Graphs and Combinatorics

, Volume 31, Issue 6, pp 2087–2102 | Cite as

Fan-Type Conditions for Spanning Eulerian Subgraphs

  • Wei-Guo Chen
  • Zhi-Hong ChenEmail author
Original Paper


For a graph \(G\), let \(\delta _F(G)=\min \{\max \{d(u), d(v)\} | \text{ for } \text{ any }~u, v\in V(G)\, \text{ with } \text{ distance }~2\}\). A graph is supereulerian if it has a spanning Eulerian subgraph. Let \(p>0\), \(g>2\) and \(\epsilon \) be given nonnegative numbers. Let \(\mathcal{Q}\) be the family of non-supereulerian graphs with order at most \(5(p-2)\). In this paper, we prove that for a 3-edge-connected graph \(G\) of order \(n\), if \(G\) satisfies a Fan-type condition \(\delta _F(G)\ge \frac{n}{(g-2)p}-\epsilon \) and \(n\) is sufficiently large, then \(G\) is supereulerian if and only if \(G\) is not contractible to a graph in \(\mathcal{Q}\). Results on best possible values of \(p\) and \(\epsilon \) for such graphs to either be supereulerian or be contractible to the Petersen graph are given.


Spanning Eulerian subgraphs Reduction method  Fan-Type condition 



The authors would like to thank the referees for their comments which help to improve the presentation of the paper.


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Copyright information

© Springer Japan 2015

Authors and Affiliations

  1. 1.Guangdong Economic Information CenterGuangzhouPeople’s Republic of China
  2. 2.Butler UniversityIndianapolisUSA

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