A note on fractional ID-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e35" altimg="si6.svg"><mml:mrow><mml:mo>[</mml:mo><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:math>-factor-critical covered graphs

Zhou, Sizhong; Liu, Hongxia; Xu, Yan

doi:10.1016/j.dam.2021.03.004

Cited by 36 publications

(6 citation statements)

References 23 publications

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“…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: 84%

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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Section: Introductionmentioning

confidence: 84%

Isolated toughness and path-factor uniform graphs

2021

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“…Fan et al [21] showed a lower bound for the adjacency spectral radius to ensure a graph admitting an odd [1, 𝑏]-factor. More other results on graph factors can be found in [22][23][24][25][26][27][28][29][30][31][32].…”

Section: Introductionmentioning

confidence: 88%

Characterizing an odd [1, b]-factor on the distance signless Laplacian spectral radius

Zhou¹,

Liu²

2023

RAIRO-Oper. Res.

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Let G be a connected graph of even order n. An odd [1,b]-factor of G is a spanning subgraph F of G such that dF(v) ∈ {1,3,5,··· ,b} for any v ∈ V (G), where b is positive odd integer. The distance matrix D(G) of G is a symmetric real matrix with (i,j)-entry being the distance between the vertices vi and vj. The distance signless Laplacian matrix Q(G) of G is defined by Q(G) = Tr(G) + D(G), where Tr(G) is the diagonal matrix of the vertex transmissions in G. The largest eigenvalue η1(G) of Q(G) is called the distance signless Laplacian spectral radius of G. In this paper, we verify sharp upper bounds on the distance signless Laplacian spectral radius to guarantee the existence of an odd [1,b]-factor in a graph; we provide some graphs to show that the bounds are optimal.

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“…In Zhou [12] and Zhou et al [14], the setting was different from the previous situations in which there is a difference ∆ between g(x) and f (x) for every vertex…”

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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A note on fractional ID-[a,b]-factor-critical covered graphs

Cited by 36 publications

References 23 publications

Isolated toughness and path-factor uniform graphs

Isolated toughness and path-factor uniform graphs

Characterizing an odd [1, b]-factor on the distance signless Laplacian spectral radius

The Extension Degree Conditions for Fractional Factor

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