A Note on the 3-Edge-Connected Supereulerian Graphs

Abstract:

      For two integers $l>0$ and $k\geq 0$, define $C(l,k)$ to be the family of 2-edge connected graphs such that a graph $G\in C(l,k)$ if and only if for every bond $S\subseteq E(G)$ with $|S|\leq 3$, each component of $G-S$ has order at least $(|V(G)|-k)/l$. In this note we prove that if a 3-edge-connected simple graph $G$ is in $C(10,3)$, then $G$ is supereulerian if and only if $G$ cannot be contracted to the Petersen graph. Our result extends an earlier result in [Supereulerian graphs and Petersen graph. JCMCC 1991, 9: 79-89] by Chen.