Facial colorings using Hall’s Theorem

A vertex coloring of a plane graph is ℓ -facial if every two distinct vertices joined by a facial walk of length at most ℓ receive distinct colors. It has been conjectured that every plane graph has an ℓ -facial coloring with at most 3 ℓ + 1 colors. We improve the currently best known bound and show that every plane graph has an ℓ -facial coloring with at most ⌊ 7 ℓ / 2 ⌋ + 6 colors. Our proof uses the standard discharging technique, however, in the reduction part we have successfully applied Hall’s Theorem, which seems to be quite an unusual approach in this area.