Random Even Graphs

Random Even Graphs

We study a random even subgraph of a finite graph G with a general edge-weight p ∈ (0, 1). We demonstrate how it may be obtained from a certain random-cluster measure on G, and we propose a sampling algorithm based on coupling from the past. A random even subgraph of a planar lattice undergoes a phase transition at the parameter-value 12pc, where pc is the critical point of the q = 2 random-clu...

Random even graphs and the Ising model

We explore the relationship between the Ising model with inverse temperature β, the q = 2 random-cluster model with edge-parameter p = 1− e−2β , and the random even subgraph with edge-parameter 12p. For a planar graph G, the boundary edges of the + clusters of the Ising model on the planar dual of G forms a random even subgraph of G. A coupling of the random even subgraph of G and the q = 2 ran...

An Extremal Problem For Random Graphs And The Number Of Graphs With Large Even-Girth

We study the maximal number of edges a C2k-free subgraph of a random graph Gn;p may have, obtaining best possible results for a range of p = p(n). Our estimates strengthen previous bounds of F uredi 12] and Haxell, Kohayakawa, and Luczak 13]. Two main tools are used here: the rst one is an upper bound for the number of graphs with large even-girth, i.e., graphs without short even cycles, with a...

Even cycles in graphs

Even pairs in Artemis graphs

Everett et al. [2] defined a stretcher to be a graph whose edge set can be partitioned into two disjoint triangles and three vertex disjoint paths, each with an endpoint in both triangles. They also conjectured that graphs with no odd hole, no antihole and no stretcher (called Artemis graphs) may be reduced to a clique by successive contractions of even pairs. To date, no proof exists that Arte...