J ul 2 01 3 Is the five - flow conjecture almost false ?

The number of nowhere zero ZQ flows on a graph G can be shown to be a polynomial in Q, defining the flow polynomial ΦG(Q). According to Tutte’s five-flow conjecture, ΦG(5) > 0 for any bridgeless G. A conjecture by Welsh that ΦG(Q) has no real roots for Q ∈ (4,∞) was recently disproved by Haggard, Pearce and Royle. These authors conjectured the absence of roots for Q ∈ [5,∞). We study the real roots of ΦG(Q) for a family of non-planar cubic graphs known as generalised Petersen graphs G(m,k). We show that the modified conjecture on real flow roots is also false, by exhibiting infinitely many real flow roots Q > 5 within the class G(nk, k). In particular, we compute explicitly the flow polynomial of G(119, 7), showing that it has real roots at Q ≈ 5.0000197675 and Q ≈ 5.1653424423. We moreover prove that the graph families G(6n, 6) and G(7n, 7) possess real flow roots that accumulate at Q = 5 as n → ∞ (in the latter case from above and below); and that Qc(7) ≈ 5.2352605291 is an accumulation point of real zeros of the flow polynomials for G(7n, 7) as n → ∞.