Edge colorings and circular flows on regular graphs

Let ${\varphi }_c(G)$ be the circular flow number of a bridgeless graph $G$. In [Edge-colorings and circular flow numbers of regular graphs, J. Graph Theory 79 (2015) 1-7] it was proved that, for every $t\ge 1$, $G$ is a bridgeless $(2t+1)$-regular graph with $\phi_c(G) \in {2+\frac{1}{t}, 2 + \frac{2}{2t-1}}$ifandonlyif$G$hasaperfectmatching$M$suchthat$G-M$isbipartite\mathrm{.}Thisimpliesthat$G$isaclass1graph\mathrm{.}For$t=1$, all graphs with circular flow number bigger than 4 are class 2 graphs. We show for all $t \geq 1$,that$2 + \frac{2}{2t-1} = \inf { \phi_c(G)\colon G $\text{ is a }$ (2t+1) $\text{-regular class }$ 2 $\text{ graph}$}$. This was conjectured to be true in [Edge-colorings and circular flow numbers of regular graphs, J. Graph Theory 79 (2015) 1-7]. Moreover we prove that $\inf \{{\varphi }_c(G)\text{\hspace{0.17em}⁣}:G$ is a $(2t+1)$-regular class $1$ graph with no perfect matching whose removal leaves a bipartite graph$\}=2+\frac{2}{2t-1}$. We further disprove the conjecture that every $(2t+1)$-regular class $1$ graph has circular flow number at most $2+\frac{2}{t}$.