Partial Graph Orientations and the Tutte Polynomial release_5pwhodphtjagdlqczdy3t3wzgu

by Spencer Backman

Released as a article .

2014  

Abstract

Gessel and Sagan investigated the Tutte polynomial, T(x,y) using depth first search, and applied their techniques to show that the number of acyclic partial orientations of a graph is 2^gT(3,1/2). We provide a short deletion-contraction proof of this result and demonstrate that dually, the number of strongly connected partial orientations is 2^n-1T(1/2,3). We then prove that the number of partial orientations modulo cycle reversals is 2^gT(3,1) and the number of partial orientations modulo cut reversals is 2^n-1T(1,3). To prove these results, we introduce cut and cycle minimal partial orientations which provide distinguished representatives for partial orientations modulo cut and cycle reversals. These extend classes of total orientations introduced by Gioan, and Greene and Zaslavksy, and we highlight a close connection with graphic and cographic Lawrence ideals. We conclude with edge chromatic generalizations of the quantities presented, which allow for a new interpretation of the reliability polynomial for all probabilities, p with 0 < p <1/2.
In text/plain format

Archived Files and Locations

application/pdf  389.6 kB
file_qaf6wz6gfjgixgxkrtlc7rpvpe
arxiv.org (repository)
web.archive.org (webarchive)
Read Archived PDF
Preserved and Accessible
Type  article
Stage   submitted
Date   2014-09-02
Version   v2
Language   en ?
arXiv  1408.3962v2
Work Entity
access all versions, variants, and formats of this works (eg, pre-prints)
Catalog Record
Revision: e339a4e3-03a5-42ce-8eed-4fc2b3bdd86b
API URL: JSON