The cycles of a graph give a natural cyclic ordering to their edge-sets, and these orderings are consistent in that two edges are adjacent in one cycle if and only if they are adjacent in every cycle in which they appear together. An orderable matroid is one whose set of circuits admits such a consistent ordering. In this paper, we consider the question of determining which matroids are orderable. Although we are able to answer this question for non-binary matroids, it remains open for binary matroids. We give examples to provide insight into the potential difficulty of this question in general. We also show that, by requiring that the ordering preserves the three arcs in every theta-graph restriction of a binary matroid $M$, we guarantee that $M$ is orderable if and only if $M$ is graphic.
@article{10_37236_11117,
author = {Cameron Crenshaw and James Oxley},
title = {Ordering circuits of matroids},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {4},
doi = {10.37236/11117},
zbl = {1510.05024},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11117/}
}
TY - JOUR
AU - Cameron Crenshaw
AU - James Oxley
TI - Ordering circuits of matroids
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/11117/
DO - 10.37236/11117
ID - 10_37236_11117
ER -
%0 Journal Article
%A Cameron Crenshaw
%A James Oxley
%T Ordering circuits of matroids
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/11117/
%R 10.37236/11117
%F 10_37236_11117
Cameron Crenshaw; James Oxley. Ordering circuits of matroids. The electronic journal of combinatorics, Tome 29 (2022) no. 4. doi: 10.37236/11117
