While reading for my graph theory class, I came across a short - yet curious - proof for the following theorem: if a graph G is critically k−colorable then . Here is the proof to the claim:
Suppose (for a contradiction) that G is k-critical and that satisfies . Then has a coloring, and this coloring extends to a - coloring of G. This yields a contradiction.
Everything in this proof makes sense besides one particular item: in the second line, what does the author mean by extending a coloring from a subgraph of G to the whole graph? In addition, why does have a coloring that extends to all of G (if that makes any sense)? I understand this may seem like an easy Google search but to be honest I can't find anything helpful and figured someone could provide some insight.
Note that this is not a homework question but simply for going beyond what I am learning in class.