An improved upper bound on the maximum degree of terminal-pairable complete graphs

TL;DR

This paper improves the upper bound on the maximum degree for which complete graphs are terminal-pairable with respect to any demand multigraph, challenging a previous conjecture and advancing understanding of graph packing problems.

Contribution

It provides a tighter upper bound on the degree threshold for terminal-pairability in complete graphs, disproving a longstanding conjecture.

Findings

Established a new upper bound on elta(n)

Disproved the previous conjecture on terminal-pairability

Enhanced understanding of demand graph embeddings in complete graphs

Abstract

A graph $G$ is terminal-pairable with respect to a demand multigraph $D$ on the same vertex set as $G$, if there exists edge-disjoint paths joining the end vertices of every demand edge of $D$. In this short note, we improve the upper bound on the largest $\Delta(n)$ with the property that the complete graph on $n$ vertices is terminal-pairable with respect to any demand multigraph of maximum degree at most $\Delta(n)$. This disproves a conjecture originally stated by Csaba, Faudree, Gy\'arf\'as, Lehel and Schelp.