Global cycle properties in graphs with large minimum clustering coefficient

Global Cycle Properties in Graphs with Large Minimum Clustering Coefficient

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 coefficien...

Cycle lengths in graphs with large minimum degree

Our main result is the following theorem. Let k 2 be an integer, G be a graph of su¢ ciently large order n; and (G) n=k: Then: (i)G contains a cycle of length t for every even integer t 2 [4; (G) + 1] : (ii) If G is nonbipartite then G contains a cycle of length t for every odd integer t 2 [2k 1; (G) + 1] ; unless k 6 and G belongs to a known exceptional class.

Generating Random Graphs with Tunable Clustering Coefficient

Global minimum cuts in surface embedded graphs

We give a deterministic algorithm to find the minimum cut in a surface-embedded graph in near-linear time. Given an undirected graph embedded on an orientable surface of genus g, our algorithm computes the minimum cut in gO(g)n log log n time, matching the running time of the fastest algorithm known for planar graphs, due to Łącki and Sankowski, for any constant g. Indeed, our algorithm calls Ł...

Properties of Clustering Coefficient Inrandom Agent