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丛宇
2026-03-28 12:13:03 +08:00
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@@ -8,7 +8,7 @@ Let $M=(E,\mathcal B)$ be a rank-$r$ matroid whose ground set decomposes into t
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
::: {.Problem #prob-rainbowbasecover}
What is the minimum number of rainbow bases needed to cover $E$?
:::
@@ -21,7 +21,7 @@ Currently known bounds:
::: {.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 that cover each element exactly twice.
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 $M$.
@@ -43,9 +43,35 @@ Let $M_1$ and $M_2$ be two matroids on the same ground set.
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.
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 $G=(A\Delta B, F)$.