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Yu Cong
2025-11-04 22:44:10 +08:00
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@@ -1,10 +1,11 @@
\documentclass[a4paper]{article} \documentclass[a4paper,11pt]{article}
\usepackage{chao} \usepackage{chao}
\DeclareMathOperator{\supp}{supp}
\title{Cogirth of Perturbed Graphic Matroids} \title{Cogirth of Perturbed Graphic Matroids}
\author{} \author{}
\date{} \date{}
\begin{document} \begin{document}
% \maketitle \maketitle
We want to use basepacking on perturbed graphic matroid. Basepacking works for deletion closed matroid classes with constant cogirth-packing gap. We want to use basepacking on perturbed graphic matroid. Basepacking works for deletion closed matroid classes with constant cogirth-packing gap.
PGMs are closed under deletion. It remains to show PGMs have constant gap. PGMs are closed under deletion. It remains to show PGMs have constant gap.
@@ -85,8 +86,8 @@ Let $A$ be the binary representation of $M$. Note that the cocircuit space of $M
Now we consider the hitting sets. It follows from definitions that the minimum hitting set of $\{B-e |\forall B ,\forall e\in B\cap C(B,f)\}$ is exactly the minimum cocircuit of $(M+f)/f$, which is the minimum set in the cocircuit space with $yv=0$. So the minimum hitting set of $B-e$ is always an even cocycle. Now we consider the hitting sets. It follows from definitions that the minimum hitting set of $\{B-e |\forall B ,\forall e\in B\cap C(B,f)\}$ is exactly the minimum cocircuit of $(M+f)/f$, which is the minimum set in the cocircuit space with $yv=0$. So the minimum hitting set of $B-e$ is always an even cocycle.
Consider a set $X$ which fails to hit all $\{C(B,f)|\forall B\}$. Then there must be a set $F\subseteq E\setminus X$ that span the new element $f$. On matrix, this is equivalent to the existence of $\chi_F$ that $A\chi_F=v$. Using Farkas' lemma on $\mathbb F_2$, this implies that there dose not exists $y$ that ${\rm supp}(yA)\in X\land yv=1$. (chatGPT says one can use Farkas lemma on finite field like this, need to verify). Consider a set $X$ which fails to hit all $\{C(B,f)|\forall B\}$. Then there must be a set $F\subseteq E\setminus X$ that span the new element $f$. On matrix, this is equivalent to the existence of $\chi_F$ that $A\chi_F=v$. Using Farkas' lemma on $\mathbb F_2$, this implies that there dose not exists $y$ that $\supp(yA)\in X\land yv=1$. (chatGPT says one can use Farkas lemma on finite field like this, need to verify).
So $X$ hits all $\{C(B,f)|\forall B\}$ iff there is such a $y$. Minimizing $X$ pushes ${\rm supp}(yA)$ which is an odd cocycle. Hence, the minimum hitting set for $C(B,f)$ is always an odd cocycle. So $X$ hits all $\{C(B,f)|\forall B\}$ iff there is such a $y$. Minimizing $X$ pushes $\supp(yA)$ which is an odd cocycle. Hence, the minimum hitting set for $C(B,f)$ is always an odd cocycle.
The theorem then follows directly from the fact that any cocycle is either odd or even. The theorem then follows directly from the fact that any cocycle is either odd or even.