Distance-constrained labellings of Cartesian products of graphs

An $L(h_1, h_2, \ldots, h_l)$-labelling of a graph $G$ is mapping $\phi: V(G) \rightarrow \{0, 1, 2, \ldots\}$ such that for $1\le i\le l$ and each pair vertices $u, v$ at distance $i$, we have $|\phi(u) - \phi(v)| \geq h_i$. The span $\phi$ the difference between largest smallest labels assigned to by $\phi$, $\lambda_{h_1, h_l}(G)$ defined as minimum over all h_l)$-labellings $G$. In this paper study $\lambda_{h, 1}$ Cartesian products graphs, where $(h, 1)$ an $l$-tuple with $l \ge 3$. We prove that, under certain natural conditions, value three related invariants on $H$ which product $l$ graphs attain common lower bound. particular, chromatic number $l$-th power equals bound plus one. further obtain sandwhich theorem extends result family subgraphs contain subgraph $H$. All these results apply in particular class Hamming graphs: if $q_1\ge \cdots q_d\ge 2$ $3\le l\le d$ then $H=H_{q_1,q_2,\ldots ,q_d}$ satisfies $\lambda_{q_l,1,\ldots,1}(H) = q_1q_2\ldots q_l-1$ whenever $q_1q_2\ldots q_{l-1}>3(q_{l-1}+1)q_l\ldots q_d$. settles case open problem powers hypercubes.