**Originator(s):** William Tutte

**Conjecture/Question:**
Every bridgeless graph has a nowhere-zero 5-flow.

**Definitions/Background/motivation:**
A *bridgeless* graph is a graph with no cut-edge.
For planar graphs, the conjecture is equivalent to the statement that
all planar graphs are 5-colorable, so the conjecture generalizes that
statement.

**Comments/Partial results:**
Kilpatrick [K] and Jaeger [J] proved that every bridgeless graph is 8-flowable.
Seymour [Se] proved that every bridgeless graph is 6-flowable.

Celmins proved that a smallest counterexample to 5-flowability must be a cyclically 5-edge-connected snark with girth at least 7. Steinberg [St] proved the conjecture for graphs embeddable in the projective plane.

The conjecture is implied by a conjecture of Archdeacon [1984] and Jaeger [1988] that bridgeless graphs have orientable cycle double covers of order at most 5.

**References:**

[Se] Seymour, P. D. Nowhere-zero $6$-flows. J. Combin. Theory Ser. B 30 (1981),
no. 2, 130--135.

[St] Steinberg, Richard Tutte's $5$-flow conjecture for the projective plane.
J. Graph Theory 8 (1984), no. 2, 277--289.

**Keywords:**