Extremal problems of Erdős, Faudree, Schelp and Simonovits on paths and cycles

Abstract

For positive integers n>d≥k, let ϕ(n,d,k) denote the least integer ϕ such that every n-vertex graph with at least ϕ vertices of degree at least d contains a path on k+1 vertices. Many years ago, Erdős, Faudree, Schelp and Simonovits proposed the study of the function ϕ(n,d,k), and conjectured that for any positive integers n>d≥k, it holds that ϕ(n,d,k)≤⌊k−12⌋⌊nd+1⌋+ϵ, where ϵ=1 if k is odd and ϵ=2 otherwise. In this paper we determine the values of the function ϕ(n,d,k) exactly. This confirms the above conjecture of Erdős et al. for all positive integers k≠4 and in a corrected form for the case k=4. Our proof utilizes, among others, a lemma of Erdős et al. [3], a theorem of Jackson [6], and a (slight) extension of a very recent theorem of Kostochka, Luo and Zirlin [7], where the latter two results concern maximum cycles in bipartite graphs. Moreover, we construct examples to provide answers to two closely related questions raised by Erdős et al.