$, then $|E_{1}| + |E_{2}| + \max \{ \Delta (G_{1}), \Delta (G_{2}) \} \leq 3n - 7$ is sufficient for $G_{1}$ and $G_{2}$ to pack. We prove that, up to an additive constant, \.{Z}ak's conjecture is correct. Namely, there is a constant $C$ such that if $\Delta(G_1),\Delta(G_2) \leq n-2$ and $|E_{1}| + |E_{2}| + \max \{ \Delta(G_{1}), \Delta(G_{2}) \} \leq 3n - C$, then $G_{1}$ and $G_{2}$ pack. In order to facilitate induction, we prove a stronger result on list packing.