Abstract.
It is known that the functions F G (k) and I G (k) evaluating the numbers of nowhere-zero ℤ k - and k-flows in a graph G, respectively, are polynomials of k. If X is a totally cyclic orientation of G, then the number of integral flows having values 1,…,k−1 on the arcs of X can be evaluated by a polynomial I X (k). F G (k) and I G (k) can be expressed as sums of I X (k). In this paper we show that the value I X (k) is positive for every totally cyclic orientation X of G if and only if k is greater than or equal to the maximum cardinality of an elementary edge-cut of G.
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Acknowledgments. This paper was finished during visiting School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia.
Mathematics Subject Classification (1991): 05C15, 05C20
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Kochol, M. A Theorem About Elementary Cuts and Flow Polynomials. Graphs and Combinatorics 19, 389–392 (2003). https://doi.org/10.1007/s00373-002-0516-6
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DOI: https://doi.org/10.1007/s00373-002-0516-6