Cyclic coloring of plane graphs with maximum face size 16 and 17

Maximum face-constrained coloring of plane graphs

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-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)...

A note on face coloring entire weightings of plane graphs

Given a weighting of all elements of a 2-connected plane graph G = (V,E, F ), let f(α) denote the sum of the weights of the edges and vertices incident with the face α and also the weight of α. Such an entire weighting is a proper face colouring provided that f(α) 6= f(β) for every two faces α and β sharing an edge. We show that for every 2-connected plane graph there is a proper face-colouring...

Coloring plane graphs with independent crossings

We show that every plane graph with maximum face size four whose all faces of size four are vertex-disjoint is cyclically 5-colorable. This answers a question of Albertson whether graphs drawn in the plane with all crossings independent are 5-colorable.