Long Alternating Cycles in Edge-Colored Complete Graphs

  • Hao Li
  • Guanghui Wang
  • Shan Zhou
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4613)


Let \(K_n^c\) denote a complete graph on n vertices whose edges are colored in an arbitrary way. And let \(\Delta (K_n^c)\) denote the maximum number of edges of the same color incident with a vertex of \(K_n^c\). A properly colored cycle (path) in \(K_n^c\), that is, a cycle (path) in which adjacent edges have distinct colors is called an alternating cycle (path). Our note is inspired by the following conjecture by B. bollobás and P. Erdős(1976): If \(\Delta(K_n^c)<\lfloor n/2\rfloor\), then \(K_n^c\) contains an alternating Hamiltonian cycle. We prove that if \(\Delta (K_n^c)< \lfloor n/2\rfloor\), then \(K_n^c\) contains an alternating cycle with length at least \(\lceil \frac{n+2}{3}\rceil+1\).


alternating cycle color degree edge-colored graph 


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  1. 1.
    Alon, N., Gutin, G.: Properly colored Hamiltonian cycles in edge colored complete graphs. Random Structures and Algorithms 11, 179–186 (1997)zbMATHCrossRefGoogle Scholar
  2. 2.
    Bang-Jensen, J., Gutin, G.: Alternating cycles and paths in edge-colored multigraphs: a survey. Discrete Math., pp. 165–166, pp. 39–60 (1997)Google Scholar
  3. 3.
    Bang-Jensen, J., Gutin, G., Yeo, A.: Properly coloured Hamiltonian paths in edge-colored complete graphs. Discrete Appl. Math. 82, 247–250 (1998)zbMATHCrossRefGoogle Scholar
  4. 4.
    Bollobás, B., Erdös, P.: Alternating Hamiltonian cycles, Israel. J. Math. 23, 126–131 (1976)zbMATHGoogle Scholar
  5. 5.
    Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. Macmillan Press, New York (1976)Google Scholar
  6. 6.
    Caccetta, L., Häggkvist, R.: On minimal digraphs with given girth. In: Proceedings, Ninth S-E Conference on Combinatorics, Graph Theory and Computing, pp. 181–187 (1978)Google Scholar
  7. 7.
    Chen, C.C., Daykin, D.E.: Graphs with Hamiltonian cycles having adjacent lines different colors. J. Combin. Theory Ser. B21, 135–139 (1976)CrossRefGoogle Scholar
  8. 8.
    Chow, W.S., Manoussakis, Y., Megalakaki, O., Spyratos, M., Tuza, Z.: Paths through fixed vertices in edge-colored graphs. J. des Mathematqiues, Informatique et Science, Humaines 32, 49–58 (1994)Google Scholar
  9. 9.
    Dorninger, D.: On permutations of chromosomes. In: Contributions to General Algebra, vol. 5, Verlag Hölder-Pichler-Tempsky, Wien, Teubner, Stuttgart, pp. 95–103 (1987)Google Scholar
  10. 10.
    Dorninger, D.: Hamiltonian circuits determing the order of chromosomes. Discrete Appl. Math. 50, 159–168 (1994)zbMATHCrossRefGoogle Scholar
  11. 11.
    Dorninger, D., Timischl, W.: Geometrical constraints on Bennett’s predictions of chromosome order. Heredity 58, 321–325 (1987)Google Scholar
  12. 12.
    Grossman, J.W., Häggkvist, R.: Alternating cycles in edge-partitioned graphs. J. Combin. Theory Ser. B 34, 77–81 (1983)CrossRefGoogle Scholar
  13. 13.
    Häggkvist, R.: A talk at Intern. Colloquium on Combinatorics and Graph Theory at Balatonlelle, Hungary (July 15-19, 1996)Google Scholar
  14. 14.
    Hahn, G., Thomassen, C.: Path and cycle sub-Ramsey numbers and edge-coloring conjecture. Discrete Math. 62(1), 29–33 (1986)zbMATHCrossRefGoogle Scholar
  15. 15.
    Shearer, J.: A property of the colored complete graph. Discrete Math. 25, 175–178 (1979)zbMATHCrossRefGoogle Scholar
  16. 16.
    Li, H., Wang, G.: Color degree and alternating cycles in edge-colored graphs, RR L.R.I NO1461(2006)Google Scholar
  17. 17.
    Yeo, A.: Alternating cycles in edge-coloured graphs. J. Combin. Theory Ser. B 69, 222–225 (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Hao Li
    • 1
    • 3
  • Guanghui Wang
    • 1
    • 2
  • Shan Zhou
    • 1
    • 3
  1. 1.Laboratoire de Recherche en Informatique, UMR 8623, C.N.R.S-Université de Paris-sud, 91405-Orsay cedexFrance
  2. 2.School of Mathematics and System Science, Shandong University, 250100 Jinan, ShandongChina
  3. 3.School of Mathematics and Statistics, Lanzhou University, 730000 LanzhouChina

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