Tight toughness bounds for path-factor critical avoidable graphs

• Published in: AKCE international journal of graphs and combinatorics Vol. 21; no. 2; pp. 167 - 170
• Main Authors: Wang, Wenqi, Dai, Guowei
• Format: Journal Article
• Language: English
• Published: Taylor & Francis Group 03.05.2024
• Subjects: 05C38; factor critical avoidable graph; Graph; isolated toughness; path-factor; toughness
• ISSN: 0972-8600; 2543-3474
• DOI: 10.1080/09728600.2024.2314512

Given a graph G and an integer [Formula: see text], a spanning subgraph H of G is called a [Formula: see text]-factor of G if every component of H is a path with at least k vertices. A graph G is [Formula: see text]-factor avoidable if for every edge [Formula: see text], G has a [Formula: see text]-factor excluding e. A graph G is said to be [Formula: see text]-factor critical avoidable if the graph [Formula: see text] is [Formula: see text]-factor avoidable for any [Formula: see text] with [Formula: see text]. Here we study the sharp bounds of toughness and isolated toughness conditions for the existence of [Formula: see text]-factor critical avoidable graphs. In view of graph theory approaches, this paper mainly contributes to verify that (i) An [Formula: see text]-connected graph is [Formula: see text]-factor critical avoidable if its toughness [Formula: see text]; (ii) An [Formula: see text]-connected graph is [Formula: see text]-factor critical avoidable if its isolated toughness [Formula: see text].