Global cycle properties in graphs with large minimum clustering coefficient

Abstract

Let P be a graph property. A graph G is said to be locally P (closed locally P) if the subgraph induced by the open neighbourhood (closed neighbourhood, respectively) of every vertex in G has property P. The clustering coefficient of a vertex is the proportion of pairs of its neighbours that are themselves neighbours. The minimum clustering coefficient of G is the smallest clustering coefficient among all vertices of G. Let H be a subgraph of a graph G and let S ⊆ V (H). We say that H is a strongly induced subgraph of G with attachment set S, if H is an induced subgraph of G and the vertices of V (H) − S are not incident with edges that are not in H. A graph G is fully cycle extendable if every vertex of G lies in a triangle and for every nonhamiltonian cycle C of G, there is a cycle of length |V (C)| + 1 that contains the vertices of C. A complete 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 forbidden strongly induced subgraphs (with specified attachment sets). Moreover, it is shown that all locally connected graphs with ∆ ≤ 6 and sufficiently large minimum clustering coefficient are weakly pancylic, thereby proving Ryjáček's conjecture for this class of graphs.

Global properties of graphs with local degree conditions

“…In either case there is a vertex v j , where j ∈ {2, t − 2}, such that v j ∼ {x, v 0 , v 1 }. So, by Lemma 3.2 (7), d ≥ 8. Case 3 Suppose d = 8.…”

“…by the open neighbourhood of every vertex in G is an isometric subgraph of G. It was shown in [6] that the problem of deciding whether a locally isometric graph is hamiltonian is NP-complete for graphs with maximum degree at most 8. Locally connected graphs that are sufficiently 'locally dense' were introduced in [7]. The clustering coefficient of a vertex in a graph is the proportion of pairs of neighbours of the vertex that are themselves neighbours (see [23]).…”

“…The clustering coefficient of a vertex in a graph is the proportion of pairs of neighbours of the vertex that are themselves neighbours (see [23]). The minimum clustering coefficient of a graph G is the smallest clustering coefficient of its vertices, taken over all vertices (see [7]). It was shown in [7], that even for connected locally connected graphs with minimum clustering coefficient as large as 1/2, hamiltonicity of the graph is not guaranteed.…”

“…The minimum clustering coefficient of a graph G is the smallest clustering coefficient of its vertices, taken over all vertices (see [7]). It was shown in [7], that even for connected locally connected graphs with minimum clustering coefficient as large as 1/2, hamiltonicity of the graph is not guaranteed. Nevertheless, it was shown that many of these graphs have a rich cycle structure.…”