On uniquely 3-colorable planar graphs

A k-chromatic graph G is called uniquely k-colorable if every k-coloring of the vertex set V of G induces the same partition of V into k color classes. There is an innnite class C of uniquely 4-colorable planar graphs obtained from the K 4 by repeatedly inserting new vertices of degree 3 in triangular faces. In this paper we are concerned with the well-known conjecture (see 6]) that every uniquely 4-colorable planar graph belongs to C. We shall show that a minimal counterexample to this conjecture is 5-connected.