Note on a new coloring number of a graph
Peter Horák, Jozef Širáň · Journal of Graph Theory · 1980
Abstract The distance coloring number Xd(G) of a graph G is the minimum number n such that every vertex of G can be assigned a natural number m ≤ n and no two vertices at distance i are both assigned i. It is proved that for any natural number n there exists a graph G with Xd(G) = n.