Graph Homomorphisms, Circular Colouring, and Fractional Covering by H-cuts

TL;DR

This paper explores the properties of a graph parameter related to homomorphisms and coloring, extending results to specific graph classes and addressing conjectures on fractional covering and chromatic numbers.

Contribution

It analyzes the parameter on circular complete graphs and extends findings to K_4-minor-free graphs and graphs with bounded average degree, also addressing conjectures on cubical chromatic numbers.

Findings

Extended properties of the parameter to circular complete graphs K_{p/q} for 2 <= p/q <= 3

Applied results to K_4-minor-free graphs and graphs with bounded maximum average degree

Resolved two conjectures on fractional covering by cuts and cubical chromatic numbers

Abstract

A graph homomorphism is a vertex map which carries edges from a source graph to edges in a target graph. The instances of the Weighted Maximum H-Colourable Subgraph problem (MAX H-COL) are edge-weighted graphs G and the objective is to find a subgraph of G that has maximal total edge weight, under the condition that the subgraph has a homomorphism to H; note that for H=K_k this problem is equivalent to MAX k-CUT. Farnqvist et al. have introduced a parameter on the space of graphs that allows close study of the approximability properties of MAX H-COL. Specifically, it can be used to extend previously known (in)approximability results to larger classes of graphs. Here, we investigate the properties of this parameter on circular complete graphs K_{p/q}, where 2 <= p/q <= 3. The results are extended to K_4-minor-free graphs and graphs with bounded maximum average degree. We also consider connections with Samal's work on fractional covering by cuts: we address, and decide, two conjectures concerning cubical chromatic numbers.