Subgraphs with orthogonal factorizations in graphs

Zhou, Sizhong; Zhang, Tao; Xu, Zurun

doi:10.1016/j.dam.2019.12.011

Cited by 24 publications

(4 citation statements)

References 24 publications

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“…Zhou, Wu and Bian [24] obtained a result on the existence of (P ≥3 , n)-factor-critical covered graphs. Some recent advances on graph factors can be found in Zhou, Liu and Xu [21], Zhou [16][17][18], Zhou, Zhang and Xu [26], Wang and Zhang [10][11][12], Yuan and Hao [14], Anstee and Nam [1], Egawa, Kano and Yokota [2]. In this article, we study the existence of (P ≥2 , n)-factor-critical covered graphs and (P ≥3 , n)-factor-critical covered graphs, and derive the following two results.…”

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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“…Some results on the relationships between isolated toughness and graph factors are obtained by Gao, Liang and Chen [6]. For many other results on graph factors, see [1,12,14,16,19,21,23]. In this paper, we study (P ≥k , n)-factor critical avoidable graphs and get some sufficient conditions for graphs to be (P ≥k , n)factor critical avoidable graphs depending on toughness and isolated toughness, which are given in Sections 2 and 3.…”

Section: Introductionmentioning

confidence: 95%

Some results on path-factor critical avoidable graphs

Zhou¹

2020

Discuss. Math. Graph Theory

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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. We write P ≥k = {P i : i ≥ k}. Then a P ≥k-factor of G means a path factor in which every component admits at least k vertices, where k ≥ 2 is an integer. A graph G is called a P ≥k-factor avoidable graph if for any e ∈ E(G), G admits a P ≥k-factor excluding e. A graph G is called a (P ≥k , n)-factor critical avoidable graph if for any Q ⊆ V (G) with |Q| = n, G − Q is a P ≥k-factor avoidable graph. Let G be an (n + 2)-connected graph. In this paper, we demonstrate that (i) G is a (P ≥2 , n)-factor critical avoidable graph if tough(G) > n+2 4 ; (ii) G is a (P ≥3 , n)-factor critical avoidable graph if tough(G) > n+1 2 ; (iii) G is a (P ≥2 , n)-factor critical avoidable graph if I(G) > n+2 3 ; (iv) G is a (P ≥3 , n)factor critical avoidable graph if I(G) > n+3 2. Furthermore, we claim that these conditions are sharp.

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“…Zhou et al [28] showed two existence theorems for graphs to admit component factors. Zhou et al [27], Sun and Zhou [12] and Wang and Zhang [15] investigated the existence of edge-disjoint factors in graphs and gained some results for graphs to possess edge-disjoint factors. Other results on graph factors can be found in [14,21,23,25,26,29].…”

Section: Introductionmentioning

confidence: 99%

Isolated toughness and path-factor uniform graphs

2021

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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.

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Subgraphs with orthogonal factorizations in graphs

Cited by 24 publications

References 24 publications

Isolated toughness for path factors in networks

Isolated toughness for path factors in networks

Some results on path-factor critical avoidable graphs

Isolated toughness and path-factor uniform graphs

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