Even cycles in graphs

Even cycles in graphs

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.

Vanishing ideals over graphs and even cycles

Let X be an algebraic toric set in a projective space over a finite field. We study the vanishing ideal, I(X), of X and show some useful degree bounds for a minimal set of generators of I(X). We give an explicit combinatorial description of a set of generators of I(X), when X is the algebraic toric set associated to an even cycle or to a connected bipartite graph with pairwise vertex disjoint e...

Finding Even Cycles Even Faster

We describe efficient algorithms for finding even cycles in undirected graphs. Our main results are the following: (i) For every k ≥ 2, there is an O(V ) time algorithm that decides whether an undirected graph G = (V,E) contains a simple cycle of length 2k and finds one if it does. (ii) There is an O(V ) time algorithm that finds a shortest even cycle in an undirected graph G = (V,E).

A Note on Graphs Without Short Even Cycles

In this note, we show that any n-vertex graph without even cycles of length at most 2k has at most 12n 1+1/k + O(n) edges, and polarity graphs of generalized polygons show that this is asymptotically tight when k ∈ {2, 3, 5}.