Graphs and Combinatorics

, Volume 28, Issue 6, pp 901–905 | Cite as

A Note on Heterochromatic C 4 in Edge-Colored Triangle-Free Graphs

  • Guanghui Wang
  • Hao Li
  • Yan Zhu
  • Guizhen LiuEmail author
Original Paper


Let G be an edge-colored graph. A heterochromatic cycle of G is a cycle in which any pair of edges have distinct colors. Let d c (v), named the color degree of a vertex v, be defined as the maximum number of edges incident with v, that have distinct colors. In this paper, we prove that if G is an edge-colored triangle-free graph of order n ≥ 9 and \({d^c(v) \geq \frac{(3-\sqrt{5})n}{2}+1}\) for each vertex v of G, G has a heterochromatic C 4.


Heterochromatic cycle Color neighborhood Color degree 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albert, M., Frieze, A., Reed, B.: Multicolored Hamilton cycles. Electron. J. Comb. 2, R10 (1995) (research paper)Google Scholar
  2. 2.
    Bondy J.A., Murty U.S.R.: Graph Theory with Applications. Macmillan Press[M], New York (1976)zbMATHGoogle Scholar
  3. 3.
    Broersma H.J., Li X., Woeginger G., Zhang S.: Paths and cycles in colored graphs. Aust. J. Comb. 31, 297–309 (2005)MathSciNetGoogle Scholar
  4. 4.
    Brualdi R.A., Shen J.: Disjoint cycles in Eulerian digraphs and the diameter of interchange graphs. J. Comb. Theory Ser. B 85(2), 189–196 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Chen, H., Li, X.: Long heterochromatic paths in edge-colored graphs. Electron. J. Comb. 12(1), R33 (2005) (research paper)Google Scholar
  6. 6.
    Jørgensen L.K.: Girth 5 graphs from relative difference sets. Discret. Math. 293, 177–184 (2005)CrossRefGoogle Scholar
  7. 7.
    Kano M., Li X.: Monochromatic and heterochromatic subgraphs in edge-colored graphs—a survey. Graph Comb. 24(4), 237–263 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Li H., Li X., Liu G., Wang G.: The heterochromatic matchings in edge-colored bipartite graph. Ars comb. 93, 129–139 (2009)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Li, H., Wang, G: Color degree and heterochromatic cycles in edge colored graphs. Eur. J. Comb. (to appear)Google Scholar
  10. 10.
    Li H., Wang G.: Color degree and heterochromatic matchings in edge-colored bipartite graphs. Utilitas Math. 77, 145–154 (2008)zbMATHGoogle Scholar
  11. 11.
    Suzuki K.: A necessary and sufficient condition for the existence of a heterochromatic spanning tree in a graph. Graph Comb. 22, 261–269 (2006)zbMATHCrossRefGoogle Scholar
  12. 12.
    Wang, G., Li, H.: Heterochromatic matchings in edge-colored graphs. Electron. J. Comb. 15, #R138 (2008) (research paper)Google Scholar

Copyright information

© Springer 2011

Authors and Affiliations

  1. 1.School of MathematicsShandong UniversityJinanChina
  2. 2.Laboratoire de Recherche en InformatiqueUMR 8623, C.N.R.S.-Université de Paris-sudOrsay cedexFrance
  3. 3.Department of MathematicsEast China University of Science and TechnologyShanghaiChina

Personalised recommendations