Global cycle properties in graphs with large minimum clustering coefficient

TL;DR

This paper characterizes the cycle structures of certain locally connected graphs with high clustering coefficients and bounded degree, proving a conjecture about their pancyclic properties.

Contribution

It provides a complete structural characterization of locally connected graphs with minimum clustering coefficient 1/2 and maximum degree at most 6, and proves Ryjacek's conjecture for this class.

Findings

Graphs are fully cycle extendable under given conditions.

All such graphs are weakly pancyclic.

The paper confirms Ryjacek's conjecture for this class.

Abstract

The clustering coefficient of a vertex in a graph is the proportion of neighbours of the vertex that are adjacent. The minimum clustering coefficient of a graph is the smallest clustering coefficient taken over all vertices. A complete structural characterization of those locally connected graphs, with minimum clustering coefficient 1/2 and maximum degree at most 6, that are fully cycle extendable is given in terms of strongly induced subgraphs with given attachment sets. Moreover, it is shown that all locally connected graphs with minimum clustering coefficient 1/2 and maximum degree at most 6 are weakly pancyclic, thereby proving Ryjacek's conjecture for this class of locally connected graphs.