Graph Color Extensions: When Hadwiger's Conjecture and Embeddings Help

Suppose G is r-colorable and P ⊆ V (G) is such that the components of G[P ] are far apart. We show that any (r + s)-coloring of G[P ] in which each component is s-colored extends to an (r + s)-coloring of G. If G does not contract to K5 or is planar and s ≥ 2, then any (r + s − 1)-coloring of P in which each component is s-colored extends to an (r + s − 1)-coloring of G. This result uses the Four Color Theorem and its equivalence to Hadwiger’s Conjecture for k = 5. For s = 2 this provides an affirmative answer to a question of Thomassen. Similar results hold for coloring arbitrary graphs embedded in both orientable and non-orientable surfaces.