Global properties of graphs with local degree conditions

TL;DR

This paper investigates how local degree conditions like Dirac and Ore influence global graph properties such as connectivity, diameter, and planarity, revealing insights into the structure of graphs satisfying these local constraints.

Contribution

It provides a comprehensive analysis of the global properties of graphs with local Dirac and Ore conditions, highlighting their structural implications.

Findings

Graphs with local Dirac conditions tend to be highly connected.

Local Ore conditions influence the cycle structure of graphs.

Certain global properties like planarity are affected by local degree constraints.

Abstract

Let P be a graph property. A graph is locally P if the subgraph induced by the open neighbourhood of every vertex has property P. A graph has the Dirac condition if the minimum degree of every vertex is at least half the order of the graph and it satisfies the Ore condition if the sum of the degrees of any pair of non-adjacent vertices is at least the order of the graph. In this paper we study global properties of graphs that possess the local Dirac and Ore conditions. We focus on the connectivity, edge-connectivity, diameter, planarity and cycle structure of these graphs.