Toward \.Zak's conjecture on graph packing

TL;DR

This paper advances the understanding of graph packing by proving a near-optimal bound related to Zak's conjecture, showing that under certain degree and edge conditions, two graphs can be packed, with implications for graph theory.

Contribution

The authors prove that Zak's conjecture holds up to an additive constant by establishing a stronger list packing result, improving the bounds for graph packing conditions.

Findings

Proved that Zak's conjecture is correct up to a constant additive term.

Established a new bound involving degrees and edges for graph packing.

Developed a stronger list packing theorem to facilitate the proof.

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.