Winding number and circular 4-coloring of signed graphs
share
Concerning the recent notion of circular chromatic number of signed graphs, for each given integer k we introduce two signed bipartite graphs, each on 2k^2-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, M_ℓ(C_2k+1). In the course of proving our result, we also obtain a simple proof of the fact that M_ℓ(C_2k+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.
READ FULL TEXT