Orientations of graphs

Carsten Thomassen
Technical University of Denmark

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Content: Let $\Gamma$ be an abelian group and $F$ a set of elements of $\Gamma$. We consider the following general problem: Given a graph $G$, is it possible to give every edge a direction and an element from $F$ such that, at every vertex, the in-sum equals the out-sum. Tutte's flow conjectures and the $(2+\epsilon)$-flow conjecture are instances of this problem. We discuss other instances as well and show when sufficiently large edge-connectivity guarantees an $F$-flow.

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