Total-Coloring of Plane Graphs with Maximum Degree Nine

The central problem of the total-colorings is the total-coloring conjecture, which asserts that every graph of maximum degree ∆ admits a (∆+2)-total-coloring. Similar to edge-colorings—with Vizing’s edge-coloring conjecture—this bound can be decreased by 1 for plane graphs of higher maximum degree. More precisely, it is known that if ∆ ≥ 10, then every plane graph of maximum degree ∆ is (∆ + 1)totally-colorable. On the other hand, such a statement does not hold if ∆ ≤ 3. We prove that every plane graph of maximum degree 9 can be 10-totally-colored.