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

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