A note on two conjectures that strengthen the four colour theorem

TL;DR

This paper shows that a conjecture about signed graph coloring, which strengthens the four color theorem, implies another conjecture related to list coloring and bipartite color classes in planar graphs.

Contribution

It proves that the signed graph coloring conjecture implies the list coloring conjecture for planar graphs, establishing a logical connection between two strengthening conjectures.

Findings

Signed graph coloring conjecture implies list coloring conjecture.

Strengthens understanding of coloring properties in planar graphs.

Links two conjectures that extend the four color theorem.

Abstract

There are two conjectures concerning planar graph colourings that are strengthenings of the four colour theorem. One concerns signed graph colouring and is proposed by M\'{a}\v{c}ajov\'{a}, Raspaud and \v{S}koviera. It asserts that every signed planar graph is $4$-colourable. Another concerns list colouring and is proposed by K\"{u}ndgen and Ramamurthi which asserts that if $L$ is a $2$-list assignment of a planar graph $G$, then there is an $L$-colouring of $G$ such that each colour class induces a bipartite graph. In this note we prove that the first conjecture implies the second one.