Supereulerian Index is Stable Under Contractions and Closures

Abstract

The supereulerian index of a graph \(G\) is the smallest integer \(k\) such that the \(k\)-th iterated line graph of \(G\) is supereulerian. We first show that adding an edge between two vertices with degree sums at least three in a graph cannot increase its supereulerian index. We use this result to prove that the supereulerian index of a graph \(G\) will not be changed after either of contracting an \(A_G(F)\)-contractible subgraph \(F\) of a graph \(G\) and performing the closure operation on \(G\) (if \(G\) is claw-free). Our results extend Catlin’s remarkable theorem \([4]\) relating that the supereulericity of a graph is stable under the contraction of a collapsible subgraph.