Global cycle properties in graphs with large minimum clustering coefficient

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.