Injective Colorings with Arithmetic Constraints

An injective coloring of a graph is a vertex labeling such that two vertices sharing a common neighbor get different labels. In this work we introduce and study what we call additive colorings. An injective coloring c : V (G) → Z of a graph G is an additive coloring if for every uv, vw in E(G), c(u)+ c(w) = 2c(v). The smallest integer k such that an injective (resp. additive) coloring of a given graph G exists with k colors (resp. colors in {1, . . . , k}) is called the injective (resp. additive) chromatic number (resp. index). They are denoted by χi (G) and χ ′ a(G), respectively. In the first part of this work, we present several upper bounds for the additive chromatic index. On the one hand, we prove a super linear upper bound in terms of the injective chromatic number for arbitrary graphs, as well as a linear upper bound for bipartite graphs and trees. Complete graphs are extremal graphs for the super linear bound, while complete balanced bipartite graphs are extremal graphs for the linear bound. On the other hand, we prove a quadratic upper bound in terms of the maximum degree. In the second part, we study the computational complexity of computing χ ′ a(G). We prove that it can be Partially supported by Fondecyt 1100192, Ecos-Conicyt C09E04, Basal Program PBF 03, Beca de Doctorado Conicyt, Nucleo Milenio Información y Coordinación en Redes ICM/FIC P10-024F, French National Research Agency project AGAPE (ANR-09-BLAN-0159-03). N. Astromujoff Departamento de Matemática, Universidad de Chile, Santiago, Chile M. Chapelle · I. Todinca Laboratoire d’Informatique Fondamentale d’Orléans, Université d’Orléans, Orléans, France M. Matamala (B) Departamento de Ingeniería Matemática, Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Santiago, Chile e-mail: [email protected] J. Zamora Departamento de Matemáticas, Universidad Andrés Bello, Santiago, Chile