Partitioning vertices of 1-tough graphs into paths

Bazgan, Cristina; Harkat-Benhamdine, Amel; Li, Hao; Wózniakb, Mariusz

doi:10.1016/s0304-3975(00)00247-4

Cited by 18 publications

(4 citation statements)

References 3 publications

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“…Bazgan, Benhamdine, Li and Woźniak [5] showed a toughness condition for the existence of a P ≥3 -factor in a graph.…”

Section: Introductionmentioning

confidence: 99%

The existence of path-factor covered graphs

Dai¹

2020

Discuss. Math. Graph Theory

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A spanning subgraph H of a graph G is called a P ≥k -factor of G if every component of H is isomorphic to a path of order at least k, where k ≥ 2. A graph G is called a P ≥k -factor covered graph if there is a P ≥k -factor of G covering e for any e ∈ E(G). In this paper, we obtain two special classes of P ≥2 -factor covered graphs. We also obtain two special classes of P ≥3 -factor covered graphs. Furthermore, it is shown that these results are all sharp.

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“…Bazgan, Benhamdine, Li and Woźniak [5] showed a toughness condition for the existence of a P ≥3 -factor in a graph.…”

Section: Introductionmentioning

confidence: 99%

The existence of path-factor covered graphs

Dai¹

2020

Discuss. Math. Graph Theory

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“…Since Tutte proposed the well-known Tutte 1-factor theorem [20], path-factors of graphs [2,5,6,[8][9][10][11]16] and path-factor covered graphs [7,12,[22][23][24] have been extensively studied. More results on graph factors are referred to the survey papers and books [3,21].…”

Section: Introductionmentioning

confidence: 99%

Degree sum and restricted {P2,P5}-factor in graphs

DAI

2023

Proc. Rom. Acad. Ser. A - Math. Phys. Tech. Sci. Inf. Sci.

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"For a graph $G$, a spanning subgraph $F$ of $G$ is called a $\{P_2,P_5\}$-factor if every component of $F$ is isomorphic to $P_2$ or $P_5$, where $P_i$ denotes the path of order $i$. A graph $G$ is called a $(\{P_2,P_5\},k)$-factor critical graph if $G-V'$ contains a $\{P_2,P_5\}$-factor for any $V'\subseteq V(G)$ with $|V'|=k$. A graph $G$ is called a $(\{P_2,P_5\},m)$-factor deleted graph if $G-E'$ has a $\{P_2,P_5\}$-factor for any $E'\subseteq E(G)$ with $|E'|=m$. The degree sum of $G$ is defined by $$\sigma_{r+1}(G)=\min_{X\subseteq V(G)}\Big\{\sum_{x\in X}d_G(x): X~\mathrm{is~an~independent~set~of}~r+1~\mathrm{vertices}\Big\}.$$ In this paper, using degree sum conditions, we demonstrate that (i) $G$ is a $(\{P_2,P_5\},k)$-factor critical graph if $\sigma_{r+1}(G)>\frac{(3n+4k-2)(r+1)}{7}$ and $\kappa(G)\geq k+r$; (ii) $G$ is a $(\{P_2,P_5\},m)$-factor deleted graph if $\sigma_{r+1}(G)>\frac{(3n+2m-2)(r+1)}{7}$ and $\kappa(G)\geq\frac{5m}{4}+r$."

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“…Kaneko [7] verified that a regular graph with degree at least 2 admits a P ≥3 -factor. Bazgan et al [1] posed a sufficient condition for a graph to admit a P ≥3 -factor. 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.…”

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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Partitioning vertices of 1-tough graphs into paths

Cited by 18 publications

References 3 publications

The existence of path-factor covered graphs

The existence of path-factor covered graphs

Degree sum and restricted {P2,P5}-factor in graphs

The existence of path-factor uniform graphs with large connectivity

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