congyu/rainbow_base_cover
1
Fork 0
generated from congyu/work_with_codex
Code Issues Pull Requests Activity

Add known results and bibliography

Browse Source 
Introduce a "Known results" section discussing β(M1∩M2), state the
Aharoni–Berger conjectures and the Davies–McDiarmid theorem, and add
three bibliography entries (emlektabla16_2024, rainbow bases SIAM,
berczi_partitioning_2024).
This commit is contained in:
丛宇
2026-03-26 17:43:55 +08:00
parent 22037a5f23
commit 89a458c76c
2 changed files with 56 additions and 1 deletions
Download Patch File Download Diff File Expand all files Collapse all files

27
notes/main.md
View File

@@ -2,6 +2,8 @@
title: Matroid Rainbow Base Cover
----
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.
@@ -23,4 +25,27 @@ 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 $E$.
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 $\beta(M_1\cap M_2)\leq \max \set{\beta(M_1),\beta(M_2)}+1$.
:::
::: {.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.