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 (Delta + 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 Delta >= 10, then every plane graph of maximum degree Delta is (Delta + 1)-totally-colorable. On the other hand, such a statement does not hold if Delta <= 3. We prove that every plane graph of maximum degree 9 can be 10-totally-colored.