Two pairs of disjoint bases $\mathbf{P}_1=(R_1,B_1)$ and $\mathbf{P}_2=(R_2,B_2)$ of a matroid $M$ are called equivalent if $\mathbf{P}_1$ can be transformed into $\mathbf{P}_2$ by a series of symmetric exchanges. In 1980, White conjectured that such a sequence always exists whenever $R_1\cup B_1=R_2\cup B_2$. 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.
翻译:两对脱节基数$mathbf{P ⁇ 1=(R_1,B_1)$和$mathbf{P ⁇ 2=(R_2,B_2)美元,如果$\mathbf{P ⁇ 1美元可以通过一系列对称交换转换成$mathbf{P ⁇ 2$,那么,如果能够通过一系列对称交换来将美元转换成$mathbf{P ⁇ 1=P ⁇ 1=1=(R_1,B_1)美元和$mathbf{P ⁇ 2=(R_2)美元和美元=2\cupB_2=2美元,那么这种序列就一直存在。哈米杜恩提议加强参数,指出交换的最小长度最多为类类数。我们提议了一个Hamdoune的参数加权变量,其中交换的重量取决于交换元素的重量。我们证明若干类类类类类类类的参数:非常基本可排序的类动物、分裂型类、分裂型类、轮状和钉状的图形类。