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−1 2 ⌋⌊ n d+1 ⌋ + ǫ, where ǫ = 1 if k is odd and ǫ = 2 otherwise. In this paper we determine the value 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. Besides, we construct examples to provide answers to two closely related questions raised by Erdős et al.