Abstract
It is known that if we order the edges of a graph, or more generally, elements of a matroid, then each spanning forest, or basis, B has a subset ψ(B) of "internally passive" elements, and for each forest, or independent set, F, there is a unique basis B such that ψ(B) ⊆ F ⊆ B. In the context of matroids generally, we examine the structure of the collection of sets {ψ(B) : B is a basis}, particularly looking at the sequence of numbers of these sets of each cardinality, which we conjecture is log-concave.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Björner, Homology of matroids, preprint.
V. W. Bryant, J. E. Dawson and Hazel Perfect, Hereditary circuit spaces, Compos. Math. 37 (1978), 339–351.
H. H. Crapo, The Tutte polynomial, Aequationes Math. 3 (1969), 211–229.
J. E. Dawson, A construction for a family of sets and its application to matroids, Combinatorial Mathematics VIII, Lecture Notes in Math. 884 (Springer-Verlag, Berlin, 1981), 136–147.
J. H. Mason, Matroids: unimodal conjectures and Motzkin's theorem, Combinatorics — Proc. Conf. Combinatorial Math., Math. Inst., Oxford; (eds. D. J. A. Welsh and D. R. Woodall) (1972), 207–221.
Hazel Perfect, Notes on circuit spaces, J. Math. Anal. Appl. 54 (1976), 530–537.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1984 Springer-Verlag
About this paper
Cite this paper
Dawson, J.E. (1984). A collection of sets related to the tutte polynomial of a matroid. In: Koh, K.M., Yap, H.P. (eds) Graph Theory Singapore 1983. Lecture Notes in Mathematics, vol 1073. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073117
Download citation
DOI: https://doi.org/10.1007/BFb0073117
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-13368-1
Online ISBN: 978-3-540-38924-8
eBook Packages: Springer Book Archive