Sufficient conditions for isolated tough graphs to have path-factors

Let $G$ be a connected graph with $n$ vertices, where $n$ is a positive integer. The size of $G$ is denoted by $e(G)$. The isolated toughness of $G$, denoted by $I(G)$, is defined by

$$I(G)=min\left\{\frac{|S|}{i(G-S)}:S\subseteq V(G)\text{and}i(G-S)\ge 2\right\}$$

or $I(G)=+\mathrm{\infty }$ if $G$ is complete. A graph $G$ is called isolated $r$-tough if $I(G)\ge r$. The distance signless Laplacian matrix $\mathcal{Q}(G)$ of $G$ is defined by $\mathcal{Q}(G)=Tr(G)+\mathcal{D}(G)$, where $\mathcal{D}(G)$ denotes the distance matrix of $G$ and $Tr(G)$ is the diagonal matrix of the vertex transmissions in $G$. The largest eigenvalue of $\mathcal{Q}(G)$, denoted by $\eta (G)$, is called the distance signless Laplacian spectral radius of $G$. A $P_{\ge k}$-factor means a path factor with every component containing at least $k$ vertices, where $k$ is an integer with $k\ge 2$. In this paper, we aim to establish two tight sufficient conditions based on $e(G)$ and $\eta (G)$ to guarantee that a graph $G$ contains a $P_{\ge 2}$-factor. Let $G$ be a connected isolated $\frac{t}{2t+1}$-tough graph of order $n$, where $t\ge 1$ is an integer. Then the following two results hold.

(ⅰ) If $n\ge 6t+2$ and $e(G)\ge e(K_t\vee (K_{n-3t-1}\cup (2t+1)K_1))$, then $G$ contains a $P_{\ge 2}$-factor unless $G=K_t\vee (K_{n-3t-1}\cup (2t+1)K_1)$.

(ⅱ) If $n\ge 9t+2$ and $\eta (G)\le \eta (K_t\vee (K_{n-3t-1}\cup (2t+1)K_1))$, then $G$ contains a $P_{\ge 2}$-factor unless $G=K_t\vee (K_{n-3t-1}\cup (2t+1)K_1)$.