Narrow Results

For a set ${𝒟}$ of connected graphs, a spanning subgraph $F$ of a graph $G$ is called an ${𝒟}$-factor if each component of $F$ is isomorphic to a member of ${𝒟}$. In this paper, we obtain a sufficient condition with regard to tight toughness, isolated toughness and binding number bounds to guarantee the existence of a $\{K_{1,1},K_{1,2},C_n|n\ge 3\}$-factor, two sufficient conditions to guarantee the existence of a $P_{\ge 3}$-factor in a bipartite graph and some sufficient conditions for $\{K_{1,j}|1\le j\le k\}$-factor deleted graph and a $P_{\ge 2}$-factor deleted graph.

The binding number of a graph G is defined to be the minimum of $|N(S)|/\left|S\right|$ taken over all nonempty $S\subseteq V(G)$ such that $N(S)\ne V(G)$. Binding number, one indicator to better understand graph, is an important characteristic quantity of a graph. In this paper, the relationships between the binding number and some other graph vulnerability parameters, namely the toughness, integrity, rupture degree and scattering number, are established. Exact values for the binding numbers of wheel related graphs namely gear, helm, sunflower and friendship graph are obtained.

Let $n\ge 2$ be an integer. The $[1,n]$-factor of a graph $G$ is a spanning subgraph $F$ if $1\le {deg}_F(x)\le n$ for all $x\in V(G)$, and the $\{K_2,C_i\}$-factor is a subgraph whose each component is either $K_2$ or $C_i$. In this paper, we give the lower bounds with regard to tight toughness, isolated toughness and binding number to guarantee the existence of the $[1,n]$-factors and $\{K_2,C_i|i\ge 4\}$-factors for a graph.

For a family of connected graphs ${ℱ}$, a spanning subgraph $L$ of $G$ is called an ${ℱ}$-factor if each component of $L$ is isomorphic to a member of ${ℱ}$. In this paper, sufficient conditions with regard to tight toughness, isolated toughness and binding number bounds to guarantee the existence of the $\{P_2,C_3,P_5,{𝒯}(3)\}$-factor, $P_{\ge 2}$-factor or $\{P_2,C_{2i+1}|i\ge k\}$-factor for $k\ge 3$ are obtained.

For a set ${𝒦}$ of connected graphs, a spanning subgraph $H$ of $G$ is called a ${𝒦}$-factor if each component of $H$ is isomorphic to a member of ${𝒦}$. In this paper, some sufficient conditions with regard to tight toughness, isolated toughness and binding number bounds to guarantee the existence of the $\{K_{1,j}:1\le j\le 2k\}$-factor and $\{P_2,P_5\}$-factor for any graph are obtained.

The $\{K_{1,1},K_{1,2},\dots ,K_{1,k},{𝒯}(2k+1)\}$-factor and $\{K_{1,2},K_{1,3},K_5\}$-factor of a graph are a spanning subgraph whose each component is an element of $\{K_{1,1},K_{1,2},\dots ,K_{1,k},{𝒯}(2k+1)\}$ and $\{K_{1,2},K_{1,3},K_5\}$, respectively, where ${𝒯}(2k+1)$ is a special family of trees. In this paper, we obtain a sufficient condition in terms of tight toughness, isolated toughness and binding number bounds to guarantee the existence of a $\{K_{1,1},K_{1,2},\dots ,K_{1,k},{𝒯}(2k+1)\}$-factor and $\{K_{1,2},K_{1,3},K_5\}$-factor for any graph.

For a set ${\mathscr{H}}$ of connected graphs, a spanning subgraph $H$ of a graph $G$ is an ${\mathscr{H}}$-factor if every component of $H$ is isomorphic to some member of ${\mathscr{H}}$. In this paper, we give a criterion for the existence of tight toughness, isolated toughness and binding number bounds in a graph of a strong ${𝒮}$-star factor, $\{1,3,\dots ,2n-1\}$-factor and $f$-star factor. Moreover, we show that the bounds of the sufficient conditions are sharp.