NOTE ON A CONJECTURE OF TOFT

A conjecture of Toft [17] asserts that any 4-critical graph (or equivalently, every 4-chromatic graph) contains a fully odd subdivision of K-4. We show that if a graph G has a degree three node v such that G-v is 3-colourable, then either G is 3-colourable or it contains a fully odd K-4. This resolves Toft's conjecture in the special case where a 4-critical graph has a degree three node, which is in turn used to prove the conjecture for line-graphs. The proof is constructive and yields a polynomial algorithm which given a 3-degenerate graph either finds a 3-colouring or exhibits a subgraph that is a fully odd subdivision of K-4. (A graph is 3-degenerate if every subgraph has some node of degree at most three.)