Abstract
Given a weighted graph, let w 1, w 2, . . . ,w n denote the increasing sequence of all possible distinct spanning tree weights. In 1992, Mayr and Plaxton proved the following conjecture proposed by Kano: every spanning tree of weight w 1 is at most k−1 edge swaps away from some spanning tree of weight w k . In this paper, we extend this result for matroids. We also prove that all the four conjectures due to Kano hold for matroids provided one partitions the bases of a matroid by the weight distribution of its elements instead of their weight.
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References
Kano, M.: Maximum and kth maximal spanning trees of a weighted graph. Combinatorica 7, 205–214 (1987)
Kawamoto, T., Kajitani, Y., Shinoda, S.: On the second maximal spanning trees of a weighted graph (in Japanese). Trans. IECE of Japan 61A, 988–995 (1978)
Mayr, E.W., Plaxton, C.G.: On the spanning trees of weighted graphs. Combinatorica 12, 433–447 (1992)
Oxley, J.G.: Matroid theory. Oxford University Press, New York, 1992
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The author was partially supported by CNPq (Grant No. 302195/02-5) and ProNEx/CNPq (Grant No. 664107/97-4)
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Lemos, M. Weight Distribution of the Bases of a Matroid. Graphs and Combinatorics 22, 69–82 (2006). https://doi.org/10.1007/s00373-005-0648-6
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DOI: https://doi.org/10.1007/s00373-005-0648-6