The <i>L(j, k)</i>-Labeling Number and Circular <i>L(j,k)</i>-Labeling Number of Distance Graph D<sub>n</sub>(1,5)

For<i> j ≤ k</i>, the <i>L(j, k)</i>- labeling problem arose form the code assignment problem of the wireless network. That is, let n,j,k be non-negative real numbers with j ≤ k, an n-<i>L(j, k)</i>-labeling of a graph G is mapping f: V(G)→[0, n] such that |f(u)-f(v)| ≥ j if d(u, v)=1, and |f(u)-f(v)|≥k if t d(u, v)=2. The span of f is the difference between the maximum and minimum labeling numbers assigned by <i>f</i>. The <i>L(j, k)</i>-labeling number of graph G, denoted by λ<sub>(j,k)</sub> (G), is the minimum span of all<i> L(j, k)</i>-labeling of G. The infinite distance graph, denoted by D (d<sub>1</sub>,d<sub>2</sub>,…,d<sub>k</sub>), has the set Z of integers as a vertex set and in which two vertices i,j∈Z are adjacent if and only if |i-j|∈D. The finite distance graph, denoted by D<sub>n</sub> (d<sub>1</sub>,d<sub>2</sub>,…,d<sub>k</sub>), is the subgraph of D(d<sub>1</sub>,d<sub>2</sub>,…,d<sub>k</sub>) induced by vertices {0,1,…,n-1}. This paper determines the <i>L(j, k)</i>-labeling number and the circular <i>L(j, k)</i>-labeling number of distance graph D<sub>n</sub> (1,5) for 2j ≤ k.