Colouring negative exact-distance graphs of signed graphs

TL;DR

This paper investigates the chromatic number of negative exact-distance graphs derived from signed graphs, focusing on planar cases and their relation to generalized coloring numbers, extending concepts of graph homomorphisms.

Contribution

It introduces the study of negative exact-distance graphs of signed graphs and explores their chromatic numbers, linking them to homomorphism problems and generalized coloring numbers.

Findings

Chromatic number bounds for planar signed graphs' negative exact-distance graphs.

Relation established between these chromatic numbers and generalized coloring numbers.

Extension of homomorphism concepts to signed graph distance graphs.

Abstract

The $k$-th exact-distance graph, of a graph $G$ has $V(G)$ as its vertex set, and $xy$ as an edge if and only if the distance between $x$ and $y$ is (exactly) $k$ in $G$. We consider two possible extensions of this notion for signed graphs. Finding the chromatic number of a negative exact-distance square of a signed graph is a weakening of the problem of finding the smallest target graph to which the signed graph has a sign-preserving homomorphism. We study the chromatic number of negative exact-distance graphs of signed graphs that are planar, and also the relation of these chromatic numbers with the generalised colouring numbers of the underlying graphs. Our results are related to a theorem of Alon and Marshall about homomorphisms of signed graphs.