1. Sufficient conditions for the existence of path-factors with given properties
- Author
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Hui Qin, Guowei Dai, Yuan Chen, Ting Jin, and Yuan Yuan
- Subjects
path-factor ,sun toughness ,degree sum ,-factor-critical graph ,-factor-deleted graph ,05C38 ,Mathematics ,QA1-939 - Abstract
A spanning subgraph H of a graph G is called a [Formula: see text]-factor of G if every component of H is isomorphic to a path of order at least k, where [Formula: see text] is an integer. A graph G is called a [Formula: see text]-factor-critical graph if [Formula: see text] contains a [Formula: see text]-factor for any [Formula: see text] with [Formula: see text]. A graph G is called a [Formula: see text]-factor-deleted graph if [Formula: see text] has a [Formula: see text]-factor for any [Formula: see text] with [Formula: see text]. Intuitively, if a graph is dense enough, it will have a [Formula: see text]-factor. In this paper, we give some sufficient conditions for a graph to be a [Formula: see text]-factor-critical graph or a [Formula: see text]-factor-deleted graph. In this paper, we demonstrate that (i) G is a [Formula: see text]-factor-critical graph if its sun toughness [Formula: see text] and [Formula: see text]. (ii) G is a [Formula: see text]-factor-critical graph if its degree sum [Formula: see text] and [Formula: see text]. (iii) G is a [Formula: see text]-factor-deleted graph if its sun toughness [Formula: see text] and [Formula: see text]. (iv) G is a [Formula: see text]-factor-deleted graph if its degree sum [Formula: see text] and [Formula: see text].
- Published
- 2025
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