Toward Optimal Radio Colorings of Hypercubes via SAT-solving

19 pages•Published: June 3, 2023

Abstract

Radio 2-colorings of graphs are a generalization of vertex colorings motivated by the problem of assigning frequency channels in radio networks. In a radio 2-coloring of a graph, vertices are assigned integer colors so that the color of two vertices u and v differ by at least 2 if u and v are neighbors, and by at least 1 if u and v have a common neighbor. Our work improves the best-known bounds for optimal radio 2-colorings of small hypercube graphs, a combinatorial problem that has received significant attention in the past. We do so by using automated reasoning techniques such as symmetry breaking and Cube and Conquer, obtaining that for n = 7 and n = 8, the coding-theory upper bounds of Whittlesey et al. (1995) are not tight. Moreover, we prove the answer for n = 7 to be either 12 or 13, thus making a substantial step towards answering an open problem by Knuth (2015). Finally, we include several combinatorial observations that might be useful for further progress, while also arguing that fully determining the answer for n = 7 will require new techniques.

Keyphrases: hypercubes, radio colorings, SAT

In: Ruzica Piskac and Andrei Voronkov (editors). Proceedings of 24th International Conference on Logic for Programming, Artificial Intelligence and Reasoning, vol 94, pages 386--404

Links:	https://easychair.org/publications/paper/b8Cn
	https://doi.org/10.29007/qrmp