Abstract
Abstract 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 . 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. We also consider connections with Samal's work on fractional covering by cuts: we address, and decide, two conjectures concerning cubical chromatic numbers.
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