Dirac's map-color theorem for choosability

Abstract

It is proved that the choice number of every graph G embedded on a surface of Euler genus ε ≥ 1 and ε = 3 is at most the Heawood number H(ε) = (7 + √ 24ε + 1)/2 and that the equality holds if and only if G contains the complete graph K H(ε) as a subgraph.

Coloring Eulerian Triangulations of the Klein Bottle

“…. , v 3 keep their colors and the vertices v 4 and v 5 are recolored with the fourth and the fifth colors.…”

“…By symmetry, we can assume that w is adjacent to either v 1 , v 2 and v 3 , or to v 3 , v 4 and v 5 , see Fig. 17.…”

Five-list-coloring graphs on surfaces III. One list of size one and one list of size two

“…It follows from a theorem of Bohme at al [3] that φ can be extended to an L-coloring of G 2 and hence to an L-coloring of G.…”

Graphs with Two Crossings Are 5-Choosable

“…A general bound for choosability of graphs on surfaces, an analogue of Dirac's Map-Color Theorem, is known due to Böhme et al [1]. However, only very few graphs have choosability close to this bound.…”