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.

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.

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.

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}$.

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 Improved Predictive Ability of KPLS Regression with Memetic Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 506
Author(s):  
Jorge Daniel Mello-Román ◽  
Adolfo Hernández ◽  
Julio César Mello-Román
Keyword(s):  
Kernel Function ◽  
Linear Method ◽  
Predictive Ability ◽  
Feature Space ◽  
Memetic Algorithms ◽  
Least Squares Regression ◽  
Optimal Parameters ◽  
Number Of Components ◽  
Kernel Partial Least Squares ◽  
Dependent Variables

Kernel partial least squares regression (KPLS) is a non-linear method for predicting one or more dependent variables from a set of predictors, which transforms the original datasets into a feature space where it is possible to generate a linear model and extract orthogonal factors also called components. A difficulty in implementing KPLS regression is determining the number of components and the kernel function parameters that maximize its performance. In this work, a method is proposed to improve the predictive ability of the KPLS regression by means of memetic algorithms. A metaheuristic tuning procedure is carried out to select the number of components and the kernel function parameters that maximize the cumulative predictive squared correlation coefficient, an overall indicator of the predictive ability of KPLS. The proposed methodology led to estimate optimal parameters of the KPLS regression for the improvement of its predictive ability.

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