Isolated toughness for path factors in networks

Abstract

Let $\mathcal{H}$ be a set of connected graphs. Then an $\mathcal{H}$-factor is a spanning subgraph of $G$, whose every connected component is isomorphic to a member of the set $\mathcal{H}$. An $\mathcal{H}$-factor is called a path factor if every member of the set $\mathcal{H}$ is a path. Let $k\geq2$ be an integer. By a $P_{\geq k}$-factor we mean a path factor in which each component path admits at least $k$ vertices. A graph $G$ is called a $(P_{\geq k},n)$-factor-critical covered graph if for any $W\subseteq V(G)$ with $|W|=n$ and any $e\in E(G-W)$, $G-W$ has a $P_{\geq k}$-factor covering $e$. In this article, we verify that (\romannumeral1) an $(n+\lambda+2)$-connected graph $G$ is a $(P_{\geq2},n)$-factor-critical covered graph if its isolated toughness $I(G)>\frac{n+\lambda+2}{2\lambda+3}$, where $n$ and $\lambda$ are two nonnegative integers; (\romannumeral2) an $(n+\lambda+2)$-connected graph $G$ is a $(P_{\geq3},n)$-factor-critical covered graph if its isolated toughness $I(G)>\frac{n+3\lambda+5}{2\lambda+3}$, where $n$ and $\lambda$ be two nonnegative integers.

Path-factor critical covered graphs and path-factor uniform graphs

“…Zhou, Wu and Xu [31] demonstrated two results on the existence of (P ≥3 , n)-factor critical covered graphs depending on toughness and isolated toughness. Wang and Zhang [13] improved the previous isolated toughness condition to guarantee the existence of (P ≥3 , n)-factor critical covered graphs. The following results on (P ≥d , n)-factor critical covered graphs are known.…”

“…An (n + r + 1)-connected graph G is a (P ≥3 , n)-factor critical covered graph if its sun toughness s(G) > n+r+1 2r+1 , where n ≥ 0 and r ≥ 1 are integers. Theorem 1.4 ( [13]). Let n and λ be two nonnegative integers.…”

The existence of path-factor uniform graphs with large connectivity

“…Gao, Wang and Chen [4] obtained some results on the existence of P ≥3 -factors in graphs. Wang and Zhang [11] posed an isolated toughness condition for graphs to have P ≥3 -factors.…”

Sharp conditions on fractional ID-(g, f)-factor-critical covered graphs

“…Shiu and Liu [13] presented some sufficient conditions for a graph admitting k-factors in regular graphs. Wang and Zhang [14], Liu [11], Zhou [20], Zhou et al [26,23,22,27,29] characterized a graph with a [1, 2]-factor, and put forword some sufficient conditions for graphs to have [1,2]factors. Matsuda [12] derived some results in terms of the neighborhood condition for graphs to admit [a, b]-factors.…”