Broadcast chromatic numbers of graphs

A function pi : V -> {1,..., k} is a broadcast coloring of order k if pi(u) = pi(v) implies that the distance between a and v is more than pi(u). The minimum order of a broadcast coloring is called the broadcast chromatic number of G; and is denoted chi(b)(G). In this paper we introduce this coloring and study its properties. In particular, we explore the relationship with the vertex cover and chromatic numbers. While there is a polynomial-time algorithm to determine whether chi(b)(G) <= 3, we show that it is NP-hard to determine if chi(b)(G) <= 4. We also determine the maximum broadcast chromatic number of a tree, and show that the broadcast chromatic number of the infinite grid is finite.