Advertisement

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

Abstract

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.

Keywords

Spanning Eulerian subgraphs Reduction method  Fan-Type condition 

Notes

Acknowledgments

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

References

  1. 1.
    Bondy, J.A., Murty, U.S.R.: Graph theory with applications. American Elsevier, New York (1976)CrossRefzbMATHGoogle Scholar
  2. 2.
    Catlin, P.A.: A reduction method to find spanning Eulerian subgraphs. J. Graph Theory 12, 29–45 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Catlin, P.A.: Supereulerian graphs, collapsible graphs, and four-cycles. Congr. Numerantium 58, 233–246 (1987)MathSciNetGoogle Scholar
  4. 4.
    Catlin, P.A.: Contractions of graphs with non spanning Eulerian subgraphs. Combinatorica 8(4), 313–321 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Catlin, P.A., Han, Z., Lai, H.-J.: Graphs without spanning eulerian trails. Discrete Math. 160, 81–91 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, Z.-H.: A degree condition for spanning Eulerian subgraphs. J. Graph Theory 17, 5–21 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, Z.-H.: Supereulerian graphs, independent sets and degree-sum conditions. Discrete Math. 179, 73–87 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, Z.-H.: Fan-type conditions for collapsible graphs. Ars Combinatoria 50, 81–95 (1998)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chen, Z.-H.: Supereulerian graphs and the Petersen graph. J. Comb. Math. Comb. Comput. 9, 79–89 (1991)zbMATHGoogle Scholar
  10. 10.
    Chen, W.-G., Chen, Z.-H.: Spanning Eulerian subgraphs and Catlin’s reduced graphs. J. Comb. Math. Comb. Comput. (accepted).Google Scholar
  11. 11.
    Chen, W.-G., Chen, Z.-H.: Lai’s degree conditions for spanning and dominating closed trails (submitted).Google Scholar
  12. 12.
    Chen, W.-G., Chen, Z.-H., Lu, M.: Properties of Catlin’s reduced graphs and supereulerian graphs (submitted).Google Scholar
  13. 13.
    Chen, Z.-H., Lai, H.-J.: Collapsible graphs and matchings. J. Graph Theory 17(5), 597–605 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fan, G.H.: New sufficient conditions for cycles in graphs. J. Comb. Theory Ser. B. 37(3), 221–227 (1984)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Lai, H.-J.: Eulerian subgraphs containing given vertices and Hamiltonian line graphs. Discrete Math. 178, 93–107 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lesniak-Foster, L., Williamson, J.E.: On spanning and dominating circuits in graphs. Can. Math. Bull. 20, 215–220 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Li, X., Lei, L., Lai, H.J., Zhang, M.: Supereulerian graphs and the Petersen graphs. Acta Mathematica Sinica (English Series) 30(2), 291304 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pulleyblank, W.R.: A note on graphs spanned by eulerian graphs. J. Graph Theory 3, 309–310 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Veldman, H.J.: On dominating and spanning circuits in graphs. Discrete Math. 124, 229–239 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Japan 2015

Authors and Affiliations

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

Personalised recommendations