Total-Coloring of Plane Graphs with Maximum Degree Nine

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)...

Edge-face coloring of plane graphs with maximum degree nine

An edge-face colouring of a plane graph with edge set E and face set F is a colouring of the elements of E ∪ F so that adjacent or incident elements receive different colours. Borodin [Simultaneous coloring of edges and faces of plane graphs, Discrete Math., 128(1-3):21–33, 1994] proved that every plane graph of maximum degree ∆ > 10 can be edge-face coloured with ∆ + 1 colours. We extend Borod...

Total-colouring of plane graphs with maximum degree nine

The central problem of the total-colourings is the Total-Colouring Conjecture, which asserts that every graph of maximum degree ∆ admits a (∆ + 2)-total-colouring. Similarly to edge-colourings—with Vizing’s edge-colouring 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...

On total 9-coloring planar graphs of maximum degree seven

Given a graphG, a total k-coloring ofG is a simultaneous coloring of the vertices and edges ofGwith at most k colors. If ∆(G) is themaximum degree ofG, then no graph has a total ∆-coloring, but Vizing conjectured that every graph has a total (∆ + 2)-coloring. This Total Coloring Conjecture remains open even for planar graphs. This article proves one of the two remaining planar cases, showing th...

Maximum face-constrained coloring of plane graphs