\(L(h,k)\) labelings of \(K_n-M\) and \(K_n-P_m\) for all values of \(h\) and \(k*\)

Abstract

An \(L(h,k)\) labeling of a graph \(G\) is an integer labeling of the vertices where the labels of adjacent vertices differ by at least \(h\), and the labels of vertices that are at distance two from each other differ by at least \(k\). The span of an \(L(h,k)\) labeling \(f\) on a graph \(G\) is the largest label minus the smallest label under \(f\). The \(L(h,k)\) span of a graph \(G\), denoted \(\lambda_{h,k}(G)\), is the minimum span of all \(L(h,k)\) labelings of \(G\).