Jim Geelen and Peter Nelson proved that, for a loopless connected binary matroid $M$ with an odd circuit, if a largest odd circuit of $M$ has $k$ elements, then a largest circuit of $M$ has at most $2k-2$ elements. The goal of this note is to show that, when $M$ is $3$-connected, either $M$ has a spanning circuit, or a largest circuit of $M$ has at most $2k-4$ elements. Moreover, the latter holds when $M$ is regular of rank at least four.
@article{10_37236_11462,
author = {Manoel Lemos and James Oxley},
title = {An upper bound for the circumference of a 3-connected binary matroid},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {1},
doi = {10.37236/11462},
zbl = {1511.05031},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11462/}
}
TY - JOUR
AU - Manoel Lemos
AU - James Oxley
TI - An upper bound for the circumference of a 3-connected binary matroid
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/11462/
DO - 10.37236/11462
ID - 10_37236_11462
ER -
%0 Journal Article
%A Manoel Lemos
%A James Oxley
%T An upper bound for the circumference of a 3-connected binary matroid
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/11462/
%R 10.37236/11462
%F 10_37236_11462
Manoel Lemos; James Oxley. An upper bound for the circumference of a 3-connected binary matroid. The electronic journal of combinatorics, Tome 30 (2023) no. 1. doi: 10.37236/11462
