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 3 2 ∆ + 1 colors are sufficient to square color every planar graph of maximum degree ∆. 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∆ + 7 colors are sufficient, which improves the best known bounds when 6 ∆ 31.

$2$-distance, injective, and exact square list-coloring of planar graphs with maximum degree 4

“…In [34], La and Montassier presented a summary of the latest known results regarding the 2-distance coloring of planar graphs for different girth values, where girth of a graph G, denoted by g(G), is defined as the length of the shortest cycle. An additional more recent result is due to Bousquet et al [4]. They improved a general result, in the case of the 2-distance coloring, stating that 2∆ + 7 colors are sufficient for all planar graphs with maximum degree between 6 and 31.…”

“…Throughout the years, p-distance colorings, in particular the case when p = 2, became a focus for many researchers, see [2,26,27,42]. Only in recent years, several results were investigated with regards to the 2-distance coloring of planar graphs, see [4,13,20]. For more of the recent results see also [31,32,33].…”

Coloring, List Coloring, and Painting Squares of Graphs (and Other Related Problems)

“…Next, van den Heuvel and McGuinness [168] showed that χ 2 col (G) ⩽ 2∆ + 25; see also [64,Theorem 6.10] for a simplified proof that χ 2 col (G) ⩽ 2∆ + 34. About 20 years later, Bousquet, Deschamps, de Meyer, and Pierron [39] proved that if G is planar with ∆ ⩾ 9, then χ 2 (G) ⩽ 2∆ + 7. This is the best known bound on χ 2 (G) when 9 ⩽ ∆ ⩽ 31 (however, the proof at one point requires permuting colors, so it does not extend to list coloring or degeneracy).…”

Coloring, List Coloring, and Painting Squares of Graphs (and other related problems)

“…Recently Bousquet, Deschamps, de Meyer, and Pierron[37] proved that if G is planar with ∆ ≥ 9, then χ 2 (G) ≤ 2∆ + 7. This is the best known bound when 9 ≤ ∆ ≤ 31; however, the proof at one point requires permuting colors, so it does not extend to list coloring or degeneracy.…”