On modular monochromatic colorings of graphs

Abstract

For a nontrivial connected graph G and an integer k > 2, let c : V(G) -→ ℤ_κ be a vertex coloring of G where c(ν) ≠ 0 for at least one vertex ν of G. Then the coloring c induces a new coloring σ : V(G) -→ ℤ_κ of G defined by σ(ν) = ∑u∈N[u] C(u) where N[ν] is the closed neighborhood of ν and addition is performed in ℤ_κ. If ν(u) = σ(ν) = t ∈ ℤ_κ for every two vertices u and v in then the coloring c is called a modular monochromatic (κ, t)-coloring of G. Several results dealing with modular monochromatic (fc, 0)-colorings are presented, particularly the case where κ = 2.