On L(k,k−1,…,1) labeling of triangular lattice

An n-L(k,k−1,…,1) labeling of a simple graph G is a mapping f:V(G)→{0,1,…,n} such that |f(u)−f(v)|≥k+1−d(u,v), for all u,v∈V(G), where d(u,v) is the length of the shortest path connecting u and v. The L(k,k−1,…,1) labeling span λk(F) of a family of graphs F is the minimum n for which each G∈F admits an n-L(k,k−1,…,1) labeling. For the family L3 of all subgraphs of an infinite triangular lattice we provide upper and lower bounds of λk(L3) for general k and show that the ratio of the upper and lower bound is at most . The upper bound is given by providing an assignment algorithm to the vertices of the infinite triangular lattice.