Radio Numbers of Some Caterpillars

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 the 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 most distance m from P. This path P is called a central path. Every tree can be represented as an m-distant tree for some non-negative integer m. In this paper, we find the radio number of a class of 1-distant trees (or caterpillars).