rainbow_base_cover/notes/main.md

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title: Matroid Rainbow Base Cover
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This problem is proposed in [@emlektabla16_2024].

Let $M=(E,\mathcal B)$  be a rank-$r$ matroid whose ground set decomposes into two disjoint bases. Furthermore,
assume that $E$ is colored by $r$ colors, each color appearing exactly twice. A basis of $M$ is called rainbow if
it does not contain two elements of the same color.

::: {.Problem #prob-rainbowbasecover}
What is the minimum number of rainbow bases needed to cover $E$?
:::

Kristóf Bérczi conjectured the minimum number is 3.

Currently known bounds:

- upperbound: $\floor{\log_2 |E|}+1$ by matroid intersection;
- lowerbound: $3$ on graphic matroid of $K_4$.

::: {.Conjecture #conj-doublecover}
Let $M_1$ and $M_2$ be two matroid on the same ground set $E$. Assume that $E$ decomposes into two bases in both of them.
Then $M_1$ and $M_2$ has four common bases (allow repetition) that cover each element exactly twice.
:::

Let $M_1$ be the partition matroid of colors and let $M_2$ be $M$.
If [@conj-doublecover] is true, then 3 common bases will be enough to cover $E$.

# Known results

Let $\beta(M)$ be the covering number of matroid $M$. Given two matroids $M_1,M_2$ on the same ground set, $\beta(M_1\cap M_2)$ is the minimum number of common independent sets needed to cover the ground set.
[@conj-doublecover] is basically asking for $\beta(M_1\cap M_2)$ when the ground set $E$ is partitioned into 2 common bases of $M_1$ and $M_2$.

## Results and conjectures on $\beta(M_1\cap M_2)$

One can see that $\beta(M_1\cap M_2)\geq \min \set{\beta(M_1),\beta(M_2)}$. Aharoni and Berger showed that $\beta(M_1\cap M_2)\leq 2 \max \set{\beta(M_1),\beta(M_2)}$

::: {.Conjecture title="Aharoni and Berger"}
Let $M_1$ and $M_2$ be two matroids on the same ground set.

1. If $\beta(M_1)\neq \beta(M_2)$, then $\beta(M_1\cap M_2)=\max \set{\beta(M_1),\beta(M_2)}$.
2. If $\beta(M_1)= \beta(M_2)$, then $\beta(M_1\cap M_2)\leq \max \set{\beta(M_1),\beta(M_2)}+1$.
:::

Cases that have been verified:

- If $\beta(M_1)=\beta(M_2)=2$ then $\beta(M_1\cap M_2)=3$.^[Note that this case does not give an answer to [@prob-rainbowbasecover] since the covering in $\beta(M_1\cap M_2)$ uses common independent sets instead of common bases.]
- If $\beta(M_1)=2,\beta(M_2)=3$, then $\beta(M_1\cap M_2)\leq 4$
- ...

::: {.Theorem title="Davies and McDiarmid"}
Let $M_1$ and $M_2$ be strongly base orderable matroids on the same ground set.
Then $\beta(M_1\cap M_2)= \max \set{\beta(M_1),\beta(M_2)}$.
:::

See [@berczi_partitioning_2024] for refs. Bérczi and Schwarcz generalized SBO using the following definition.

::: {.Definition title="strongly base reorderable"}
A matroid is called **strongly base reorderable**(SBRO) if for any pair $A,B$ of bases, there exists bases $A',B'$ and a bijection $\phi:A'\setminus B'\to B'\setminus A'$ such that $A'\cap B'=A\cap B,A'\cup B'=A\cup B$, and $(A'\setminus X)\cup \phi(X)$ is a basis for every $X\subset A'\setminus B'$.
:::

::: Lemma
The class of SRBO matroids is complete(closed under taking minors, duals, direct sums, truncations and induction by directed graphs).
:::

They also showed that SBRO is sufficient for $\beta(M_1\cap M_2)=\max \set{\beta(M_1),\beta(M_2)}$.

::: Theorem
Let $M_1$ and $M_2$ be strongly base reorderable matroids on the same ground set.
Then $\beta(M_1\cap M_2)= \max \set{\beta(M_1),\beta(M_2)}$.
:::

## Circuit cover

Let $M=(E,\mathcal B)$ be a matroid and let $A,B\in \mathcal B$ be two bases of $M$.
Define a graph $G=(A\Delta B, F)$.