generated from congyu/work_with_codex
26 lines
988 B
Markdown
26 lines
988 B
Markdown
----
|
|
title: Matroid Rainbow Base Cover
|
|
----
|
|
|
|
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
|
|
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 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 $E$. |