An improved upper bound on the maximum degree of terminal-pairable 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 ∆(n) with the property that the complete graph on n vertices is terminalpairable with respect to any demand multigraph of maximum degree at most ∆(n). This disproves a conjecture originally stated by Csaba, Faudree, Gyárfás, Lehel and Schelp.

Terminal-pairability in complete bipartite graphs with non-bipartite demands

“…It is not clear if this is the extremal bound. Our attempts to adapt the proof of [17] (which improves on the trivial extremal upper bound on the demand degrees in complete graphs) to complete bipartite base graphs have been futile.…”