# Spanning Eulerian Subgraphs of Large Size

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## Abstract

A graph \(G=(V(G), E(G))\) is supereulerian if it has a spanning Eulerian subgraph. Let \(\ell (G)\) be the maximum number of edges of spanning Eulerian subgraphs of a graph *G*. Motivated by a conjecture due to Catlin on supereulerian graphs, it was shown that if *G* is an *r*-regular supereulerian graph, then \(\ell (G)\ge \frac{2}{3}|E(G)|\) when \(r\ne 5\), and \(\ell (G)> \frac{3}{5}|E(G)|\) when \(r=5\). In this paper we improve the coefficient and prove that if *G* is a 5-regular supereulerian graph, then \(\ell (G)\ge \frac{19}{30}|E(G)|+\frac{4}{3}\). For this, we first show that each graph *G* with maximum degree at most 3 has a matching with at least \(\frac{2}{7}|E(G)|\) edges and this bound is sharp. Moreover, we show that Catlin’s conjecture holds for claw-free graphs having no vertex of degree 4. Indeed, Catlin’s conjecture does not hold for claw-free graphs in general.

## Keywords

5-regular graph Supereulerian Eulerian subgraph Large size Claw-free## Notes

### Acknowledgements

The authors would like to thank the referees. The second author would also like to thank the institute for research in fundamental science (IPM). The research of the second author was in part supported by a grant from IPM (No. 96050212).

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