A Note on the Maximum Number of Edges of a Spanning Eulerian Subgraph

Abstract

A graph $G$ is supereulerian if $G$ has a spanning eulerian subgraph. We use $\mathcal{S}\mathcal{L}$ to denote the families of supereulerian graphs. In 1995, Zhi-Hong Chen and Hong-Jian Lai presented the following open problem [2, problem 8.8]: Determine

$$L=min\underset{G\in SL-\{K_1\}}{max}\{\frac{|E(H)|}{|E(G)|}:H\text{ is spanning eulerian subgroup of G}\}.$$

For a graph $G$, $O(G)$ denotes the set of all odd-degree vertices of $G$. Let $G$ be a simple graph and $|O(G)|=2k$. In this note, we show that if $G\in SL$ and $k\le 2$, then $L\ge \frac{2}{3}$.