A note on minimum degree conditions for supereulerian graphs
A graph is supereulerian if it has a spanning eulerian subgraph. The main result of the paper states that if \(G\) is a simple \(2\)-edge-connected graph of order \(n\) such that for every minimal edge cut \(S\) with \(|S|\leq 3\) every component of \(G-S\) has order at least \((n-2)/5>2\), then \(G\) is either supereulerian or belongs to one of two well-described classes of exceptions. The authors derive this result relatively easily from a result due to Catlin. As a consequence they prove that if \(G\) is a \(2\)-edge-connected graph of order \(n\) and minimum degree at least \(4\) such that \(\max\{ d(x),d(y)\}\geq (n-2)/5-1\) for every edge \(xy\) of \(G\), then \(G\) is either supereulerian or belongs to one of two well-described classes of exceptions. This is a slight improvement of a result of \textit{P. A. Catlin} and \textit{X. Li} [Adv. Math., Beijing 28, 65-70 (1999; Zbl 1054.05506)].
scientific article; zbMATH DE number 1792310