Injective Colorings with Arithmetic Constraints

Abstract

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 $$c:V(G)\rightarrow \mathbb {Z}$$c:V(G)?Z of a graph $$G$$G is an additive coloring if for every $$uv, vw$$uv,vw in $$E(G)$$E(G), $$c(u)+c(w)\ne 2c(v)$$c(u)+c(w)?2c(v). The smallest integer $$k$$k such that an injective (resp. additive) coloring of a given graph $$G$$G exists with $$k$$k colors (resp. colors in $$\{1,\ldots ,k\}$${1,?,k}) is called the injective (resp. additive) chromatic number (resp. index). They are denoted by $$\chi _i(G)$$?i(G) and $$\chi '_a(G)$$?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 $$\chi '_a(G)$$?a?(G). We prove that it can be computed in polynomial time for trees. We also prove that for bounded treewidth graphs, to decide whether $$\chi '_a(G)\le k$$?a?(G)≤k, for a fixed $$k$$k, can be done in polynomial time. On the other hand, we show that for cubic graphs it is NP-complete to decide whether $$\chi '_a(G)\le 4$$?a?(G)≤4. We also prove that for every $$\epsilon >0$$∈>0 there is a polynomial time approximation algorithm with approximation factor $$n^{1/3+\epsilon }$$n1/3+∈ for $$\chi '_a(G)$$?a?(G), when restricted to split graphs. However, unless $$\mathsf P =\mathsf NP $$P=NP, for every $$\epsilon >0$$∈>0 there is no polynomial time approximation algorithm with approximation factor $$n^{1/3-\epsilon }$$n1/3-∈ for $$\chi '_a(G)$$?a?(G), even when restricted to split graphs.