The Set of Circular Flow Numbers of Regular Graphs

This article determines the set of the circular flow numbers of regular graphs. LetF c be the set of the circular flow numbers of graphs, andF d c be the set of the circular flow numbers of d ‐regular graphs. If d is even, thenF d c = { 2 } . For d = 2 k + 1( k ≥ 1 ) it is known [6][E. Steffen, 2001] thatF 2 k + 1 c ∩ ( 2 + 1 k , 2 + 2 2 k − 1 ) = ∅ . We show thatF 2 k + 1 c = ( F c − [ 2 , 2 + 2 2 k − 1 ) ) ∪ { 2 + 1 k } . Hence, the interval ( 2 + 1 k , 2 + 2 2 k − 1 ) is the only gap for circular flow numbers of ( 2 k + 1 ) ‐regular graphs between 2 + 1 kand 5. Furthermore, if Tutte's 5‐flow conjecture is false, then it follows, that gaps for circular flow numbers of graphs in the interval [5, 6] are due for all graphs not just for regular graphs.