Weighted exchange distance of basis pairs
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Two pairs of disjoint bases 𝐏_1=(R_1,B_1) and 𝐏_2=(R_2,B_2) of a matroid M are called equivalent if 𝐏_1 can be transformed into 𝐏_2 by a series of symmetric exchanges. In 1980, White conjectured that such a sequence always exists whenever R_1∪ B_1=R_2∪ B_2. A strengthening of the conjecture was proposed by Hamidoune, stating that minimum length of an exchange is at most the rank of the matroid. We propose a weighted variant of Hamidoune's conjecture, where the weight of an exchange depends on the weights of the exchanged elements. We prove the conjecture for several matroid classes: strongly base orderable matroids, split matroids, graphic matroids of wheels, and spikes.
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