Exact Tur\'{a}n numbers of two vertex-disjoint paths

Abstract

The Tur\'{a}n number of a graph $H$ is the maximum number of edges in any graph of order $n$ that does not contain $H$ as a subgraph. In 1959, Erd\H os and Gallai obtained a sharp upper bound of Tur\'{a}n numbers for a path of arbitrary length. In 1975, Faudree and Schelp, and independently in 1977, Kopylov determined the exact values of Tur\'an numbers of paths with arbitrary length. In this paper, we determine the Tur\'{a}n number of two vertex-disjoint paths of odd order at least 4. Together with previous works, we determine the exact Tur\'an numbers of two vertex-disjoint paths completely. This confirms the first $k=2$ case of a conjecture proposed by Yuan and Zhang in 2021, which generalizes the Tur\'an number formula of paths due to Faudree-Schelp, and Kopylov in a broader setting. Our main tools include a refinement of P\'{o}sa's rotation lemma, a stability result of Kopylov's theorem on cycles, and a recent inequality on circumference, minimum degree, and clique number of a 2-connected graph.