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

Given a fixed $k$ $\in$ $\mathbb{Z}^+$ and $λ$ $\in$ $\mathbb{Z}^+$, the objective of a $λ$-$L(k, k-1, \ldots, 2, 1)$-labeling of a graph $G$ is to assign non-negative integers (known as labels) from the set $\{0, \ldots, λ-1\}$ 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 $λ$ for which there exists a $λ$-$L(k, k-1, \ldots, 2, 1)$-labeling of $G$ is known as the $L(k, k-1, \ldots, 2, 1)$-labeling number of $G$ and is denoted by $λ_k(G)$. The ratio between the upper bound and the lower bound of a $λ$-$L(k, k-1, \ldots, 2, 1)$-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 $λ$-$L(k, k-1, \ldots, 2, 1)$-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 $λ-1$ at least once.