## The Strong Circular 5-Flow Conjecture

Let *r*
2 be a real number and let G be a graph and D an
orientation of its edges. A function f: E(G)
R is a *nowhere-zero
circular r-flow* if for every vertex *v* of G, the sum of the f-values
on the D-incoming edges at *v* equals the sum of the f-values on the
D-outgoing edges at *v*, and for every edge *e* of G, the value |f(*e*)|
is between 1 and *r *- 1. Let us observe that if G admits a nowhere-zero
circular *r*-flow, then it admits an integer valued nowhere-zero *k*-flow,
where *k* is the `ceiling' of *r*.

The *circular flow index* of G is the minimum value *r* for which G
admits a nowhere-zero circular *r*-flow.

The Tutte 5-Flow Conjecture asserts that every 2-edge-connected graph admits
a nowhere-zero (circular) 5-flow. Among cubic graphs, *snarks* are
nontrivial graphs whose circular flow number is bigger than four, i.e. they do
not admit nowhere-zero 4-flows.

**Conjecture 1:**
Let G be a snark distinct from the Petersen graph. Then
the circular flow index of G is less than 5.

Conjecture 1 has been verified for some small snarks.

It is not difficult to see that the circular flow index
of the Petersen graph is equal to 5. So, Conjecture 1 claims that this is the
only snark with that high circular flow index.

Maybe Conjecture 1 can be strengthened as follows
(although I do not believe this is true, so I sustain of conjecturing it).

**Problem 2:**
Is it true that there exists a constant *r* < 5
such that every cyclically 4-edge-connected graph distinct from the Petersen
graph admits a nowhere-zero circular *r*-flow?

Send comments to Bojan.Mohar@uni-lj.si

##### Revised: april 07, 2003.