Remarks on path factors in graphs

2020 ◽  
Vol 54 (6) ◽  
pp. 1827-1834 ◽  
Author(s):  
Sizhong Zhou
Keyword(s):  
Spanning Subgraph ◽  
Critical Graph ◽  
Path Factor

A spanning subgraph of a graph is defined as a path factor of the graph if its component are paths. A P≥n-factor means a path factor with each component having at least n vertices. A graph G is defined as a (P≥n, m)-factor deleted graph if G–E′ has a P≥n-factor for every E′ ⊆ E(G) with |E′| = m. A graph G is defined as a (P≥n, k)-factor critical graph if after deleting any k vertices of G the remaining graph of G admits a P≥n-factor. In this paper, we demonstrate that (i) a graph G is (P≥3, m)-factor deleted if κ(G) ≥ 2m + 1 and bind(G) ≥  2/3 - $ \frac{3}{2}-\frac{1}{4m+4}$; (ii) a graph G is (P≥3, k)-factor critical if κ(G) ≥ k + 2 and bind(G) ≥ $ \frac{5+k}{4}$.

Some results about component factors in graphs

2019 ◽  
Vol 53 (3) ◽  
pp. 723-730 ◽  
Author(s):  
Sizhong Zhou
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Connected Graphs ◽  
Spanning Subgraph ◽  
Component Factor ◽  
Path Factor

For a set ℋ of connected graphs, a spanning subgraph H of a graph G is called an ℋ-factor of G if every component of H is isomorphic to a member ofℋ. An H-factor is also referred as a component factor. If each component of H is a star (resp. path), H is called a star (resp. path) factor. By a P≥ k-factor (k positive integer) we mean a path factor in which each component path has at least k vertices (i.e. it has length at least k − 1). A graph G is called a P≥ k-factor covered graph, if for each edge e of G, there is a P≥ k-factor covering e. In this paper, we prove that (1) a graph G has a {K1,1,K1,2, … ,K1,k}-factor if and only if bind(G) ≥ 1/k, where k ≥ 2 is an integer; (2) a connected graph G is a P≥ 2-factor covered graph if bind(G) > 2/3; (3) a connected graph G is a P≥ 3-factor covered graph if bind(G) ≥ 3/2. Furthermore, it is shown that the results in this paper are best possible in some sense.

A New Sufficient Condition for a Graph To Be (g, f, n)-Critical

2010 ◽  
Vol 53 (2) ◽  
pp. 378-384
Author(s):  
Sizhong Zhou
Keyword(s):  
Sufficient Condition ◽  
Binding Number ◽  
Minimum Value ◽  
Spanning Subgraph ◽  
Critical Graph

AbstractLet G be a graph of order p, let a, b, and n be nonnegative integers with 1 ≤ a < b, and let g and f be two integer-valued functions defined on V(G) such that a ≤ g(x) < f (x) ≤ b for all x ∈ V(G). A (g, f )-factor of graph G is a spanning subgraph F of G such that g(x) ≤ dF(x) ≤ f (x) for each x ∈ V(F). Then a graph G is called (g, f, n)-critical if after deleting any n vertices of G the remaining graph of G has a (g, f )-factor. The binding number bind(G) of G is the minimum value of |NG(X)|/|X| taken over all non-empty subsets X of V(G) such that NG(X) ≠ V(G). In this paper, it is proved that G is a (g, f, n)-critical graph ifFurthermore, it is shown that this result is best possible in some sense.

scholarly journals Isolated toughness and path-factor uniform graphs

2021 ◽  
Author(s):  
Sizhong Zhou ◽  
Zhiren Sun ◽  
Hongxia Liu
Keyword(s):  
Connected Graph ◽  
Edge Connectivity ◽  
Spanning Subgraph ◽  
K Factor ◽  
Path Factor

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.

scholarly journals Notes on the binding numbers for (a, b, k)-critical graphs

2007 ◽  
Vol 76 (2) ◽  
pp. 307-314 ◽  
Author(s):  
Sizhong Zhou ◽  
Jiashang Jiang
Keyword(s):  
Critical Graphs ◽  
Binding Number ◽  
Spanning Subgraph ◽  
Critical Graph

Let G be a graph of order n, and let a, b, k be nonnegative integers with 1 ≤ a < b. An [a, b]-factor of graph G is defined as a spanning subgraph F of G such that a ≤ dF(x) ≤ b for each x ϵ V (F). Then a graph G is called an (a, b, k)-critical graph if after deleting any k vertices of G the remaining graph of G has an [a, b]-factor. In this paper, it is proved that G is an (a, b, k)-critical graph if the binding number and Furthermore, it is showed that the result in this paper is best possible in some sense.

scholarly journals Component factors and binding number conditions in graphs

2021 ◽  
Vol 6 (11) ◽  
pp. 12460-12470
Author(s):  
Sizhong Zhou ◽  
◽  
Jiang Xu ◽  
Lan Xu ◽  
Keyword(s):  
Connected Graph ◽  
Connected Graphs ◽  
Binding Number ◽  
Spanning Subgraph ◽  
Critical Graph

