Labeling products of complete graphs with a condition at distance two (Q2706187)

Labelings of grphs are considered. For integers \(j\), \(k\) with \(1\leq k\leq j\) an \(L(j,k)\)-labeling of a graph \(G\) is a mapping from \(V(G)\) into the set of integers such that \(|L(v_2)- L(v_1)|\geq j\) if \(d_G(v_1,v_2)= 1\) and \(|L(v_2)- L(v_1)|\geq k\) if \(d_G(v_1,v_2)= 2\), where \(d_G\) is the distance of vertices in \(G\). The span of \(L\) is the difference between the largest and smallest value (label) of \(L\). The minimum span over all \(L(j,k)\)-labelings of \(G\) is the \(\lambda^j_k\)-number of \(G\) denoted by \(\lambda^j_k(G)\). The main results of the paper are the following:NEWLINENEWLINENEWLINETheorem 4.4. Let \(j\), \(k\), \(n\) and \(m\) be integers where \(2\leq n< m\) and \(j\geq k\). Then \(\lambda^j_k(K_n\times K_m)= (m-1)j+ (n-1)k\) if \(j/k> n\) and \(\lambda^j_k(K_n\times K_n)= (mn- 1)k\) if \(j/k\leq n\).NEWLINENEWLINENEWLINETheorem 4.5. Let \(j\), \(k\) and \(n\) be integers where \(2\leq n\) and \(j\geq k\). Then \(\lambda^j_k(K^2_n)= (n-1)j+ (2n-2)k\) if \(j/k> n-1\) and \(\lambda^j_k(K^2_n)= (n^2- 1)k\) if \(j/k\leq n-1\).