Graphs and Combinatorics

, Volume 35, Issue 1, pp 201–206 | Cite as

Spanning Eulerian Subgraphs of Large Size

  • Nastaran Haghparast
  • Dariush KianiEmail author
Original Paper


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.


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



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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Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Pure Mathematics, Faculty of Mathematics and Computer ScienceAmirkabir University of TechnologyTehranIran
  2. 2.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran

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