Distance-constrained labellings of Cartesian products of graphs

TL;DR

This paper investigates distance-constrained labellings of Cartesian product graphs, establishing bounds and exact values for specific labelling parameters, especially in Hamming graphs, and addresses an open problem related to hypercube powers.

Contribution

It provides new bounds and exact values for $L(h_1, h_2, \

Findings

Established a common lower bound for certain labelling invariants on Cartesian product graphs.

Proved the chromatic number of the $l$-th power of a Cartesian product graph equals this lower bound plus one.

Solved a case of the open problem on the chromatic number of powers of hypercubes.

Abstract

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