On \(L(2,1)\)-labelings of Cartesian products of paths and cycles
An \(L(2,1)\)-labeling, also often called \(\lambda\)-coloring, is a function that assigns each vertex \(v\) in a graph \(G=(V,E)\) a nonnegative integer label, such that adjacent vertices have labels that differ by at least two, and vertices at distance two have different labels. The \(L(2,1)\)-labeling number of a graph \(G\) is the smallest \(k\) such that \(G\) has an \(L(2,1)\)-labeling with labels in \(\{0,1,\ldots, k\}\). This paper looks at the \(L(2,1)\)-labeling numbers of Cartesian products of paths and cycles. These can be seen as grid graphs with a number of wrap-around edges along the borders. The paper gives a complete classification for the product of one cycle and one path; and many results for the product of two cycles.
scientific article; zbMATH DE number 2083726