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

A graph homomorphism is a vertex map which carries edges from a source graph to edgesin 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 hasmaximal total edge weight, under the condition that the subgraph has a homomorphism to H ;note that for H = Kk this problem is equivalent to MAX k-CUT. Färnqvist et al. have introduceda parameter on the space of graphs that allows close study of the approximability properties ofMAX H -COL. Specifically, it can be used to extend previously known (in)approximability resultsto larger classes of graphs. Here, we investigate the properties of this parameter on circularcomplete graphs Kp/q, where 2 ≤ p/q ≤ 3. The results are extended to K4-minor-free graphsand graphs with bounded maximum average degree. We also consider connections with Šámal’swork on fractional covering by cuts: we address, and decide, two conjectures concerning cubicalchromatic numbers.