Erdős-Gallai stability theorem for linear forests

Abstract The Erdős–Gallai Theorem states that every graph of average degree more than l − 2 contains a path of order l for l ≥ 2 . In this paper, we obtain a stability version of the Erdős–Gallai Theorem in terms of minimum degree. Let G be a connected graph of order n and F = ( ⋃ i = 1 k P 2 a i ) ⋃ ( ⋃ i = 1 l P 2 b i + 1 ) be k + l disjoint paths of order 2 a 1 , … , 2 a k , 2 b 1 + 1 , … , 2 b l + 1 , respectively, where k ≥ 0 , 0 ≤ l ≤ 2 , and k + l ≥ 2 . If the minimum degree δ ( G ) ≥ ∑ i = 1 k a i + ∑ i = 1 l b i − 1 , then F ⊆ 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.