Even factors of large size (Q2922216)

An even factor of a graph is a spanning subgraph in which each vertex has a positive even degree. The main results of this paper are: (1) If \(G\) is a graph of order \(n\) and size \(m\) admitting a 2 factor, and having no even factor of size larger that \(n\), then \(m \leq 16n/9 -1\). (2) If \(G\) is a graph of size \(m\) admitting an even factor, then \(G\) contains an even factor of size at least \(9(m+1)/16\). It is open whether the factor \(9/16\) from the latter theorem is best possible, but an example is provided demonstrating that it cannot be larger than \(4/7\).