Real flow number and the cycle rank of a graph

This article establishes a relationship between the real (circular) flow number of a graph and its cycle rank. We show that a connected graph with real flow number p / q  + 1, where p and q are two relatively prime numbers must have cycle rank at least p  +  q  − 1. A special case of this result yields that the real flow number of a 2‐connected cubic graph with chromatic index 4 and order at most 8 k  + 4 is bounded from below by 4 + 1/ k . Using this bound we prove that the real flow number of the Isaacs snark I 2 k  + 1 equals 4 + 1/ k , completing the upper bound due to Steffen [Steffen, J Graph Theory 36 (2001), 24–34]. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 11–16, 2008