**Originator(s):** William Tutte (U. Waterloo)

**Conjecture/Question:**
Every bridgeless graph containing no subdivision of the Petersen graph is
4-flowable [T2].

**Definitions/Background/motivation:**
A *bridgeless* graph is a graph with no cut-edge.

**Comments/Partial results:**
For 3-regular graphs, the statement is equivalent to the statement that
3-regular graphs containing no subdivision of the Petersen graph are
3-edge-colorable, which was proved in a series of papers by
Robertson, Sanders, Seymour, and Thomas.

Tutte [1954] proved that every graph having a *k*-face-colorable 2-cell
embedding on some orientable surface also has a nowhere-zero *k*-flow;
indeed, for planar graphs, the 4-flow Conjecture is equivalent to the Four
Color Theorem.

**References:**