Remarks on path factors in graphs

• Published in: R.A.I.R.O. Recherche opérationnelle Vol. 54; no. 6; pp. 1827 - 1834
• Main Author: Zhou, Sizhong
• Format: Journal Article
• Language: English
• Published: Paris EDP Sciences 01.11.2020
• Subjects: Apexes; Graph theory
• ISSN: 0399-0559; 1290-3868
• DOI: 10.1051/ro/2019111

A spanning subgraph of a graph is defined as a path factor of the graph if its component are paths. A P ≥ n -factor means a path factor with each component having at least n vertices. A graph G is defined as a ( P ≥ n , m )-factor deleted graph if G – E ′ has a P ≥ n -factor for every E ′ ⊆  E ( G ) with | E ′| =  m . A graph G is defined as a ( P ≥ n , k )-factor critical graph if after deleting any k vertices of G the remaining graph of G admits a P ≥ n -factor. In this paper, we demonstrate that (i) a graph G is ( P ≥3 , m )-factor deleted if κ ( G ) ≥ 2 m  + 1 and bind ( G ) ≥  2/3 - $ \frac{3}{2}-\frac{1}{4m+4}$; (ii) a graph G is ( P ≥3 , k )-factor critical if κ ( G ) ≥  k  + 2 and bind ( G ) ≥ $ \frac{5+k}{4}$.