Packing a Degree Sequence Realization With A Graph

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.