Homomorphisms of Cayley graphs and Cycle Double Covers

Radek Hušek,Robert Šámal

doi:10.37236/8456

Abstract

We study the following conjecture of Matt DeVos: If there is a graph homomorphism from a Cayley graph $\mathrm{Cay}(M, B)$ to another Cayley graph $\mathrm{Cay}(M', B')$ then every graph with an $(M,B)$-flow has an $(M',B')$-flow. This conjecture was originally motivated by the flow-tension duality. We show that a natural strengthening of this conjecture does not hold in all cases but we conjecture that it still holds for an interesting subclass of them and we prove a partial result in this direction. We also show that the original conjecture implies the existence of an oriented cycle double cover with a small number of cycles.

Highlights

For an abelian group M, an M -flow φ on a directed graph G = (V, E) is a mapping E → M such that the oriented sum around every vertex v is zero: φ(vw) − φ(uv) = 0. vw∈E uv∈EWe say that M -flow φ is an (M, B)-flow if φ(e) ∈ B for all e ∈ E
For every graph G the strong homomorphism property holds for group Zk where k is minimal such that G admits a nowhere-zero Zkflow
Observation 12 leads us to the definition of a universal mapping such that if homomorphism property (HP) or strong homomorphism property (SHP) holds for this mapping, it holds for every other mapping

Summary

Introduction

Given an abelian group M and its symmetric subset B ⊆ M we let Cay(M, B) denote the graph with vertex set M and with edges {uv : u, v ∈ M, v − u ∈ B}. The Cayley graph Cay(M, B) ∼= Kk/d is frequently denoted as the circular clique ( as circular complete graph); Cay(M , B ) ∼= Kk /d This graph is important in the study of circular coloring, the dual concept of circular flows. It is known that there is a homomorphism from Kk/d to Kk /d if and only if k/d k /d (see, for example, [11] or its references) Both sides of the conjectured implication are in this setting equivalent to k/d k /d , the conjecture holds for these combinations of groups and their subsets. We thank the anonymous referee who kindly suggested that our example using circular (2k + 1)/k-flows extends to any two circular flows

New Framework

Universal objects

Partial results

Connection to CDC