The maximum size of a nonhamiltonian graph with given order and connectivity

TL;DR

This paper refines the understanding of the maximum size of nonhamiltonian graphs with given order and connectivity, identifying extremal graphs and addressing related nontraceable graphs.

Contribution

It determines the maximum size and extremal graphs for nonhamiltonian graphs with specified order and connectivity, improving previous bounds.

Findings

Exact maximum size for given order and connectivity

Identification of extremal graphs with higher connectivity

Extension to nontraceable graphs

Abstract

Motivated by work of Erd\H{o}s, Ota determined the maximum size $g(n,k)$ of a $k$-connected nonhamiltonian graph of order $n$ in 1995. But for some pairs $n,k,$ the maximum size is not attained by a graph of connectivity $k.$ For example, $g(15,3)=77$ is attained by a unique graph of connectivity $7,$ not $3.$ In this paper we obtain more precise information by determining the maximum size of a nonhamiltonian graph of order $n$ and connectivity $k,$ and determining the extremal graphs. Consequently we solve the corresponding problem for nontraceable graphs.