The formula for Turán number of spanning linear forests

Abstract

Let F be a family of graphs. The Turán number ex(n;F) is defined to be the maximum number of edges in a graph of order n that is F-free. In 1959, Erdős and Gallai determined the Turán number of Mk+1 (a matching of size k+1) as follows: ex(n;Mk+1)=max2k+12,n2−n−k2.Since then, there has been a lot of research on Turán number of linear forests.A linear forest is a graph whose connected components are all paths or isolated vertices. Let Ln,k be the family of all linear forests of order n with k edges. In this paper, we prove that ex(n;Ln,k)=maxk2,n2−n−k−122+c,where c=0 if k is odd and c=1 otherwise. This determines the maximum number of edges in a non-Hamiltonian graph with given Hamiltonian completion number and also solves two open problems in Wang and Yang (2019) as special cases.Moreover, we show that our main theorem implies Erdős–Gallai Theorem and also gives a short new proof for it by the closure and counting techniques. Finally, we generalize our theorem to a conjecture which implies the famous Erdős Matching Conjecture.