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

Abstract

<lb>A graph homomorphism is a vertex map which carries edges from a source graph to edges<lb>in a target graph. The instances of the Weighted Maximum H-Colourable Subgraph problem<lb>(MAX H -COL) are edge-weighted graphs G and the objective is to find a subgraph of G that has<lb>maximal total edge weight, under the condition that the subgraph has a homomorphism to H ;<lb>note that for H = Kk this problem is equivalent to MAX k-CUT. Färnqvist et al. have introduced<lb>a parameter on the space of graphs that allows close study of the approximability properties of<lb>MAX H -COL. Specifically, it can be used to extend previously known (in)approximability results<lb>to larger classes of graphs. Here, we investigate the properties of this parameter on circular<lb>complete graphs Kp/q, where 2 ≤ p/q ≤ 3. The results are extended to K4-minor-free graphs<lb>and graphs with bounded maximum average degree. We also consider connections with Šámal’s<lb>work on fractional covering by cuts: we address, and decide, two conjectures concerning cubical<lb>chromatic numbers.<lb>

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Cite this paper

@article{Engstrm2009GraphHC, title={Graph Homomorphisms, Circular Colouring, and Fractional Covering by H-cuts}, author={Robert Engstr{\"{o}m and Tommy F{\"a}rnqvist and Peter Jonsson and Johan Thapper}, journal={CoRR}, year={2009}, volume={abs/0904.4600} }