Abstract: Let \(G\) and \(H\) be finite, simple graphs on the same vertex set \(V\).  A proper  \(G\)-coloring of \(H\) is a proper list-coloring of \(H\) from the lists \(N_G(v)\),  the open neighborhood of \(v \in V(G)\). 

If a \(G\)-coloring of \(H\) exists, then we say that \(H\) is \(G\)-colorable.  Further, we define a graph \(H\) to be maximally-\(G\)-colorable if \(H\) is \(G\)-colorable and for any edge \(e \in \overline{E(H)}\), \(H \cup e\) is not \(G\)-colorable.  Similarly, we define a graph \(H\) to be minimally-not-\(G\)-colorable  if \(H\) is not \(G\)-colorable and for any edge \(e \in E(H)\), \(H - e\) is \(G\)-colorable. 

In this talk, we will discuss recent results regarding maximally-\(G\)-colorable graphs \(H\)  and minimally-not-\(G\)-colorable graphs \(H\). 

This is joint work with Dr. Peter Johnson and Dr. Joseph Briggs.