Edge colorings and circular flows on regular graphs

Abstract

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 ${\varphi }_c(G)\in \{2+\frac{1}{t},2+\frac{2}{2t-1}\}$ if and only if $G$ has a perfect matching $M$ such that $G-M$ is bipartite. This implies that $G$ is a class 1 graph. For $t=1$, all graphs with circular flow number bigger than 4 are class 2 graphs. We show for all $t\ge 1$, that $2+\frac{2}{2t-1}=\inf \{{\varphi }_c(G):G\text{ is a }(2t+1)\text{-regular class }2\text{ graph}\}$. 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 $\inf \{{\varphi }_c(G):G$ 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 $2+\frac{2}{t}$.

17 pages; submitted for publication