2 research outputs found

    Interpolation theorem for a continuous function on orientations of a simple graph

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    summary:Let GG be a simple graph. A function ff from the set of orientations of GG to the set of non-negative integers is called a continuous function on orientations of GG if, for any two orientations O1O_1 and O2O_2 of GG, f(O1)f(O2)1|f(O_1)-f(O_2)|\le 1 whenever O1O_1 and O2O_2 differ in the orientation of exactly one edge of GG. We show that any continuous function on orientations of a simple graph GG has the interpolation property as follows: If there are two orientations O1O_1 and O2O_2 of GG with f(O1)=pf(O_1)=p and f(O2)=qf(O_2)=q, where p<qp<q, then for any integer kk such that p<k<qp<k<q, there are at least mm orientations OO of GG satisfying f(O)=kf(O) = k, where mm equals the number of edges of GG. It follows that some useful invariants of digraphs including the connectivity, the arc-connectivity and the absorption number, etc., have the above interpolation property on the set of all orientations of GG

    Interpolation theorem for a continuous function on orientations of a simple graph

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    Let G be a simple graph. A function f from the set of orientations of G to the set of Iron-negative integers is called a continuous function on orientations of G if, for any two orientations O-1 and O-2 of G, \f(O-1) - f(O-2)\ less than or equal to 1 whenever O-1 and O-2 differ in the orientation of exactly one edge of G. We show that any continuous function on orientations of a simple graph G has the interpolation property as follows: If there are two orientations O-1 and O-2 of G with f(O-1) = p and f(O-2) = q, where p < q, then for any integer k such that p < k < q, there are at least m orientations O of G satisfying f(O) = k, where m equals the number of edges of G. It follows that some useful invariants of digraphs including the connectivity, the arc-connectivity and the absorption number, etc., have the above interpolation property on the set of all orientations of G
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