Partial packing coloring and quasi-packing coloring of the triangular grid

Abstract

The concept of packing coloring in graph theory is motivated by the challenge of frequency assignment in radio networks. This approach entails assigning positive integers to vertices, with the requirement that for any given label (color) i, the distance between any two vertices sharing this label must exceed i. Recently, after over 20 years of intensive research, the minimal number of colors needed for packing coloring of an infinite square grid has been established to be 15. Moreover, it is known that a hexagonal grid requires a minimum of 7 colors for packing coloring, and a triangular grid is not colorable with any finite number of colors in a packing way.

Therefore, two questions come to mind: What fraction of a triangular grid can be colored in a packing model, and how much do we need to weaken the condition of packing coloring to enable coloring a triangular grid with a finite number of colors?

With the partial help of the Mixed Integer Linear Programming (MILP) solver, we have proven that it is possible to color at least 72.8% but no more than 82.2% of a triangular grid in a packing way.

Additionally, we have investigated the relaxation of packing coloring, called quasi-packing coloring, which is a special case of S-packing coloring. We have established that the S-packing chromatic number for the triangular grid, where $S=(1,1,2,3,...)$, is between 11 and 33. Furthermore, we have proven that the aforementioned sequence S is the best possible in some sense.

We have also considered the partial packing and quasi-packing coloring of an infinite hypercube and present several open problems for other classes of graphs.