Improved square coloring of planar graphs

TL;DR

This paper improves bounds on the number of colors needed for square coloring of planar graphs with small maximum degree, advancing understanding of Wegner's conjecture for specific degree ranges.

Contribution

The authors establish that $2 ext{Delta}+7$ colors suffice for square coloring of planar graphs with small maximum degree, improving previous bounds for degrees 6 to 31.

Findings

$2 ext{Delta}+7$ colors suffice for degrees 6 to 31

Improved bounds over previous results for small maximum degree

Advances understanding of Wegner's conjecture in specific cases

Abstract

Square coloring is a variant of graph coloring where vertices within distance two must receive different colors. When considering planar graphs, the most famous conjecture (Wegner, 1977) states that $\frac32\Delta+1$ colors are sufficient to square color every planar graph of maximum degree $\Delta$. This conjecture has been proven asymptotically for graphs with large maximum degree. We consider here planar graphs with small maximum degree and show that $2\Delta+7$ colors are sufficient, which improves the best known bounds when $6\leqslant \Delta\leqslant 31$.