Abstract
A graph is called claw-free if it contains no induced subgraph isomorphic to K 1,3. Matthews and Sumner proved that a 2-connected claw-free graph G is Hamiltonian if every vertex of it has degree at least (|V(G)| − 2)/3. At the workshop Camp;C (Novy Smokovec, 1993), Broersma conjectured the degree condition of this result can be restricted only to end-vertices of induced copies of N (the graph obtained from a triangle by adding three disjoint pendant edges). Fujisawa and Yamashita showed that the degree condition of Matthews and Sumner can be restricted only to end-vertices of induced copies of Z 1 (the graph obtained from a triangle by adding one pendant edge). Our main result in this paper is a characterization of all graphs H such that a 2-connected claw-free graph G is Hamiltonian if each end-vertex of every induced copy of H in G has degree at least |V(G)|/3+1. This gives an affirmative solution of the conjecture of Broersma up to an additive constant.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bedrossian, P.: Forbidden subgraph and minimum degree conditons for hamiltonicity, Ph.D. Thesis, Memphis State University, USA, 1991
Bondy, J. A., Murty, U. S. R.: Graph Theory with Applications, Macmillan, London and Elsevier, New York, 1976
Broersma, H. J.: Problem 2. In: Workshop Cycles and Colourings (Novy Smokovec, Slovakia, 1993), available on http://umv.science.upjs.sk/c&c/history/93problems.pdf
Brousek, J.: Minimal 2-connected non-Hamiltonian claw-free graphs. Discrete Math., 191, 57–64 (1998)
Brousek, J., Ryjáček, Z., Favaron, O.: Forbidden subgraphs, hamiltonicity and closure in claw-free graphs. Discrete Math., 196, 29–50 (1999)
Dirac, G. A.: Some theorems on abstract graphs. Proc. Lond. Math. Soc., 2, 69–81 (1952)
Fujisawa, J., Yamashita, T.: Degree conditions on claws and modified claws for hamiltonicity of graphs. Discrete Math., 308, 1612–1619 (2008)
Matthews, M., Sumner, D.: Longest paths and cycles in K 1,3-free graphs. J. Graph Theory, 9, 269–277 (1985)
Ryjáček, Z.: On a closure concept in claw-free graphs. J. Combin. Theory Ser. B, 70, 217–224 (1997)
Thomassen, C.: Reflections on graph theory. J. Graph Theory, 10, 309–324 (1986)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by NSFC (Grant Nos. 11271300 and 11571135), the project NEXLIZ–CZ.1.07/2.3.00/30.0038, the project P202/12/G061 of the Czech Science Foundation and by the European Regional Development Fund (ERDF), and the project NTIS - New Technologies for Information Society, European Centre of Excellence, CZ.1.05/1.1.00/02.0090
Rights and permissions
About this article
Cite this article
Čada, R., Li, B.L., Ning, B. et al. Induced subgraphs with large degrees at end-vertices for hamiltonicity of claw-free graphs. Acta. Math. Sin.-English Ser. 32, 845–855 (2016). https://doi.org/10.1007/s10114-016-4686-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-016-4686-1