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A Bijective Proof of a Theorem of Knuth

Published online by Cambridge University Press:  21 July 2010

HODA BIDKHORI  and
SHAUNAK KISHORE
HODA BIDKHORI
Affiliation:
Massachusetts Institute of Technology, Massachusetts Avenue, Cambridge, MA 02139, USA (e-mail: bidkhori@mit.edu, skishore@mit.edu)
SHAUNAK KISHORE
Affiliation:
Massachusetts Institute of Technology, Massachusetts Avenue, Cambridge, MA 02139, USA (e-mail: bidkhori@mit.edu, skishore@mit.edu)
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Abstract

The line graph G of a directed graph G has a vertex for every edge of G and an edge for every path of length 2 in G. In 1967, Knuth used the Matrix Tree Theorem to prove a formula for the number of spanning trees of G, and he asked for a bijective proof [6]. In this paper, we give a bijective proof of Knuth's formula. As a result of this proof, we find a bijection between binary de Bruijn sequences of degree n and binary sequences of length 2n−1. Finally, we determine the critical groups of all the Kautz graphs and de Bruijn graphs, generalizing a result of Levine [7].

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 1 , January 2011 , pp. 11 - 25
Copyright
Copyright © Cambridge University Press 2010

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References

[1]Berget, A., Manion, A., Maxwell, M., Potechin, A. and Reiner, V. The critical group of a line graph. arXiv:0904.1246.Google Scholar
[2]Biggs, N. L. (1999) Chip-firing and the critical group of a graph. J. Algebraic Combin. 9 25–45.CrossRefGoogle Scholar
[3]Christianson, H. and Reiner, V. (2002) The critical group of a threshold graph. Linear Algebra Appl. 349 233–244.CrossRefGoogle Scholar
[4]Holroyd, A. E., Levine, L., Mészáros, K., Peres, Y., Propp, J. and Wilson, D. B. (2008) Chip-firing and rotor-routing on directed graphs. In In and Out of Equilibrium 2, Vol. 60 of Progress in Probability, Birkhäuser, pp. 331–364.CrossRefGoogle Scholar
[5]Jacobson, B., Niedermaier, A. and Reiner, V. (2003) Critical groups for complete multipartite graphs and Cartesian products of complete graphs. J. Graph Theory 44 231–250.CrossRefGoogle Scholar
[6]Knuth, D. E. (1967) Oriented subtrees of an arc digraph. J. Combin. Theory 3 309–314.CrossRefGoogle Scholar
[7]Levine, L. (2010) Sandpile groups and spanning trees of directed line graphs. To appear in Journal of Combinatorial Theory, Series A. arXiv:0906.2809.Google Scholar
[8]Stanley, R. P. (1999) Enumerative Combinatorics, Vol. 1, Cambridge University Press.CrossRefGoogle Scholar