Distance two labelling and direct products of graphs

Abstract

An L ( 2 , 1 ) -labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that vertices at distance two get different numbers and adjacent vertices get numbers which are at least two apart. The L ( 2 , 1 ) -labelling number of a graph G, λ ( G ) , is the minimum range of labels over all L ( 2 , 1 ) -labellings of G. Given two graphs G and H, the direct product of G and H is the graph G × H with vertex set V ( G ) × V ( H ) in which two vertices ( x , y ) and ( x ′ , y ′ ) are adjacent if and only if xx ′ ∈ E ( G ) and yy ′ ∈ E ( H ) . In this paper, we completely determine the L ( 2 , 1 ) -labelling numbers of K m × K n for m , n ⩾ 2 , and K m × P n for m ⩾ 3 , n ⩾ 1 , where P n is the path of length n. The L ( 2 , 1 ) -labelling numbers of K m × C n for m ⩾ 3 and some special values of n are also determined, where C n is the cycle of length n.