# Extensions of a theorem of Erd\H{o}s on nonhamiltonian graphs

## TL;DR

This paper extends Erdős's theorem on nonhamiltonian graphs by identifying extremal graphs that maximize copies of any fixed subgraph, and provides a detailed classification of near-extremal graphs with high minimum degree and edge count.

## Contribution

It generalizes Erdős's result by showing extremal graphs maximize copies of any fixed graph and offers a stronger stability classification for graphs close to the extremal edge count.

## Key findings

- H_{n,d} maximizes copies of any fixed graph F for large n.
- Classifies nonhamiltonian graphs with high minimum degree and near-extremal edges.
- Provides a general theorem describing graphs with many copies of K_k.

## Abstract

Let $n, d$ be integers with $1 \leq d \leq \left \lfloor \frac{n-1}{2} \right \rfloor$, and set $h(n,d):={n-d \choose 2} + d^2$. Erd\H{o}s proved that when $n \geq 6d$, each nonhamiltonian graph $G$ on $n$ vertices with minimum degree $\delta(G) \geq d$ has at most $h(n,d)$ edges. He also provides a sharpness example $H_{n,d}$ for all such pairs $n,d$. Previously, we showed a stability version of this result: for $n$ large enough, every nonhamiltonian graph $G$ on $n$ vertices with $\delta(G) \geq d$ and more than $h(n,d+1)$ edges is a subgraph of $H_{n,d}$. In this paper, we show that not only does the graph $H_{n,d}$ maximize the number of edges among nonhamiltonian graphs with $n$ vertices and minimum degree at least $d$, but in fact it maximizes the number of copies of any fixed graph $F$ when $n$ is sufficiently large in comparison with $d$ and $|F|$. We also show a stronger stability theorem, that is, we classify all nonhamiltonian $n$-graphs with $\delta(G) \geq d$ and more than $h(n,d+2)$ edges. We show this by proving a more general theorem: we describe all such graphs with more than ${n-(d+2) \choose k} + (d+2){d+2 \choose k-1}$ copies of $K_k$ for any $k$.