Integer flows and cycle covers

AbstractResults related to integer flows and cycle covers are presented. A cycle cover of a graph G is a collection C of cycles of G which covers all edges of G; C is called a cycle m-cover of G if each edge of G is covered exactly m times by the members of C. By using Seymour's nowhere-zero 6-flow theorem, we prove that every bridgeless graph has a cycle 6-cover associated to covering of the edges by 10 even subgraphs (an even graph is one in which each vertex is of even degree). This result together with the cycle 4-cover theorem implies that every bridgeless graph has a cycle m-cover for any even number m ≥ 4. We also prove that every graph with a nowhere-zero 4-flow has a cycle cover C such that the sum of lengths of the cycles in C is at most |E(G)| + |V(G)| − 2, unless G belongs to a very special class of graphs

A note on 5-cycle double covers

The strong cycle double cover conjecture states that for every circuit $C$ of a bridgeless cubic graph $G$, there is a cycle double cover of $G$ which contains $C$. We conjecture that there is even a 5-cycle double cover $S$ of $G$ which contains $C$, i.e. $C$ is a subgraph of one of the five 2-regular subgraphs of $S$. We prove a necessary and sufficient condition for a 2-regular subgraph to be contained in a 5-cycle double cover of $G$

Short Cycle Covers of Cubic Graphs and Graphs with Minimum Degree Three

The Shortest Cycle Cover Conjecture of Alon and Tarsi asserts that the edges of every bridgeless graph with $m$ edges can be covered by cycles of total length at most $7m/5=1.400m$. We show that every cubic bridgeless graph has a cycle cover of total length at most $34m/21\approx 1.619m$ and every bridgeless graph with minimum degree three has a cycle cover of total length at most $44m/27\approx 1.630m$

Contractors for flows

We answer a question raised by Lov\'asz and B. Szegedy [Contractors and connectors in graph algebras, J. Graph Theory 60:1 (2009)] asking for a contractor for the graph parameter counting the number of B-flows of a graph, where B is a subset of a finite Abelian group closed under inverses. We prove our main result using the duality between flows and tensions and finite Fourier analysis. We exhibit several examples of contractors for B-flows, which are of interest in relation to the family of B-flow conjectures formulated by Tutte, Fulkerson, Jaeger, and others.Comment: 22 pages, 1 figur