Winding number and circular 4-coloring of signed graphs

TL;DR

This paper introduces specific bipartite signed graphs with a circular chromatic number of 4, linking algebraic topology and graph coloring, and provides elementary proofs for related coloring properties.

Contribution

It constructs bipartite signed graphs with prescribed properties and offers elementary proofs connecting algebraic topology to graph coloring.

Findings

Constructed bipartite signed graphs with shortest negative cycle of length 2k.

Proved these graphs have circular chromatic number 4.

Provided elementary proofs relating algebraic topology to graph coloring.

Abstract

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, $M_{\ell}(C_{2k+1})$. In the course of proving our result, we also obtain a simple proof of the fact that $M_{\ell}(C_{2k+1})$ 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.