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1 Fundamenta Informaticae

1 2024

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Two sufficient conditions for graphs to admit path factors

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Zhou S. , Wu J.

Fundamenta Informaticae

2024

tom Vol. 191, nr 1

67--77

Let A be a set of connected graphs. Then a spanning subgraph A of G is called an A-factor if each component of A is isomorphic to some member of A. Especially, when every graph in A is a path, A is a path factor. For a positive integer d ≥ 2, we write P≥d = {Pi|i ≥ d}. Then a P≥d-factor means a path factor in which every component admits at least d vertices. A graph G is called a (P≥d, m)-factor deleted graph if G − E′ admits a P≥d-factor for any E′ ⊆ E(G) with |E′| = m. A graph G is called a (P≥d, k)-factor critical graph if G − Q has a P≥d-factor for any Q ⊆ V (G) with |Q| = k. In this paper, we present two degree conditions for graphs to be (P≥3, m)-factor deleted graphs and (P≥3, k)-factor critical graphs. Furthermore, we show that the two results are best possible in some sense.