The square of a planar cubic graph is $7$-colorable

TL;DR

This paper proves Wegner's 1977 conjecture that the square of any planar cubic graph can be colored with at most 7 colors, confirming the optimality of this bound.

Contribution

It provides a proof for Wegner's longstanding conjecture, establishing the 7-colorability of the square of all planar cubic graphs.

Findings

Confirmed Wegner's 1977 conjecture

Proved 7-colorability is optimal

Established the square of any planar cubic graph is 7-colorable

Abstract

We prove the conjecture made by G.Wegner in 1977 that the square of every planar, cubic graph is $7$-colorable. Here, $7$ cannot be replaced by $6$.