Packing a Degree Sequence Realization With A Graph

TL;DR

This paper advances understanding of graph packing by establishing new degree sequence conditions under which a graph with a given degree sequence can pack with another graph, partially confirming the BEC conjecture.

Contribution

It proves a new bound for degree sequences ensuring packing with a given graph and confirms the BEC conjecture in specific cases involving unigraphs and forests.

Findings

Established a new degree sequence bound for graph packing.

Confirmed the BEC conjecture for certain classes involving unigraphs and forests.

Showed the existing bound is not sharp in most cases.

Abstract

Two simple $n$-vertex graphs $G_{1}$ and $G_{2}$, with respective maximum degrees $\Delta_{1}$ and $\Delta_{2}$, are said to pack if $G_{1}$ is isomorphic to a subgraph of the complement of $G_{2}$. The BEC conjecture by Bollob\'{a}s, Eldridge, and Catlin, states that if $(\Delta_{1}+1)(\Delta_{2}+1)\leq n+1$, then $G_{1}$ and $G_{2}$ pack. The BEC conjecture is true when $\Delta_{1}=2$ and has been confirmed for a few other classes of graphs with various conditions on $\Delta_{1}$, $\Delta_{2}$, or $n$. We show that if \[(\Delta_{1}+1)(\Delta_{2}+1)\leq n+\min\{\Delta_{1},\Delta_{2}\},\] then there exists a simple graph with an identical degree sequence as $G_{1}$ that packs with $G_{2}$. However, except for a few cases, we show that this bound is not sharp. As a consequence of our work, we confirm the BEC conjecture if $G_{1}$ is the vertex disjoint union of a unigraph and a forest $F$ such that either $F$ has at least $\Delta_{2}+1$ components or at most $2\Delta_{2}-1$ edges.