Winding number and circular 4-coloring of signed graphs
Résumé
Concerning the recent notion of circular chromatic number of signed graphs, for each given integer k we introduce two signed bipartite graphs, each on 2k2 − k + 1 vertices, having shortest negative cycle of length 2k, and the circular chromatic number 4. Each of the construction can be viewed as a bipartite analogue of the generalized Mycielski graphs on
odd cycles, Mℓ(C2k+1). In the course of proving our result, we also obtain a simple proof of the fact that Mℓ(C2k+1) and some similar quadrangulations of the projective plane have circular chromatic number 4. These proofs have the advantage that they illuminate, in an elementary manner, the strong relation
between algebraic topology and graph coloring problems.
Domaines
Mathématique discrète [cs.DM]
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CircularChormaticProjectivePlanar4-Face-30June2023.pdf (394.07 Ko)
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