Two sufficient conditions for the existence of path factors in graphs

Zhou, Sizhong; Yang, Fan; Xu, Lan

doi:10.24200/sci.2018.5151.1122

Cited by 18 publications

(19 citation statements)

References 23 publications

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“…Some results on factor deleted graphs see [3,4,19], and some results on factor critical graphs see [2,5,15,16,21]. Many authors [1,6,10,11,20,22,23] studied the existence of path factors in graphs. In this paper, we study the existence of (P ≥3 , m)-factor deleted graphs and (P ≥3 , k)-factor critical graphs and obtain some sufficient conditions for graphs to be (P ≥3 , m)-factor deleted graphs and (P ≥3 , k)-factor critical graphs depending on binding numbers, which are shown in Sections 2 and 3.…”

Section: Introductionmentioning

confidence: 99%

Remarks on path factors in graphs

Zhou

2020

RAIRO-Oper. Res.

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A spanning subgraph of a graph is defined as a path factor of the graph if its component are paths. A $P_{\geq n}$-factor means a path factor with each component having at least $n$ vertices. A graph $G$ is defined as a $(P_{\geq n},m)$-factor deleted graph if $G-E'$ has a $P_{\geq n}$-factor for every $E'\subseteq E(G)$ with $|E'|=m$. A graph $G$ is defined as a $(P_{\geq n},k)$-factor critical graph if after deleting any $k$ vertices of $G$ the remaining graph of $G$ admits a $P_{\geq n}$-factor. In this paper, we demonstrate that (\romannumeral1) a graph $G$ is $(P_{\geq3},m)$-factor deleted if $\kappa(G)\geq2m+1$ and $bind(G)\geq\frac{3}{2}-\frac{1}{4m+4}$; (\romannumeral2) a graph $G$ is $(P_{\geq3},k)$-factor critical if $\kappa(G)\geq k+2$ and $bind(G)\geq\frac{5+k}{4}$.

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Section: Introductionmentioning

confidence: 99%

Remarks on path factors in graphs

Zhou

2020

RAIRO-Oper. Res.

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“…The concept of fractional (g, f , n , m)critical deleted graph can be regarded as a combination of fractional (g, f , n )-critical graph and fractional (g, f , m)-deleted graph, i.e., a graph G is fractional (g, f , n , m)-critical deleted if after n vertices are deleted from G, the rest of the graph is a fractional (g, f , m)-deleted graph. For some recent advances on the existence of a fractional factor in various settings, we can refer to Gao et al [7][8][9][10]12], Liu et al [18], and Zhou et al [34,35,37,[39][40][41][42][43].…”

Section: Background and Conceptsmentioning

confidence: 99%

Parameters and fractional factors in different settings

Gao

Guirao

2019

J Inequal Appl

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In computer networks, the feasibility of data transmission can be measured in terms of fractional factor model, and specifically the framework of fractional critical deleted graph can express the setting when certain sites and channels are unavailable at a special time. We study the relationship between some parameters in graphs and the existence of fractional (g, f )-factor in various settings here. Our main contributions are three-fold: first, a connectivity condition for a graph to be fractional (g, f , n , m)-critical deleted is determined; second, the relationship between independence number and fractional ID-(g, f , m)-deleted graphs is studied; third, an isolated toughness bound for fractional (g, f )-factors is given in balanced bipartite graph setting. Furthermore, by showing counterexamples we explain that bounds for parameters are tight in some sense, and corresponding conditions in all fractional factor settings are discussed as well. Finally, two open problems are proposed for future research. MSC: 05C70

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“…When functions g and f take special value for all vertices, the fractional (g, f, n ′ , m)-critical deleted graph becomes various names which are presented in Table 1. Several recent contributions in this topic were presented in Zhou et al [10], [11], [13], [15] and [16], and Gao et al [3], [4], [5] and [7], Knor et al [8], and Liu et al [9].…”

Section: Introductionmentioning

confidence: 99%

The Extension Degree Conditions for Fractional Factor

Gao

Wang

Guirao

2020

Acta. Math. Sin.-English Ser.

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In Gao's previous work, the authors determined several graph degree conditions of a graph which admits fractional factor in particular settings. It was revealed that these degree conditions are tight if b = f (x) = g(x) = a for all vertices x in G. In this paper, we continue to discuss these degree conditions for admitting fractional factor in the setting that several vertices and edges are removed and there is a difference ∆ between g(x) and f (x) for every vertex x in G. These obtained new degree conditions reformulate Gao's previous conclusions, and show how ∆ acts in the results. Furthermore, counterexamples are structured to reveal the sharpness of degree conditions in the setting f (x) = g(x) + ∆.

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Two sufficient conditions for the existence of path factors in graphs

Abstract: The two results are best possible in some conditions.

Cited by 18 publications

References 23 publications

Remarks on path factors in graphs

Remarks on path factors in graphs

Parameters and fractional factors in different settings

The Extension Degree Conditions for Fractional Factor

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