Short cycle covers of graphs and nowhere-zero flows

Abstract

A shortest cycle cover of a graph G is a family of cycles which together cover all the edges of G and the sum of their lengths is minimum. In this article we present upper bounds to the length of shortest cycle covers, associated with the existence of two types of nowhere-zero flows-circular flows and Fano flows. Fano flows, or Fano colorings, are nowhere-zero ℤ23 -flows on cubic graphs, with certain restrictions on the flow values meeting at a vertex. Such flows are conjectured to exist on every bridgless cubic graph. © 2010 Wiley Periodicals, Inc.