No-hole $\lambda$-$L(k,k-1,\dots ,2,1)$-labeling for Square Grid

Abstract

Given a fixed $k$ $\in$ ${\mathbb{Z}}^{+}$ and $\lambda$ $\in$ ${\mathbb{Z}}^{+}$, the objective of a $\lambda$-$L(k,k-1,\dots ,2,1)$-labeling of a graph $G$ is to assign non-negative integers (known as labels) from the set $\{0,\dots ,\lambda -1\}$ to the vertices of $G$ such that the adjacent vertices receive values which differ by at least $k$, vertices connected by a path of length two receive values which differ by at least $k-1$, and so on. The vertices which are at least $k+1$ distance apart can receive the same label. The smallest $\lambda$ for which there exists a $\lambda$-$L(k,k-1,\dots ,2,1)$-labeling of $G$ is known as the $L(k,k-1,\dots ,2,1)$-labeling number of $G$ and is denoted by ${\lambda }_k(G)$. The ratio between the upper bound and the lower bound of a $\lambda$-$L(k,k-1,\dots ,2,1)$-labeling is known as the approximation ratio. In this paper a lower bound on the value of the labeling number for square grid is computed and a formula is proposed which yields a $\lambda$-$L(k,k-1,\dots ,2,1)$-labeling of square grid, with approximation ratio at most $\frac{9}{8}$. The labeling presented is a no-hole one, i.e., it uses each label from $0$ to $\lambda -1$ at least once.