Braodcast Chromatic Numbers of Graphs

A function π : V → {1, . . . , k} is a broadcast coloring of order k if π(u) = π(v) implies that the distance between u and v is more than π(u). The minimum order of a broadcast coloring is called the broadcast chromatic number of G, and is denoted χb(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 χb(G) ≤ 3, we show that it is NP-hard to determine if χb(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.