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The zero-sum 6-flow conjecture

Let GG be a finite undirected graph. A zero-sum flow assigns a nonzero integer to each edge so that the sum of the assignments on all edges incident with every vertex is zero. A zero-sum kk-flow is a zero-sum flow whose values lie in {±1,…,±(k−1)}\{\pm1,\ldots,\pm(k-1)\}. Zero-sum 6-flow conjecture. If a graph GG admits a zero-sum flow, then it admits a zero-sum 6-flow. This is equivalent to Bouchet's conjecture by the theorem stated in the paper and is presented as an open problem; the paper proves the special case of 5-regular graphs.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The zero-sum 6-flow conjecture

    Let GG be a graph. A zero-sum flow is a function f:E(G)→R∖{0}f:E(G)\to\mathbb R\setminus\{0\} such that the values on the edges incident with each vertex sum to zero. A zero-sum kk-flow is a zero-sum flow with integer values satisfying 1≤∣f(e)∣≤k−11\leq |f(e)|\leq k-1 for every edge ee. Zero-Sum Conjecture. If GG admits a zero-sum flow, then it admits a zero-sum 66-flow. This is equivalent to the signed-graph 66-flow conjecture for all-negative signed graphs, and is presented as its zero-sum formulation; the source gives no resolution.

    source: Tomáš Kaiser, Edita Rollová and Robert Lukot'ka, “Nowhere-zero flows in signed graphs: A survey”, arXiv:1608.06944 (2016).

Sources & referencesView supporting material

Primary source

Fan Yang and Xiangwen Li, “Zero-sum 6-flows in 5-regular graphs”, arXiv:1601.07813 (2016).

graph-flowssigned-graphszero-sum-flows

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