Narrow Results

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.

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.

Let $A$ be a family of connected graphs. A spanning subgraph $H$ of $G$ is called an $A$-factor (component factor) of $G$ if each component of $H$ is in $A$. In this paper, we study the component factors of the Cartesian product of graphs. Here, we take $A=\{K_{1,n},C_4\}$ and show that every connected graph $G\cong H_1\square H_2$ has a $\{K_{1,n},C_4\}$-factor where $1\le n\le t$ and $t$ is the maximum degree of an induced subgraph $K_{1,t}$ in $H_1$ or $H_2$. Also, we characterize graphs $G\cong H_1\square H_2$ having a $C_4$-factor.