Extremal problems of Erd\H{o}s, Faudree, Schelp and Simonovits on paths and cycles

TL;DR

This paper determines the exact values of the function (n,d,k) related to paths in graphs, confirming a longstanding conjecture for all but one case, and extends related theorems on bipartite graph cycles.

Contribution

It exactly computes (n,d,k), confirming Erd
51s et al.'s conjecture for all ka0a0 4, and extends existing theorems on bipartite graph cycles.

Findings

Confirmed the conjecture for all ka0a0 4 with a correction for k=4.

Determined (n,d,k) exactly for all positive integers n,d,k.

Extended theorems on maximum cycles in bipartite graphs.

Abstract

For positive integers $n>d\geq k$, let $\phi(n,d,k)$ denote the least integer $\phi$ such that every $n$-vertex graph with at least $\phi$ vertices of degree at least $d$ contains a path on $k+1$ vertices. Many years ago, Erd\H{o}s, Faudree, Schelp and Simonovits proposed the study of the function $\phi(n,d,k)$, and conjectured that for any positive integers $n>d\geq k$, it holds that $\phi(n,d,k)\leq \lfloor\frac{k-1}{2}\rfloor\lfloor\frac{n}{d+1}\rfloor+\epsilon$, where $\epsilon=1$ if $k$ is odd and $\epsilon=2$ otherwise. In this paper we determine the values of the function $\phi(n,d,k)$ exactly. This confirms the above conjecture of Erd\H{o}s et al. for all positive integers $k\neq 4$ and in a corrected form for the case $k=4$. Our proof utilizes, among others, a lemma of Erd\H{o}s et al. \cite{EFSS89}, a theorem of Jackson \cite{J81}, and a (slight) extension of a very recent theorem of Kostochka, Luo and Zirlin \cite{KLZ}, 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 Erd\H{o}s et al.