Even graphs

Even cycles in 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...

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...

Even Cycles in Directed Graphs

It is proved that every strongly connected directed graph with n nodes and at least ⌊(n + 1)/4⌋ edges must contain an even cycle. This is best possible, and the structure of extremal graphs is discussed.

Strongly even cycle decomposable graphs

We introduce the notion of an Eulerian graph being strongly evencycle decomposable, and note that a few basic classes have this property. Our main results are that several fundamental composition operations that preserve the property of being Eulerian also preserve the property of being strongly even-cycle decomposable. As an application, our composition operations easily imply that all Euleria...