Global properties of graphs with local degree conditions

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.

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