Extending total colorings in planar graphs

Abstract

We initiate the study of total-coloring extensions, and focus our attention on planar graphs, asking: ``When can a total-$k$-coloring of some subgraph $H$ of a planar graph $G$ be extended to a total-$k$-coloring of $G$?'' We prove that if $H$ is a matching, then any total-$(\Delta+3)$-coloring of $H$ in $G$ extends to $G$ provided $\Delta\geq 28$; this number of colors is best-possible without introducing a distance condition on $H$. We also prove that if $H$ is a set of distance-3 cliques then any total-$(\Delta+1)$-coloring of $H$ extends to $G$ provided $\Delta\geq 27$; this distance condition cannot be lowered.