# Spanning Eulerian Subgraphs of Large Size

- 63 Downloads

## 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).

## References

- 1.Biedl, T., Demaine, E.D., Duncan, C.A., Fleischer, R., Kobourov, S.G.: Tight bounds on maximal and maximum matchings. Discret. Math.
**285**(1), 7–15 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. American Elsevier, New York (1976)CrossRefzbMATHGoogle Scholar
- 3.Cheng, J., Zhang, C.-Q., Zhu, B.-X.: Even factors of graphs. J. Combin. Optim.
**33**, 1343–1353 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Lai, H.-J.: Lecture notes on supereulerian graphs and related topics, unpublished notes (1996)Google Scholar
- 5.Lai, H.-J., Chen, Z.-H.: Even subgraphs of a graph, Combinatorics, Graph Theory and Algorithms, New Issues Press, Kalamazoo, pp. 221–226 (1999)Google Scholar
- 6.Li, D., Li, D., Mao, J.: On maximum number of edges in a spanning eulerian subgraph. Discret. Math.
**274**(1), 299–302 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Pulleyblank, W.R.: A note on graphs spanned by eulerian graphs. J. Graph Theory
**3**(3), 309–310 (1976)MathSciNetCrossRefzbMATHGoogle Scholar