Multi-level and antipodal labelings for certain classes of circulant graphs

A radio k-labeling c of a graph G is a mapping c : V ( G ) → Z + ∪ { 0 } such that d ( u ; v ) + | c ( u ) − c ( v ) | ≥ k + 1 for every two distinct vertices u and v of G, where d(u; v) is the distance between any two vertices u and v of G. The span of a radio k-labeling c is denoted by sp(c) and defined as max { | c ( u ) − c ( v ) | : u ; v ∈ V ( G ) } . The radio labeling is a radio k-labeling when k = d i a m ( G ) . In other words, a radio labeling is a one-to-one function f from V ( G ) to Z + ∪ { 0 } such that | c ( u ) − c ( v ) | ≥ d i a m ( G ) + 1 − d ( u ; v ) for any pair of vertices u, v in G. The radio number of G expressed by rn(G), is the lowest span taken over all radio labelings of the graph. For k = d i a m ( G ) − 1 , a radio k- labeling is called a radio antipodal labeling. An antipodal labeling for a graph G is a function c : V ( G ) → { 0 ; 1 ; 2 ; . . . } such that d ( u ; v ) + | c ( u ) − c ( v ) | ≥ d i a m ( G ) for all u ; v ∈ V ( G ) . The radio antipodal number for G denoted by an(G), is the minimum span of an antipodal labeling admitted by G. In this paper, we investigate the exact value of the radio number and radio antipodal number for the circulant graphs G ( 4 m k + 2 m ; { 1 ; 2 m } ) , when m ≥ 3 is odd. Furthermore, we also determine the lower bound of the radio number for the circulant graphs G ( 4 m k + 2 m ; { 1 ; 2 m } ) , when m ≥ 2 is even.