Extensions of Erd\H{o}s's 1962 theorem on non-Hamiltonian graphs

TL;DR

This paper extends Erdős's 1962 theorem on non-Hamiltonian graphs, providing generalized extremal results, solving open problems, and introducing new proof techniques for related graph properties.

Contribution

It offers a unified generalization of Erdős's theorem and spectral analogs, solving several open problems and extending results on Hamiltonian properties.

Findings

Generalized Erdős's theorem for non-Hamiltonian graphs.

Solved open problems from 2016 related to extremal graph properties.

Extended results to Hamiltonian-connected graphs and Hamilton cycles.

Abstract

For a positive integer $k$, a graph property $\mathcal{H}$, and a graph parameter $\mathcal{P}$, let $\operatorname{ex}_{\mathcal{P}}(n, \mathcal{H}; \delta \geq k)$ denote the maximum value of $\mathcal{P}$ over all $n$-vertex graphs with minimum degree at least $k$ that do not possess the property $\mathcal{H}$. The corresponding extremal families are denoted by $\operatorname{EX}_{\mathcal{P}}(n, \mathcal{H}; \delta \geq k)$. For two disjoint graphs $H_1$ and $H_2$, let $H_1 \cup H_2$ denote their (disjoint) union, i.e., the graph with vertex set $V(H_1) \cup V(H_2)$ and edge set $E(H_1) \cup E(H_2)$; and let $H_1 \vee H_2$ denote their join. In 1962, Erd\H{o}s established a classical theorem on the maximum number of edges in a non-Hamiltonian graph of given order and minimum degree. Motivated by recent work on feasible graph parameters in \cite{Ai2023}, we prove several extensions of Erd\H{o}s's 1962 theorem on non-Hamiltonian graphs. The first result gives a common generalization of the extremal theorem due to Erd\H{o}s and its spectral analogs. As direct applications, we obtain complete solutions to open problems raised in the literature since 2016, thereby improving nearly all related prior results in this direction. Our proof technique differs somewhat from those in \cite{MR3539577,MR3556876}. We also prove an analog theorem for the Hamiltonian-connected property and obtain a result which extends the theorem of F\"{u}redi, Kostochka, and Luo \cite{MR3843180} on Hamilton cycles.