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) \ \mbox{and} \ i(G-S)\geq2\right\} $

or $ I(G) = +\infty $ if $ G $ is complete. A graph $ G $ is called isolated $ r $-tough if $ I(G)\geq 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_{\geq k} $-factor means a path factor with every component containing at least $ k $ vertices, where $ k $ is an integer with $ k\geq2 $. 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_{\geq2} $-factor. Let $ G $ be a connected isolated $ \frac{t}{2t+1} $-tough graph of order $ n $, where $ t\geq1 $ is an integer. Then the following two results hold.

(ⅰ) If $ n\geq6t+2 $ and $ e(G)\geq e(K_t\vee(K_{n-3t-1}\cup(2t+1)K_1)) $, then $ G $ contains a $ P_{\geq2} $-factor unless $ G = K_t\vee(K_{n-3t-1}\cup(2t+1)K_1) $.

(ⅱ) If $ n\geq9t+2 $ and $ \eta(G)\leq\eta(K_t\vee(K_{n-3t-1}\cup(2t+1)K_1)) $, then $ G $ contains a $ P_{\geq2} $-factor unless $ G = K_t\vee(K_{n-3t-1}\cup(2t+1)K_1) $.