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

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, 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 with prescribed order and minimum degree. Motivated by recent work on feasible graph parameters in \cite{ALNS2023}, 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 analogues. 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.