Some results about component factors in graphs

• Published in: R.A.I.R.O. Recherche opérationnelle Vol. 53; no. 3; pp. 723 - 730
• Main Author: Zhou, Sizhong
• Format: Journal Article
• Language: English
• Published: Paris EDP Sciences 01.07.2019
• Subjects: 05C38; 05C70; 90B10; Apexes; binding number; component factor; component factor covered graph; Decision trees; Graph; Graph theory; Graphs; Integers
• ISSN: 0399-0559; 1290-3868
• DOI: 10.1051/ro/2017045

For a set ℋ of connected graphs, a spanning subgraph H of a graph G is called an ℋ-factor of G if every component of H is isomorphic to a member ofℋ. An H-factor is also referred as a component factor. If each component of H is a star (resp. path), H is called a star (resp. path) factor. By a P≥ k-factor (k positive integer) we mean a path factor in which each component path has at least k vertices (i.e. it has length at least k − 1). A graph G is called a P≥ k-factor covered graph, if for each edge e of G, there is a P≥ k-factor covering e. In this paper, we prove that (1) a graph G has a {K1,1,K1,2, … ,K1,k}-factor if and only if bind(G) ≥ 1/k, where k ≥ 2 is an integer; (2) a connected graph G is a P≥ 2-factor covered graph if bind(G) > 2/3; (3) a connected graph G is a P≥ 3-factor covered graph if bind(G) ≥ 3/2. Furthermore, it is shown that the results in this paper are best possible in some sense.