# Journal of Combinatorics

## Volume 7 (2016)

### Number 2–3

Guest editors: Rong Luo and Cun-Quan Zhang

### Toward Żak’s conjecture on graph packing

Pages: 307 – 340

DOI: https://dx.doi.org/10.4310/JOC.2016.v7.n2.a6

#### Authors

Ervin Gyori (Alfréd Rényi Institute of Mathematics, Budapest, Hungary; and Department of Mathematics, Central European University, Budapest, Hungary)

Alexandr Kostochka (Department of Mathematics, University of Illinois, Urbana, Il., U.S.A.; and Sobolev Institute of Mathematics, Novosibirsk, Russia)

Andrew McConvey (Department of Mathematics, University of Illinois, Urbana, Il., U.S.A.)

Derrek Yager (Department of Mathematics, University of Illinois, Urbana, Il., U.S.A.)

#### 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, Żak proved that if $\Delta(G_{1}), \Delta(G_{2}) \leq n-2$ and $|E_{1}| + |E_{2}| + \max\{ \Delta(G_{1}), \Delta(G_{2}) \} \leq3n - 96n^{3/4} - 65$, 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 the weaker condition $|E_{1}| + |E_{2}| + \max\{ \Delta (G_{1}), \Delta(G_{2}) \} \leq3n - 7$ is sufficient for $G_{1}$ and $G_{2}$ to pack. We prove that, up to an additive constant, Żak’s conjecture is correct. Namely, there is a constant $C$ such that if $\Delta(G_1),\Delta(G_2) \leq n-2$ and $|E_{1}| + |E_{2}| + \max\{ \Delta(G_{1}), \Delta(G_{2}) \} \leq3n - C$, then $G_{1}$ and $G_{2}$ pack. In order to facilitate induction, we prove a stronger result on list packing.

#### Keywords

graph packing, maximum degree, edge sum, list coloring

#### 2010 Mathematics Subject Classification

05C35, 05C70

Published 23 February 2016