Distance-constrained labellings of Cartesian products of graphs

Abstract

An L(h1,h2,…,hl)-labelling of a graph G is a mapping ϕ:V(G)→{0,1,2,…} such that for 1≤i≤l and each pair of vertices u,v of G at distance i, we have |ϕ(u)−ϕ(v)|≥hi. The span of ϕ is the difference between the largest and smallest labels assigned to the vertices of G by ϕ, and λh1,h2,…,hl(G) is defined as the minimum span over all L(h1,h2,…,hl)-labellings of G.In this paper we study λh,1,…,1 for Cartesian products of graphs, where (h,1,…,1) is an l-tuple with l≥3. We prove that, under certain natural conditions, the value of this and three related invariants on a graph H which is the Cartesian product of l graphs attain a common lower bound. In particular, the chromatic number of the lth power of H equals this lower bound plus one. We further obtain a sandwich theorem which extends the result to a family of subgraphs of H which contain a certain subgraph of H. All these results apply in particular to the class of Hamming graphs: if q1≥⋯≥qd≥2 and 3≤l≤d then the Hamming graph H=Hq1,q2,…,qd satisfies λql,1,…,1(H)=q1q2…ql−1 whenever q1q2…ql−1>3(ql−1+1)ql…qd. In particular, this settles a case of the open problem on the chromatic number of powers of the hypercubes.