A bijective proof of a theorem of knuth

  • Authors:

  • Hoda Bidkhori;Shaunak Kishore

  • Affiliations:

  • Massachusetts institute of technology, massachusetts avenue, cambridge, ma 02139, usa (e-mail: bidkhori@mit.edu, skishore@mit.edu);Massachusetts institute of technology, massachusetts avenue, cambridge, ma 02139, usa (e-mail: bidkhori@mit.edu, skishore@mit.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2011

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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].