Sun Toughness Conditions for P 2 and P 3 Factor Uniform and Factor Critical Avoidable Graphs

The security of a network is closely related to the structure of the network graph. The denser the network graph structure is, the better it can resist attacks. Toughness and isolated toughness are used to characterize the vulnerable programs of the network which have been paid attention from mathematics and computer scholars. On this basis, considering the particularity of the sun component structures, sun toughness was introduced in mathematics and applied to computer networks. From the perspective of modern graph theory, this paper presents the sun toughness conditions of the path factor uniform graph and the path factor critical avoidable graph in P ≥ 2 -factor and P ≥ 3 -factor settings. Furthermore, examples show that the given boundaries are sharp.

Some degree conditions for P≥k-factor covered graphs

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.

Isolated toughness and path-factor uniform graphs

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

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

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.