On maximum number of edges in a spanning eulerian subgraph

A graph is supereulerian if it has a spanning eulerian subgraph. It has been an open problem to determine the infimum L of the ratio of the maximum size of a spanning eulerian subgraph H of a graph G to the size of G, among all nontrivial supereulerian graphs. Catlin once thought that L could be 23. In this note, we present infinite families of graphs to show that L should be less than 23. Moreover, we show that when restricted to r-regular supereulerian graphs, if r≠5, then L⩾23 and if r=5, then L>35.