L(j,k)- and circular L(j,k)-labellings for the products of complete graphs
Let j and k be two positive integers with j >= k. An L(j,k)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that the difference between labels of any two adjacent vertices is at least j, and the difference between labels of any two vertices that are at distance two apart is at least k. The minimum range of labels over all L(j,k)-labellings of a graph G is called the lambda(j,k)-number of G, denoted by lambda(j,k) (G). A sigma(j,k)-circular labelling with span m of a graph G is a function f:V(G) --> {0,1,..., m-1} such that vertical bar f(u)-f(v)vertical bar (m) j if u and v are adjacent; and vertical bar f(u)-f(v)vertical bar (m) >= k if u and v are at distance two apart, where vertical bar x vertical bar (m) = min{vertical bar x vertical bar,m-vertical bar x vertical bar}. The minimum m such that there exists a sigma(j,k)-circular labelling with span m for G is called the sigma(j,k) -number of G and denoted by sigma (j,k) (G). The lambda (j,k) -numbers of Cartesian products of two complete graphs were determined by Georges, Mauro and Stein ((2000) SIAM J Discret Math 14:28-35). This paper determines the lambda(j,k)-numbers of direct products of two complete graphs and the sigma(j,k)-numbers of direct products and Cartesian products of two complete graphs.
lambda(j,k) cartesian product direct product sigma(j,k)-number