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Update notes.tex

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Yu Cong
2025-11-04 22:44:10 +08:00
parent 82d8bf7ac3
commit 94de1d1f19
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notes.tex
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@@ -1,10 +1,11 @@
\documentclass[a4paper]{article}
\documentclass[a4paper,11pt]{article}
\usepackage{chao}
\DeclareMathOperator{\supp}{supp}
\title{Cogirth of Perturbed Graphic Matroids}
\author{}
\date{}
\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.
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.
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).
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.
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 $\supp(yA)$ 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.