Seymour’s Exact Conjecture for planar edge-coloring

Prove that every planar graph G is ⌈η′(G)⌉-edge-colorable, where η′(G) is the fractional chromatic index of G.

Background

Seymour’s Exact Conjecture generalizes the subcubic planar setting and, if true, would imply Vizing’s planar conjecture as well as the Four-Color Theorem. It connects integral edge-colorings with their fractional counterparts via the fractional chromatic index.

The authors mention this broader conjecture to situate their augmentation results within the larger landscape of planar edge-coloring theory.

References

Generalizing \cref{conj:Groetzsch}, Seymour's Exact Conjecture states that every planar graph~$G$ is $\lceil \eta'(G) \rceil$-edge-colorable, where~$\eta'(G)$ denotes the fractional chromatic index of~$G$.

— Recognition Complexity of Subgraphs of k-Connected Planar Cubic Graphs  (2401.05892 - Goetze et al., 2024) in Section 6 (Discussion)