Edge colorings and circular flows on regular graphs

D. Mattiolo, E. Steffen, ArXiv:2001.02484 (2020).

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Let $\phi_c(G)$ 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 $t \geq 1$, $G$ is a bridgeless $(2t+1)$-regular graph with $\phi_c(G) \in \{2+\frac{1}{t}, 2 + \frac{2}{2t-1}\}$ 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 $t \geq 1$, 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\{ \phi_c(G)\colon 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 $2+\frac{2}{t}$.
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arXiv:2001.02484
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Mattiolo D, Steffen E. Edge colorings and circular flows on regular graphs. arXiv:200102484. 2020.
Mattiolo, D., & Steffen, E. (2020). Edge colorings and circular flows on regular graphs. ArXiv:2001.02484.
@article{Mattiolo_Steffen_2020, title={Edge colorings and circular flows on regular graphs}, journal={arXiv:2001.02484}, author={Mattiolo, Davide and Steffen, Eckhard}, year={2020} }
Mattiolo, Davide, and Eckhard Steffen. “Edge Colorings and Circular Flows on Regular Graphs.” ArXiv:2001.02484, 2020.
D. Mattiolo and E. Steffen, “Edge colorings and circular flows on regular graphs,” arXiv:2001.02484. 2020.
Mattiolo, Davide, and Eckhard Steffen. “Edge Colorings and Circular Flows on Regular Graphs.” ArXiv:2001.02484, 2020.

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