Toughness, Isolated Toughness and Path Factors in Graphs

Zhou, Sizhong; Wu, Jiancheng; Xu, Yan

doi:10.1017/s0004972721000952

Cited by 41 publications

(15 citation statements)

References 22 publications

(18 reference statements)

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“…Zhou [19], Zhou, Bian and Pan [20], Zhou, Sun and Liu [23], Gao and Wang [3], Zhou, Sun and Bian [22] discussed the existence of P ≥2 -factors and P ≥3 -factors with given properties in graphs. Zhou, Wu and Xu [25] posed two sufficient conditions for a graph to be a (P ≥3 , n)-factor-critical covered graph. Gao, Wang and Chen [4] showed a binding number condition for a graph to be a (P ≥3 , n)-factor-critical covered graph.…”

Section: Theorem 14 ( [15]mentioning

confidence: 99%

Isolated toughness for path factors in networks

Wang

Zhang

2022

RAIRO-Oper. Res.

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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.

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Section: Theorem 14 ( [15]mentioning

confidence: 99%

Isolated toughness for path factors in networks

Wang

Zhang

2022

RAIRO-Oper. Res.

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“…Kano, Lu and Yu [10] claimed that a graph with i(G − X) ≤ 2 3 |X| for any X ⊆ V (G) has a P ≥3 -factor. Zhou, Bian and Pan [20], Zhou, Wu and Bian [26], Zhou, Wu and Xu [28], Zhou [18] investigated the existence of P ≥3 -factors in graphs with given properties. Gao, Wang and Chen [4] obtained some results on the existence of P ≥3 -factors in graphs.…”

Section: Introductionmentioning

confidence: 99%

The existence of path-factor uniform graphs with large connectivity

Zhou

Bian

2022

RAIRO-Oper. Res.

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A path-factor is a spanning subgraph F of G such that every component of F is a path with at least two vertices. Let k ≥ 2 be an integer. A P ≥ k -factor of G means a path factor in which each component is a path with at least k vertices. A graph G is a P ≥ k -factor covered graph if for any e ∈ E ( G ), G has a P ≥ k -factor covering e . A graph G is called a P ≥ k -factor uniform graph if for any e 1 ,e 2 ∈ E ( G ) with e 1 6= e 2 , G has a P ≥ k -factor covering e 1 and avoiding e 2 . In other words, a graph G is called a P ≥ k -factor uniform graph if for any e ∈ E ( G ), G − e is a P ≥ k -factor covered graph. In this paper, we present two sufficient conditions for graphs to be P ≥3 -factor uniform graphs depending on binding number and degree conditions. Furthermore, we show that two results are best possible in some sense.

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“…Bazgan et al [4] put forward a toughness condition for a graph to have a P ≥3 -factor. Zhou, Bian and Pan [5], Zhou, Sun and Liu [6], Zhou, Wu and Bian [7], Zhou, Wu and Xu [8], Zhou [9,10] obtained some results on P ≥3 -factors in graphs with given properties. Johansson [11] presented a sufficient condition for a graph to have a path-factor.…”

Section: Introductionmentioning

confidence: 99%

Some sufficient conditions for path-factor uniform graphs

Zhou¹,

Sun²,

Liu³

2022

Preprint

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For a set H of connected graphs, a spanning subgraph H of G is called an H-factor of G if each component of H is isomorphic to an element of H. A graph G is called an H-factor uniform graph if for any two edges e1 and e2 of G, G has an H-factor covering e1 and excluding e2. Let each component in H be a path with at least d vertices, where d ≥ 2 is an integer. Then an H-factor and an H-factor uniform graph are called a P ≥d -factor and a P ≥d -factor uniform graph, respectively. In this article, we verify that (i) a 2-edge-connected graph G is a P ≥3 -factor uniform graph if δ(G) > α(G)+4 2 ; (ii) a (k + 2)-connected graph G of order n with n ≥ 5k + 3 − 3 5γ−1 is a P ≥3 -factor uniform graph if |NG(A)| > γ(n − 3k − 2) + k + 2 for any independent set A of G with |A| = ⌊γ(2k + 1)⌋, where k is a positive integer and γ is a real number with 1 3 ≤ γ ≤ 1.

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Toughness, Isolated Toughness and Path Factors in Graphs

Abstract: A graph G is called a $(P_{\geq n},k)$ -factor-critical covered graph if for any $Q\subseteq V(G)$ with $|Q|=k$ and any $e\in E(G-Q)$ , $G-Q$ has a $P_{\geq n}$ -factor covering e. We demonstrate that (i) a $(k+1)$ -connected graph G with at least $k+3$ vertices is a … Show more

Cited by 41 publications

References 22 publications

Isolated toughness for path factors in networks

Isolated toughness for path factors in networks

The existence of path-factor uniform graphs with large connectivity

Some sufficient conditions for path-factor uniform graphs

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