Remarks on component factors in graphs

Abstract

For a family of connected graphs $\mathcal{F}$, a spanning subgraph $H$ of a graph $G$ is called an $\mathcal{F}$-factor of $G$ if its each component is isomorphic to an element of $\mathcal{F}$. In particular, $H$ is called an $\mathcal{S}_k$-factor of $G$ if $\mathcal{F}=\{K_{1,1},K_{1,2},...,K_{1,k}\}$, where integer $k\geq2$; $H$ is called a $P_{\geq 3}$-factor of $G$ if every component in $\mathcal{F}$ is a path of order at least three. As an extension of $\mathcal{S}_k$-factors, the induced star-factor (i.e., $\mathcal{IS}_k$-factor) is a spanning subgraph each component of which is an induced subgraph isomorphic to some graph in $\mathcal{F}=\{K_{1,1},K_{1,2},...,K_{1,k}\}$. In this paper, we firstly prove that a graph $G$ has an $\mathcal{S}_k$-factor if and only if its isolated toughness $I(G)\geq \frac{1}{k}$. Secondly, we prove that a planar graphs $G$ has an $\mathcal{S}_{2}$-factors if its minimum degree $\delta(G)\geq3$. Thirdly, we give two sufficient conditions for graphs with $\mathcal{IS}_k$-factors by toughness and minimum degree, respectively. Additionally, we obtain three special classes of graphs admitting $P_{\geq3}$-factors.

Degree sum and restricted {P2,P5}-factor in graphs

“…Since Tutte proposed the well-known Tutte 1-factor theorem [20], path-factors of graphs [2,5,6,[8][9][10][11]16] and path-factor covered graphs [7,12,[22][23][24] have been extensively studied. More results on graph factors are referred to the survey papers and books [3,21].…”

Sufficient conditions for graphs with {P2, P5}-factors

“…Since Tutte proposed the well-known Tutte 1-factor theorem [20], path-factors of graphs [3,7,8,13,17] and path-factor covered graphs [6,9,23] have attracted a great deal of attention. More results on graph factors are referred to the survey papers and books [2,19,22].…”