On the chromatic numbers of signed triangular and hexagonal grids
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A signed graph is a simple graph with two types of edges. Switching a vertex v of a signed graph corresponds to changing the type of each edge incident to v. A homomorphism from a signed graph G to another signed graph H is a mapping φ: V(G) → V(H) such that, after switching any number of the vertices of G, φ maps every edge of G to an edge of the same type in H. The chromatic number χ_s(G) of a signed graph G is the order of a smallest signed graph H such that there is a homomorphism from G to H. We show that the chromatic number of signed triangular grids is at most 10 and the chromatic number of signed hexagonal grids is at most 4.
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