Unique four-colorings of planar graphs and the like… I am doing research into a particular graph coloring problem and wonder if someone can direct me to published work that bears on what I’m studying.
It is known that planar graphs that are uniquely four-colorable belong to a class populated by $K_4$ (the complete graph of order $4$, a.k.a, the "tetrahedron") and any graph derived from $K_4$ by inserting vertices into triangles. There are several interesting properties of such graphs, not the least of which that they have following properties:


*

*They are plane triangulations.

*Every vertex is adjacent to vertices of the other three colors. 

*In every coloring, every Kempe chain is a tree.  

*In every coloring, there is one and only one Kempe chain for each color-pair.


I am not interested in uniquely four-colorable planar graphs per se, but in the class of planar graphs that are NOT uniquely four-colorable but which satisfy the properties above and also  


*

*In any drawing in the plane in which the graph is represented as a set of vertices and edges within an "outermost" triangle T, there are no vertices inside any triangle other than T (that is, no "separating triangles").

*They are minimum-degree $5$.


The "icosahedron" is the only member of this class that I’ve been able to construct/locate.  Does anyone know of work that has been done on this problem?  Does anyone care to speculate about whether there are any others? 

This continues to be part of the research I'm doing on graph coloring.  If one relaxes the condition of "minimum degree $5$" to "$4$-connected" (that is, the triangulation $T$ must be $4$-connected rather than minimum degree $5$) but insists that the graph still be $4$-connected after an edge joining vertices of at least degree 5 is deleted to form a near-triangulation $G$, then, apart from the icosahedron, which we've previously identified as satisfying all of the conditions, there is a $T$ of order $9$ for which there are two distinct colorings, each of which is comprised of six Kempe chains, one for each of the six possible color-pairs.  Of course, all the Kempe chains are trees or there wouldn't be only six.  The following graph illustrates the two distinct colorings from which the Kempe chain claim can be verified easily. 
