Isolated toughness and path-factor uniform graphs

Abstract

A $P_{\geq k}$-factor of a graph $G$ is a spanning subgraph of $G$ whose components are paths of order at least $k$. We say that a graph $G$ is $P_{\geq k}$-factor covered if for every edge $e\in E(G)$, $G$ admits a $P_{\geq k}$-factor that contains $e$; and we say that a graph $G$ is $P_{\geq k}$-factor uniform if for every edge $e\in E(G)$, the graph $G-e$ is $P_{\geq k}$-factor covered. In other words, $G$ is $P_{\geq k}$-factor uniform if for every pair of edges $e_1,e_2\in E(G)$, $G$ admits a $P_{\geq k}$-factor that contains $e_1$ and avoids $e_2$. In this article, we testify that (\romannumeral1) a 3-edge-connected graph $G$ is $P_{\geq2}$-factor uniform if its isolated toughness $I(G)>1$; (\romannumeral2) a 3-edge-connected graph $G$ is $P_{\geq3}$-factor uniform if its isolated toughness $I(G)>2$. Furthermore, we explain that these conditions on isolated toughness and edge-connectivity in our main results are best possible in some sense.

The existence of path-factor uniform graphs with large connectivity

“…A 2-edge-connected graph G is a P ≥3 -factor uniform graph if its binding number bind(G) > 5 3 . Some other results on P ≥3 -factor uniform graphs can be found in Zhou, Sun and Liu [25], S. Zhou, Sun and Bian [24], Hua [5]. In this paper, we characterize P ≥3 -factor uniform graphs with respect to the vertex degree or the binding number, and obtain the following two results.…”

Toughness, Isolated Toughness and Path Factors in Graphs

“…The relationships between isolated toughness and graph factors were discussed by Yang et al [13], Zhou et al [18], Zhou et al [19], Gao et al [4] and Gao and Wang [5]. Zhou et al [21] gave an isolated toughness condition for the existence of P ≥3 -factor covered graphs.…”

Toughness and binding number bounds of star-like and path factor

“…A graph 𝐺 is named 𝑃 ≥𝑘 -factor uniform if for every pair of edges 𝑒 1 , 𝑒 2 ∈ 𝐸(𝐺), 𝐺 admits a 𝑃 ≥𝑘 -factor that contains 𝑒 1 and avoids 𝑒 2 . Recently, Zhou et al [18] proved that a graph 𝐺 is 𝑃 ≥2 -factor uniform when 𝐺 is 3-edge-connected graph and 𝐼(𝐺) > 1. Moreover, they also obtained that 𝐺 is 𝑃 ≥3 -factor uniform for 3-edge-connected graph 𝐺 and 𝐼(𝐺) > 2.…”