Abstract
A k-tree of a graph is a spanning tree with maximum degree at most k. We give sufficient conditions for a graph G to have a k-tree with specified leaves: Let k,s, and n be integers such that k≥2, 0≤s≤k, and n≥s+1. Suppose that (1) G is (s+1)-connected and the degree sum of any k independent vertices of G is at least |G|+(k−1)s−1, or (2) G is n-connected and the independence number of G is at most (n−s)(k−1)+1. Then for any s specified vertices of G, G has a k-tree containing them as leaves. We also discuss the sharpness of the results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bondy, J.A., Murty, U.S.R.: Graph theory with applications, North Holland, New York, 1981
Chvátal, V., Erdös, P.: A note on Hamilton circuits. Discrete Math. 2, 111–113 (1972)
Hall, P.: On representatives of subsets. J London Math. Soc. 10, 26–30 (1935)
Neumann-Lara, V., Rivera-Campo, E.: Spanning trees with bounded degrees. Combinatorica 11, 55–61 (1991)
Ore, O.: Notes on Hamilton circuits. Amer. Math. Monthly 67, 55 (1960)
Ore, O.: Hamilton connected graphs. J. Math. Pures Appl. 42, 21–27 (1963)
Win, S.: Existenz von Gerüsten mit vorgeschriebenem Maximalgrad in Graphen. Abh Math Seminar Univ Hamburg 43, 263–267 (1975)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists, 15740077, 2005
This research was partially supported by the Japan Society for the Promotion of Science for Young Scientists.
Rights and permissions
About this article
Cite this article
Matsuda, H., Matsumura, H. On a k-Tree Containing Specified Leaves in a Graph. Graphs and Combinatorics 22, 371–381 (2006). https://doi.org/10.1007/s00373-006-0660-5
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s00373-006-0660-5