A radio labeling of a graph G is a mapping f : and d(u, v) the distance between u and v in G. The radio number of G is the smallest integer k such that G has a radio labeling f with max{f (v) : v ∈ V (G)} = k. We give a necessary and sufficient condition for a lower bound on the radio number of trees to be achieved, two other sufficient conditions for the same bound to be achieved by a tree, and an upper bound on the radio number of trees. Using these, we determine the radio number for three families of trees.
Abstract. A hamiltonian coloring c of a graph G of order p is an assignment of colors to the vertices of G such that D(u, v) + |c(u)−c(v)| ≥ p − 1 for every two distinct vertices u and v of G, where D(u, v) denotes the detour distance between u and v. The value hc(c) of a hamiltonian coloring c is the maximum color assigned to a vertex of G. The hamiltonian chromatic number, denoted by hc(G), is the min{hc(c)} taken over all hamiltonian coloring c of G. In this paper, we present a lower bound for the hamiltonian chromatic number of block graphs and give a sufficient condition to achieve the lower bound. We characterize symmetric block graphs achieving this lower bound. We present two algorithms for optimal hamiltonian coloring of symmetric block graphs.
Let G = (V(G), E(G)) be a connected graph. For integers j ≥ k, L( j, k)-labeling of a graph G is an integer labeling of the vertices in V such that adjacent vertices receive integers which differ by at least j and vertices which are at distance two apart receive labels which differ by at least k. In this paper we discuss L(2, 1)-labeling (or distance two labeling) in the context of some graph operations.
A radio labeling of a graph is a function from the vertex set to the set of nonnegative integers such that , where and are diameter and distance between and in graph , respectively. The radio number of is the smallest number such that has radio labeling with . We investigate radio number for total graph of paths.
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