Even and Odd Cycles Passing a Given Edge or a Vertex

TL;DR

This paper establishes sufficient conditions for the existence of even or odd cycles passing through specific vertices or edges in 2-connected and 2-edge connected graphs, expanding understanding of cycle structures in such graphs.

Contribution

It provides new sufficient conditions for the existence of even or odd cycles passing through given vertices or edges in 2-connected and 2-edge connected graphs.

Findings

Every edge in a 2-connected k-regular graph with k ≥ 3 is contained in an even cycle.

In a 2-edge connected graph, a vertex with odd degree is contained in an even cycle.

Conditions for the existence of odd or even circuits passing through specific vertices or edges.

Abstract

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, $k \geq 3$, 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.