Lower Bounds for Identifying Codes in some Infinite Grids

An $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and unique. On a finite graph, the density of a code is $|C|/|V(G)|$, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We present new lower bounds for densities of codes for some small values of $r$ in both the square and hexagonal grids.

Formation of Infinite Grids with Chiral Square Compartments by Self-Assembly of Achiral 1,1'-Biphenyl-2,2',6,6'-tetracarboxylic Acids. Effect of 4,4'-Substitution on the Grid Stacking

The achiral 1,1'-biphenyl-2,2',6,6'-tetracarboxylic acid as well as its 4,4'-dibromo and 4,4'-dinitro derivative self-assemble in crystal under formation of a two-dimensional grid set up from hydrogen-bonded cyclotetrameric compartments which are individually chiral (D4 symmetry). The roughly square cavities (ca 5.6 Å) of individual compartments partly accommodate the perpendicularly oriented benzene rings of the neighbouring grid, in dependence on the nature of the 4,4'-substituents. As a consequence, the grids are stacked in a staggered (nonconcatenated) manner.