Abstract
It is shown by Luo and Zhao (J Graph Theory 73:469–482, 2013) that an overfull \(\Delta \)-critical graph with n vertices that satisfies \(\Delta \ge \frac{n}{2}\) is Hamiltonian. If Hilton’s overfull subgraph conjecture (Chetwynd and Hilton 100:303–317, 1986) was proved to be true, then the above result could be said that any \(\Delta \)-critical graph with n vertices that satisfies \(\Delta \ge \frac{n}{2}\) is Hamiltonian. Since the overfull subgraph conjecture is still open, the natural question is how to directly prove a \(\Delta \)-critical graph with n vertices that satisfies \(\Delta \ge \frac{n}{2}\) is Hamiltonian. Luo and Zhao (J Graph Theory 73:469–482, 2013) show that a \(\Delta \)-critical graph with n vertices that satisfies \(\Delta \ge \frac{6n}{7}\) is Hamiltonian. In this paper, by developing new lemmas for critical graphs, we show that if G is a \(\Delta \)-critical graph with n vertices satisfying \(\Delta \ge \frac{4n}{5}\), then G is Hamiltonian.
Similar content being viewed by others
References
Beineke, L.W., Fiorini, S.: On small graphs critical with respect to edge-colourings. Discret. Math. 16, 109–121 (1976)
Bokal, D., Brinkmann, G., Grünewald, S.: Chromatic-index-critical graphs of orders 13 and 14. Discret. Math. 300, 16–29 (2005)
Bondy, J.A.: Large cycles in graphs. Discret. Math. 1, 121–132 (1971)
Bondy, J.A., Chvátal, V.: A method in graph theory. Discret. Math. 15, 111–135 (1976)
Brinkmann, G., Steffen, E.: 3- and 4- critical graphs of small even order. Discret. Math. 169, 193–197 (1997)
Brinkmann, G., Steffen, E.: Chromatic-index-critical graphs of orders 11 and 12. Eur. J. Comb. 19, 889–900 (1998)
Chetwynd, A.G., Yap, H.P.: Chromatic index critical graphs of order 9. Discret. Math. 47, 23–33 (1983)
Chetwynd, A.G., Hilton, A.J.W.: Star multigraphs with three vertices of maximum degree. Math. Proc. Camb. Philos. Soc. 100, 303–317 (1986)
Chvátal, V.: On Hamilton’s ideals. J. Comb. Theory Ser. B 12, 163–168 (1972)
Dirac, G.A.: Some theorems on abstract graphs. Proc. Lond. Math. Soc. 2, 69–81 (1952)
Fiorini, S.: On the chromatic index of a graph, III: uniquely edge-colourable graphs. Quart. J. Math. Oxf. 26, 129–140 (1975)
Fiorini, S.: Some remarks on a paper by Vizing on critical graphs. Math. Proc. Camb. Phil. Soc. 77, 475–483 (1975)
Fiorini, S., Wilson, R.J.: Edge colorings of graphs. Pitman, San Francisco (1977)
Grünewald, S., Steffen, E.: Independent sets and 2-factors in edge-chromatic-critical graphs. J. Graph Theory 45, 113–118 (2004)
Jensen, T.R., Toft, B.: Graph Coloring Problems. Wiley, New York (1995)
Luo, R., Zhao, Y.: Finding \(\Delta (\Sigma )\) for the surface \(\Sigma \) of characteristic \(\chi (\Sigma ) =-5\). J. Graph Theory 68, 148–168 (2011)
Luo, R., Zhao, Y.: A sufficient condition for edge chromatic critical graphs to be Hamiltonian-An approach to Vizing’s 2-Factor Conjecture. J. Graph Theory 73, 469–482 (2013)
Sanders, D., Zhao, Y.: Planar graphs of maximum degree seven are class I. J. Comb. Theory Ser. B 83, 201–212 (2001)
Steibitz, M., Scheide, D., Toft, B., Favrholdt, L.: Graph Edge Coloring: Vizing’s theorem and Goldberg’s Conjecture. Wiley, New York (2012)
Vizing, V.G.: Critical graphs with a given chromatic index (in Russian). Diskret. Analiz. 5, 9–17 (1965)
Vizing, V.G.: The chromatic class of a multigraph (in Russian), Kibernetika 3, 29-39 (1965). English translation in Cybernetics and System Analysis 1 32-41
Vizing, V.G.: Some unsolved problems in graph theory (in Russian), Uspekhi Mat.Nauk.23(6), 117–134 (1968); or (in English) Russian Math. Surveys 23 (6) (1968) 125–141
Author information
Authors and Affiliations
Corresponding author
Additional information
Zhengke Miao: Research was partially supported by NSF-China grant: NSFC 11171288 and 11571149.
Rights and permissions
About this article
Cite this article
Luo, R., Miao, Z. & Zhao, Y. Hamiltonian Cycles in Critical Graphs with Large Maximum Degree. Graphs and Combinatorics 32, 2019–2028 (2016). https://doi.org/10.1007/s00373-016-1698-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-016-1698-7