Radio Numbers of Certain m-Distant Trees
Radio coloring of a graph G with diameter d is an assignment f of positive integers to the vertices of G such that |f(u)-f(v)|≥1+d-d(u,v), where u and v are any two distinct vertices of G and d(u,v) is the distance between u and v. The number max {f(u):u∈V(G)} is called the span of f. The minimum of spans over all radio colorings of G is called radio number of G, denoted by rn(G). An m-distant tree T is a tree in which there is a path P of maximum length such that every vertex in V(T)∖V(P) is at the most distance m from P. This path P is called a central path. For every tree T, there is an integer m such that T is a m-distant tree. In this paper, we determine the radio number of some m-distant trees for any positive integer m≥2, and as a consequence of it, we find the radio number of a class of 1-distant trees (or caterpillars).