Sufficient conditions for the existence of a path‐factor which are related to odd components

Abstract

In this article, we are concerned with sufficient conditions for the existence of a {P2,P2k+1}‐factor. We prove that for k≥3, there exists εk>0 such that if a graph G satisfies 0true∑0≤j≤k−1c2j+1(G−X)≤εk|X| for all X⊆V(G), then G has a {P2,P2k+1}‐factor, where cifalse(G−Xfalse) is the number of components C of G−X with |V(C)|=i. On the other hand, we construct infinitely many graphs G having no {P2,P2k+1}‐factor such that 0true∑0≤j≤k−1c2j+1(G−X)≤32k+14172k−78|X| for all X⊆V(G).

The existence of path-factor covered graphs

“…Since Tutte proposed the well known Tutte 1-factor theorem [15], there are many results on graph factors [2,3,8,9,16] and P ≥k -factors in claw-free graphs and cubic graphs [4,12,13]. More results on graph factors can be found in the survey papers and books in [2,14,18].…”

Component factors in $K_{1,r}$-free graphs

“…Since Tutte proposed the well known Tutte 1-factor theorem [18], there are many results on graph factors [2,3,8,9,19], S n -factors [7,13,14,22], and P ≥k -factors in claw-free graphs and cubic graphs [4,12,15]. More results on graph factors can be found in the survey papers and books in [2,17,24].…”

Sufficient conditions for graphs with {P2, P5}-factors

“…In particular, the existence problem of a {P 2 , P 2k+1 }-factor is NP-complete for k ≥ 2. As {P 2 , P 2k+1 }-factor is a useful tool for finding large matchings, Egawa, Furuya and Ozeki [12] investigated the existence of {P 2 , P 2k+1 }-factors and obtained the following theorem.…”