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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 …