# Cycle Decompositions and Double Covers of Graphs

## Document Type

Presentation Abstract

10-16-1997

## Abstract

Informally, an Euler tour of a connected graph G is a tracing of the edges of G, with the conditions that you must start and finish at the same vertex, you must trace over each edge exactly once, and you must not lift your pencil from the page. A characterization of graphs that have Euler tours was given by Leonhard Euler in 1736: a graph has an Euler tour if and only if it is connected and all its vertices have even degree. The paper in which Euler gives this characterization is considered to be the first paper in the area of mathematics that we now know as graph theory.

Another way to characterize graphs with an Euler tour is the following: a graph G has an Euler tour if and only if G is connected and admits a cycle decomposition. A cycle decomposition of a graph G is simply a partition of the edge set of G into cycles. There are a number of ways to generalize the notion of a cycle decomposition of a graph; the one that we will concern ourselves with is cycle double covers. A cycle double cover of a graph G is a collection of cycles, C such that every edge of G lies in precisely two cycles of C. The one obvious necessary condition that is required for a graph to have a cycle double cover is that the graph be bridgeless, and in his well known Cycle Double Cover Conjecture, P.D. Seymour asserts that this condition is also sufficient.

This talk will report on the status of the cycle double cover conjecture, as well as provide a survey of results and conjectures concerning cycle double covers and cycle decompositions.