The existence of path-factor covered graphs

Dai, Guowei

doi:10.7151/dmgt.2353

Cited by 14 publications

(7 citation statements)

References 17 publications

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“…The first author of this paper has also verified that a claw-free graph G is a P ≥3 -factor covered graph if δ(G) ≥ 3 [6]. We extend the above result and obtain a minimum degree condition for a K 1,r -free graph being a P ≥3 -factor covered graph.…”

Section: Remarksupporting

confidence: 62%

Component factors in $K_{1,r}$-free graphs

Dai¹,

Zhang²,

Zhang³

2020

Preprint

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Section: Remarksupporting

confidence: 62%

Component factors in $K_{1,r}$-free graphs

Dai¹,

Zhang²,

Zhang³

2020

Preprint

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“…Since the well-known Tutte 1-factor theorem [10] was proposed, there are many results about component-factors, see [5,9,13,14], etc.…”

Section: Introductionmentioning

confidence: 99%

Remarks on component factors in graphs

Dai

2022

RAIRO-Oper. Res.

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For a family of connected graphs $\mathcal{F}$, a spanning subgraph $H$ of a graph $G$ is called an $\mathcal{F}$-factor of $G$ if its each component is isomorphic to an element of $\mathcal{F}$. In particular, $H$ is called an $\mathcal{S}_k$-factor of $G$ if $\mathcal{F}=\{K_{1,1},K_{1,2},...,K_{1,k}\}$, where integer $k\geq2$; $H$ is called a $P_{\geq 3}$-factor of $G$ if every component in $\mathcal{F}$ is a path of order at least three. As an extension of $\mathcal{S}_k$-factors, the induced star-factor (i.e., $\mathcal{IS}_k$-factor) is a spanning subgraph each component of which is an induced subgraph isomorphic to some graph in $\mathcal{F}=\{K_{1,1},K_{1,2},...,K_{1,k}\}$. In this paper, we firstly prove that a graph $G$ has an $\mathcal{S}_k$-factor if and only if its isolated toughness $I(G)\geq \frac{1}{k}$. Secondly, we prove that a planar graphs $G$ has an $\mathcal{S}_{2}$-factors if its minimum degree $\delta(G)\geq3$. Thirdly, we give two sufficient conditions for graphs with $\mathcal{IS}_k$-factors by toughness and minimum degree, respectively. Additionally, we obtain three special classes of graphs admitting $P_{\geq3}$-factors.

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“…Recently, Zhou [21] and Dai [7] obtained some classes of P ≥2 -factor covered graphs, respectively. Theorem 1.5.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1.6. (Dai [7]) Let G be a connected graph of order at least two. Then G is a P ≥2 -factor covered graph if one the following holds:…”

Section: Introductionmentioning

confidence: 99%

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Some degree conditions for 𝒫_≥k-factor covered graphs

et al. 2021

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A spanning subgraph of a graph $G$ is called a path-factor of $G$ if its each component is a path. A path-factor is called a $\mathcal{P}_{\geq k}$-factor of $G$ if its each component admits at least $k$ vertices, where $k\geq2$. Zhang and Zhou [\emph{Discrete Mathematics}, \textbf{309}, 2067-2076 (2009)] defined the concept of $\mathcal{P}_{\geq k}$-factor covered graphs, i.e., $G$ is called a $\mathcal{P}_{\geq k}$-factor covered graph if it has a $\mathcal{P}_{\geq k}$-factor covering $e$ for any $e\in E(G)$. In this paper, we firstly obtain a minimum degree condition for a planar graph being a $\mathcal{P}_{\geq 2}$-factor and $\mathcal{P}_{\geq 3}$-factor covered graph, respectively. Secondly, we investigate the relationship between the maximum degree of any pairs of non-adjacent vertices and $\mathcal{P}_{\geq k}$-factor covered graphs, and obtain a sufficient condition for the existence of $\mathcal{P}_{\geq2}$-factor and $\mathcal{P}_{\geq 3}$-factor covered graphs, respectively.

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The existence of path-factor covered graphs

Cited by 14 publications

References 17 publications

Component factors in $K_{1,r}$-free graphs

Component factors in $K_{1,r}$-free graphs

Remarks on component factors in graphs

Some degree conditions for 𝒫_≥k-factor covered graphs

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The existence of path-factor covered graphs

Cited by 14 publications

References 17 publications

Component factors in $K_{1,r}$-free graphs

Component factors in $K_{1,r}$-free graphs

Remarks on component factors in graphs

Some degree conditions for 𝒫≥k-factor covered graphs

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Some degree conditions for 𝒫_≥k-factor covered graphs