Mapping sparse signed graphs to (K2k,M) $({K}_{2k},M)$

Abstract

AbstractA homomorphism of a signed graph to is a mapping of vertices and edges of to (respectively) vertices and edges of such that adjacencies, incidences, and the signs of closed walks are preserved. We observe in this work that, for , the ‐coloring problem of a given graph can be captured by homomorphism to from a signed bipartite graph that is built from . Here assigns a negative sign to the edges of a perfect matching and a positive sign to the rest. Motivated by these reformulations and in connection with results on 3‐colorings of planar graphs, such as Grötzsch's theorem, we prove that any signed graph with the maximum average degree strictly less than admits a homomorphism to . For , we show that the maximum average degree being strictly less than 3 would suffice for a signed graph to admit a homomorphism to . Both of these bounds are tight. We discuss applications of our work to signed planar graphs and its connection to the study of homomorphisms of 2‐edge‐colored graphs. Among a number of interesting questions that are left open, a notable one is a possible extension of Steinberg's conjecture for the class of signed bipartite planar graphs.