The maximum size of a graph with prescribed order, circumference and minimum degree

TL;DR

This paper refines the understanding of the maximum size of graphs with given order, circumference, and minimum degree, revealing that the maximum is sometimes achieved by graphs with higher minimum degree than prescribed.

Contribution

It determines the maximum size of graphs with specified order, circumference, and minimum degree, improving previous bounds and addressing cases where the maximum is attained by graphs with higher minimum degree.

Findings

Exact maximum size for given parameters is identified.

The maximum size may be achieved by graphs with higher minimum degree.

Results extend to clique versions and longest path problems.

Abstract

Erd\H{o}s determined the maximum size of a nonhamiltonian graph of order $n$ and minimum degree at least $k$ in 1962. Recently, Ning and Peng generalized. Erd\H{o}s' work and gave the maximum size $h(n,c,k)$ of graphs with prescribed order $n$, circumference $c$ and minimum degree at least $k.$ But for some triples $n,c,k,$ the maximum size is not attained by a graph of minimum degree $k.$ For example, $h(15,14,3)=77$ is attained by a unique graph of minimum degree $7,$ not $3.$ In this paper we obtain more precise information by determining the maximum size of a graph with prescribed order, circumference and minimum degree. Consequently we solve the corresponding problem for longest paths. All these results on the size of graphs have clique versions.