Even factors of graphs

Abstract

An even factor of a graph is a spanning subgraph in which each vertex has a positive even degree. Favaron and Kouider (J Gr Theory 77:58–67, 2014) showed that if a simple graph G has an even factor, then it has an even factor F with $|E(F)| \ge \frac{7}{16} (|E(G)| + 1)$. This ratio was improved to $\frac{4}{7}$ recently by  Chen and Fan (J Comb Theory Ser B 119:237–244, 2016), which is the best possible. In this paper, we take the set of vertices of degree 2 (say $V_{2}(G)$) into consideration and further strengthen this lower bound. Our main result is to show that for any simple graph G having an even factor, G has an even factor F with $|E(F)| \ge \frac{4}{7} (|E(G)| + 1)+\frac{1}{7}|V_{2}(G)|$.