Packings in bipartite prisms and hypercubes

TL;DR

This paper investigates the properties of packings in bipartite prisms and hypercubes, establishing bounds and relationships that advance understanding of graph packings and their applications to coloring problems.

Contribution

It provides new bounds for 2-packings and open packings in hypercubes, and characterizes the injective colorability of hypercubes, including the smallest non-perfect case.

Findings

Established lower bounds for 2-packings in hypercubes.

Proved a relation between open packings in bipartite graphs and their prisms.

Identified that Q9 is the smallest hypercube not perfectly injectively colorable.

Abstract

The $2$-packing number $\rho_2(G)$ of a graph $G$ is the cardinality of a largest $2$-packing of $G$ and the open packing number $\rho^{\rm o}(G)$ is the cardinality of a largest open packing of $G$, where an open packing (resp. $2$-packing) is a set of vertices in $G$ no two (closed) neighborhoods of which intersect. It is proved that if $G$ is bipartite, then $\rho^{\rm o}(G\Box K_2) = 2\rho_2(G)$. For hypercubes, the lower bounds $\rho_2(Q_n) \ge 2^{n - \lfloor \log n\rfloor -1}$ and $\rho^{\rm o}(Q_n) \ge 2^{n - \lfloor \log (n-1)\rfloor -1}$ are established. These findings are applied to injective colorings of hypercubes. In particular, it is demonstrated that $Q_9$ is the smallest hypercube which is not perfect injectively colorable. It is also proved that $\gamma_t(Q_{2^k}\times H) = 2^{2^k-k}\gamma_t(H)$, where $H$ is an arbitrary graph with no isolated vertices.