Erdos-Gallai Stability Theorem for Linear Forests

The Erd\H{o}s-Gallai Theorem states that every graph of average degree more than $l-2$ contains a path of order $l$ for $l\ge 2$. In this paper, we obtain a stability version of the Erd\H{o}s-Gallai Theorem in terms of minimum degree. Let $G$ be a connected graph of order $n$ and $F=(\bigcup_{i=1}^kP_{2a_i})\bigcup(\bigcup_{i=1}^lP_{2b_i+1})$ be $k+l$ disjoint paths of order $2a_1, \ldots, 2a_{k}, 2b_1+1, \ldots, 2b_l+1,$ respectively, where $k\ge 0$, $0\le l\le 2$, and $k+l\geq 2$. If the minimum degree $\delta(G)\ge \sum_{i=1}^ka_i+\sum_{i=1}^lb_i-1$, then $F\subseteq G$ except several classes of graphs for sufficiently large $n$, which extends and strengths the results of Ali and Staton for an even path and Yuan and Nikiforov for an odd path.