The existence of path-factor uniform graphs with large connectivity

A path-factor in a graph G is a spanning subgraph F of G such that every component of F is a path. Let d and n be two nonnegative integers with d ≥ 2. A P ≥ d -factor of G is its spanning subgraph each of whose components is a path with at least d vertices. A graph G is called a P ≥ d -factor covered graph if for any e ∈ E ( G ), G admits a P ≥ d -factor containing e . A graph G is called a ( P ≥ d ,n )-factor critical covered graph if for any N ⊆ V ( G ) with | N | = n , the graph G − N is a P ≥ d -factor covered graph. A graph G is called a P ≥ d -factor uniform graph if for any e ∈ E ( G ), the graph G − e is a P ≥ d -factor covered graph. In this paper, we verify the following two results: (i) An ( n + 1)-connected graph G of order at least n + 3 is a ( P ≥3 ,n )-factor critical covered graph if G satisfies; (ii) Every regular graph G with degree r ≥ 2 is a

Section: Theorem 13 ( [29]mentioning

confidence: 99%

Path-factor critical covered graphs and path-factor uniform graphs

2022

“…Gao and Wang [1] improved Zhou and Sun's result on 𝑃 ≥3 -factor uniform graphs. Zhou and Bian [19] showed two sufficient conditions for the existence of a 𝑃 ≥3 -factor uniform graph. Zhou et al [22] provided an isolated toughness condition for a graph to possess a 𝑃 ≥3 -factor uniform graph.…”

Section: Theorem 2 ([12]mentioning

confidence: 99%

Sun toughness and path-factor uniform graphs

Liu

2022

A path-factor is a spanning subgraph $F$ of $G$ such that each component of $F$ is a path of order at least two. Let $k$ be an integer with $k\geq2$. A $P_{\geq k}$-factor is a spanning subgraph of $G$ whose components are paths of order at least $k$. A graph $G$ is called a $P_{\geq k}$-factor covered graph if for any edge $e$ of $G$, $G$ admits a $P_{\geq k}$-factor covering $e$. A graph $G$ is called a $P_{\geq k}$-factor uniform graph if for any two distinct edges $e_1$ and $e_2$ of $G$, $G$ has a $P_{\geq k}$-factor covering $e_1$ and excluding $e_2$. In this article, we claim that (\romannumeral1) a 4-edge-connected graph $G$ is a $P_{\geq3}$-factor uniform graph if its sun toughness $s(G)\geq1$; (\romannumeral2) a 4-connected graph $G$ is a $P_{\geq3}$-factor uniform graph if its sun toughness $s(G)>\frac{4}{5}$.

“…Liu and Li [4] characterized a graph or a bipartite graph with a 1-factor by virtue of its distance signless Laplacian spectral radius. Lots of researchers presented some sufficient conditions on various parameters to guarantee the existence of [1,2]-factors in graphs, such as the neighborhood condition [5], the degree conditions [6][7][8], the binding number [8,9], the independence number [10,11], the isolated toughness [12][13][14][15], and the sun toughness [16]. Zhou [17] derived some sufficient conditions for graphs to possess [1,2]-factors.…”

Section: Introductionmentioning

confidence: 99%

“…(i) For 𝑛 ≥ 𝑏 + 3 and (𝑏, 𝑛) / ∈ { (1,6), (1,8), (1, 10)}, if 𝜂 1 (𝐺) < 𝜃(𝑛), then 𝐺 has an odd [1, 𝑏]-factor, where 𝜃(𝑛) is the largest root of 𝑥 3 − (5𝑛 + 2𝑏 − 7)𝑥 2 + (8𝑛 2 + (2𝑏 − 27)𝑛 + 4𝑏(𝑏 + 1) + 24)𝑥 − 4𝑛 3 + 22𝑛 2 − (4𝑏 2 + 8𝑏 + 42)𝑛 + 6𝑏 2 + 14𝑏 + 28 = 0.…”

Section: Introductionmentioning

confidence: 99%

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

Zhou¹,

Liu²

2023

Self Cite

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.