<abstract><p>Let $ G $ be a graph. For a set $ \mathcal{H} $ of connected graphs, an $ \mathcal{H} $-factor of a graph $ G $ is a spanning subgraph $ H $ of $ G $ such that every component of $ H $ is isomorphic to a member of $ \mathcal{H} $. A graph $ G $ is called an $ (\mathcal{H}, m) $-factor deleted graph if for every $ E'\subseteq E(G) $ with $ |E'| = m $, $ G-E' $ admits an $ \mathcal{H} $-factor. A graph $ G $ is called an $ (\mathcal{H}, n) $-factor critical graph if for every $ N\subseteq V(G) $ with $ |N| = n $, $ G-N $ admits an $ \mathcal{H} $-factor. Let $ m $, $ n $ and $ k $ be three nonnegative integers with $ k\geq2 $, and write $ \mathcal{F} = \{P_2, C_3, P_5, \mathcal{T}(3)\} $ and $ \mathcal{H} = \{K_{1, 1}, K_{1, 2}, \cdots, K_{1, k}, \mathcal{T}(2k+1)\} $, where $ \mathcal{T}(3) $ and $ \mathcal{T}(2k+1) $ are two special families of trees. In this article, we verify that (i) a $ (2m+1) $-connected graph $ G $ is an $ (\mathcal{F}, m) $-factor deleted graph if its binding number $ bind(G)\geq\frac{4m+2}{2m+3} $; (ii) an $ (n+2) $-connected graph $ G $ is an $ (\mathcal{F}, n) $-factor critical graph if its binding number $ bind(G)\geq\frac{2+n}{3} $; (iii) a $ (2m+1) $-connected graph $ G $ is an $ (\mathcal{H}, m) $-factor deleted graph if its binding number $ bind(G)\geq\frac{2}{2k-1} $; (iv) an $ (n+2) $-connected graph $ G $ is an $ (\mathcal{H}, n) $-factor critical graph if its binding number $ bind(G)\geq\frac{2+n}{2k+1} $.</p></abstract>

scholarly journals The Existence of a Path-Factor without Small Odd Paths

10.37236/5817 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Yoshimi Egawa ◽  
Michitaka Furuya
Keyword(s):  
Number Of Components ◽  
Spanning Subgraph ◽  
Path Factor

A $\{P_{2},P_{5}\}$-factor of a graph is a spanning subgraph of the graph each of whose components is isomorphic to either $P_{2}$ or $P_{5}$, where $P_{n}$ denote the path of order $n$. In this paper, we show that if a graph $G$ satisfies $c_{1}(G-X)+\frac{2}{3}c_{3}(G-X)\leq \frac{4}{3}|X|+\frac{1}{3}$ for all $X\subseteq V(G)$, then $G$ has a $\{P_{2},P_{5}\}$-factor, where $c_{i}(G-X)$ is the number of components $C$ of $G-X$ with $|V(C)|=i$. Moreover, it is shown that above condition is sharp.

scholarly journals Some degree conditions for P≥k-factor covered graphs

2021 ◽  
Author(s):  
Guowei Dai ◽  
Zan-Bo Zhang ◽  
Yicheng Hang ◽  
Xiaoyan Zhang
Keyword(s):  
Planar Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Discrete Mathematics ◽  
Sufficient Condition ◽  
Degree Condition ◽  
Spanning Subgraph ◽  
K Factor ◽  
The Relationship ◽  
Path Factor

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.

scholarly journals A Recursive Formula for the Reliability of a r-Uniform Complete Hypergraph and Its Applications

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Ke Zhang ◽  
Haixing Zhao ◽  
Zhonglin Ye ◽  
Lixin Dong
Keyword(s):  
Recurrence Relations ◽  
Recursive Formula ◽  
Finite Graph ◽  
Hard Problem ◽  
Np Hard ◽  
Spanning Subgraph ◽  
Reliability Polynomial ◽  
Np Hard Problem

The reliability polynomial R(S,p) of a finite graph or hypergraph S=(V,E) gives the probability that the operational edges or hyperedges of S induce a connected spanning subgraph or subhypergraph, respectively, assuming that all (hyper)edges of S fail independently with an identical probability q=1-p. In this paper, we investigate the probability that the hyperedges of a hypergraph with randomly failing hyperedges induce a connected spanning subhypergraph. The computation of the reliability for (hyper)graphs is an NP-hard problem. We provide recurrence relations for the reliability of r-uniform complete hypergraphs with hyperedge failure. Consequently, we determine and calculate the number of connected spanning subhypergraphs with given size in the r-uniform complete hypergraphs.

Vertices of degree 6 in a 6-contraction critical graph

2001 ◽  
Vol 10 ◽  
pp. 2-6
Author(s):  
Kiyoshi Ando ◽  
Atsushi Kaneko ◽  
Ken-ichi Kawarabayashi
Keyword(s):  
Critical Graph

scholarly journals Neighborhood condition for all fractional (g, f, n′, m)-critical deleted graphs

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 544-553 ◽  
Author(s):  
Wei Gao ◽  
Yunqing Zhang ◽  
Yaojun Chen
Keyword(s):  
Data Transmission ◽  
Transmission Networks ◽  
Neighborhood Conditions ◽  
Transmission Rates ◽  
High Data ◽  
Critical Graph ◽  
Network Research ◽  
Neighborhood Condition ◽  
Fractional Factor ◽  
The Relationship

Abstract In data transmission networks, the availability of data transmission is equivalent to the existence of the fractional factor of the corresponding graph which is generated by the network. Research on the existence of fractional factors under specific network structures can help scientists design and construct networks with high data transmission rates. A graph G is named as an all fractional (g, f, n′, m)-critical deleted graph if the remaining subgraph keeps being an all fractional (g, f, m)-critical graph, despite experiencing the removal of arbitrary n′ vertices of G. In this paper, we study the relationship between neighborhood conditions and a graph to be all fractional (g, f, n′, m)-critical deleted. Two sufficient neighborhood conditions are determined, and furthermore we show that the conditions stated in the main results are sharp.

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