Total colouring of plane graphs with maximum degree nine

Lukasz Kowalik (1) , Jean-Sébastien Sereni (2, 3) , Riste Skrekovski (4)

1 Institute of Informatics [Warsaw]
2 KAM - Department of Applied Mathematics (KAM)
3 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications
4 FMF - Faculty of Mathematics and Physics [Ljubljana]

The central problem of the total-colorings is the total-coloring conjecture, which asserts that every graph of maximum degree D admits a (D+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 D > 9, then every plane graph of maximum degree D is (D+1)-totally-colorable. On the other hand, such a statement does not hold if D < 4. We prove that every plane graph of maximum degree 9 can be 10-totally-colored.

Mots clés: total-coloring; planar graph; discharging method

Domaines: Mathématique discrète [cs.DM]

Lukasz Kowalik, Jean-Sébastien Sereni, Riste Skrekovski. Total colouring of plane graphs with maximum degree nine. SIAM Journal on Discrete Mathematics, 2008, 22 (4), pp.1462--1479. ⟨10.1137/070688389⟩. ⟨hal-00487320⟩