  In 1880, P. G. Tait showed that the four colour theorem is equivalent to the
assertion that every 3-regular planar graph without cut-edges is
3-edge-colourable, and in 1891, J. Petersen proved that every 3-regular graph
with at most two cut-edges has a 1-factor. In this paper, we introduce the
notion of collapsing all edges of a 1-factor of a 3-regular planar graph,
thereby obtaining what we call a vertex-oriented 4-regular planar graph. We
also introduce the notion of o-colouring a vertex-oriented 4-regular planar
graph, and we prove that the four colour theorem is equivalent to the assertion
that every vertex-oriented 4-regular planar graph without nontransversally
oriented cut-vertex (VOGWOC in short) is 3-o-colourable. This work proposes an
alternative avenue of investigation in the search to find a more conceptual
proof of the four colour theorem, and we are able to prove that every VOGWOC is
o-colourable (although we have not yet been able to prove 3-o-colourability).

Combinatorics