Improved square coloring of planar graphs

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 32Δ+1 colors are sufficient to square color every planar graph of maximum degree Δ. This conjecture has been proven asymptotically for graphs with large maximum degree. We consider here planar graphs with small maximum degree and show that 2Δ+7 colors are sufficient, which improves the best known bounds when 6⩽Δ⩽31.