Graph Color Extensions: When Hadwiger's Conjecture and Embeddings Help
Suppose $G$ is $r$-colorable and $P \subseteq V(G)$ 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 $K_5$ or is planar and $s \geq 2$, 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.