Wegner’s square-coloring conjecture for planar graphs with maximum degree Δ ≥ 4

Prove Wegner’s conjecture for planar graphs with maximum degree Δ ≥ 4: for every planar graph G with maximum degree Δ, establish that the square chromatic number satisfies χ2(G) ≤ Δ + 5 when Δ ∈ {4,5,6,7} and χ2(G) ≤ ⌊3Δ/2⌋ + 1 when Δ ≥ 8.

Background

Square k-coloring is equivalent to packing (2^k)-coloring, and χ2(G) denotes the minimum k such that the square of G is k-colorable. Wegner (1977) conjectured the precise bounds of χ2(G) for planar graphs with maximum degree Δ, giving a piecewise formula: 7 for Δ=3, Δ+5 for Δ∈{4,5,6,7}, and ⌊3Δ/2⌋+1 for Δ≥8.

The case Δ=3 has been settled by Thomassen, and independently by Hartke, Jahanbekam, and Thomas. For Δ=4, the current best upper bound is 12 due to Bousquet, Deschamps, de Meyer, and Pierron; asymptotically, χ2(G) ≤ (3/2)Δ + o(Δ) is known. Despite these advances, the conjecture itself is unresolved for all Δ≥4.