Even and Odd Cycles Passing a Given Edge or a Vertex

الأصل · EN

In this paper we provide some sufficient conditions for the existence of an odd or even cycle that passing a given vertex or an edge in 2-connected or 2-edge connected graphs. We provide some similar conditions for the existence of an odd or even circuit that passing a given vertex or an edge in 2-edge connected graphs. We show that if G is a 2-connected k-regular graph, k ≥ 3, then every edge of G is contained in an even cycle. We also prove that in a 2-edge connected graph, if a vertex has odd degree, then there is an even cycle containing this vertex.