rainbow_base_cover/notes/counterexample-search-2026-03-26.md

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Counterexample Search Notes

Goal

Try to find a counterexample to the rainbow-base-cover conjecture:

If a rank-r matroid on 2r elements decomposes into two disjoint bases, and the elements are colored by r colors each used twice, then three rainbow bases cover the ground set.

I also tracked the stronger special case of the double-cover conjecture in the partition-matroid setting:

For the rainbow bases, are there always four of them covering each element exactly twice?

Search Design

Encoding

Fix a pairing of the 2r elements into color classes. A rainbow basis chooses exactly one element from each pair, so for fixed coloring there are at most 2^r rainbow candidates.

For a matroid M, I tested:

1. whether M has two disjoint bases;
2. for each pairing, which of the 2^r transversals are actual bases;
3. whether some 3 rainbow bases cover all 2r elements;
4. whether some 4 rainbow bases cover every element exactly twice.

This was implemented in search_rainbow_counterexample.py.

Exhaustive part

Sage's AllMatroids(2r, r) is available up to r = 4, so I checked all unlabeled rank-r matroids on 2r elements for r = 1,2,3,4, and all pairings of the ground set:

• r = 1: 1 pairing
• r = 2: 3 pairings
• r = 3: 15 pairings
• r = 4: 105 pairings

Additional rank-5 probes

Since Sage does not exhaust all rank-5 matroids on 10 elements, I also sampled random linear matroids over small fields, and exhaustively checked some graphic families:

• random rank-5 linear matroids over GF(2), GF(3), GF(4), GF(5);
• all simple graphic matroids coming from 8-edge graphs on 5 vertices.

Results

Exhaustive search for r \le 4

No counterexample appeared.

rank r	matroids checked	qualifying matroids	max observed minimum cover	any 3-cover failure?	any 4-double-cover failure?
1	2	1	2	no	no
2	7	3	2	no	no
3	38	17	3	no	no
4	940	730	3	no	no

So:

• the rainbow-cover conjecture holds for every matroid in Sage's complete database with 2r \le 8;
• in the same range, the stronger partition-matroid double-cover statement also holds.

Explicit rank-4 witness requiring 3 bases

The first rank-4 example I found with minimum cover number exactly 3 is

• matroid: all_n08_r04_#493
• pairing: \bigl((0,5),(1,4),(2,3),(6,7)\bigr)

For this pairing there are exactly 8 rainbow bases:

(0,1,3,6), (0,1,3,7), (0,2,4,6), (0,2,4,7), (1,2,5,6), (1,2,5,7), (3,4,5,6), (3,4,5,7).

This instance still has:

• minimum rainbow cover size = 3;
• a 4-rainbow exact double cover.

So the search really is reaching the sharp bound 3, not just easy cases with cover number 2.

Graphic search

The known lower-bound example K_4 is reproduced computationally:

• all simple graphs on 4 vertices with 6 edges: 1 graph checked;
• maximum minimum cover number: 3;
• no 3-cover failure;
• no 4-double-cover failure.

I also checked all simple 8-edge graphs on 5 vertices:

• graphs checked: 45;
• connected graphs: 45;
• qualifying graphic matroids: 45;
• maximum minimum cover number: 3;
• no 3-cover failure;
• no 4-double-cover failure.

One concrete simple graphic rank-4 witness with minimum cover 3 is the graph with edge set

\bigl((0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3)\bigr)

and pairing

\bigl((0,3),(1,5),(2,4),(6,7)\bigr).

Random rank-5 linear search

No sampled rank-5 linear matroid produced a counterexample.

Finished runs:

field	samples	distinct matroids checked	qualifying matroids	max observed minimum cover	any 3-cover failure?	any 4-double-cover failure?
GF(2)	200	200	92	3	no	no
GF(2)	500	500	245	3	no	no
GF(3)	200	200	180	3	no	no
GF(4)	200	200	99	3	no	no
GF(5)	200	200	200	2	no	no

Observations

Small-rank evidence is strong

Up through rank 4, the search is exhaustive, not heuristic. In that range I found no obstruction even to the stronger four-basis exact double-cover property.

Rank 4 already has nontrivial tight examples

The bound 3 is still best possible in rank 4: there are pairings where 2 rainbow bases do not suffice, but 3 do.

The partition-matroid special case looks robust

At least computationally, the partition-matroid case of the double-cover conjecture behaves better than expected:

• exhaustive success for all matroids on 8 elements of rank 4;
• no failures in the rank-5 linear samples over four small fields;
• no failures in the checked graphic families.

What I would try next

More targeted rank-5 search

The next most plausible places to look are:

1. exhaustive or semi-exhaustive graphic search on 10 edges and 6 vertices;
2. sparse paving matroids of rank 5 on 10 elements;
3. biased random linear constructions, especially sparse or highly structured matrices rather than dense uniform random ones;
4. direct search for the stronger four-basis exact double-cover failure, since that might break before the three-cover statement does.

Structural reformulation

For a fixed pairing, the rainbow bases form a subset F \subseteq \{0,1\}^r. Then:

• $3$-cover means there exist x,y,z \in F such that in every coordinate, not all of x_i,y_i,z_i are equal;
• exact $4$-double-cover means there exist x_1,x_2,x_3,x_4 \in F such that every coordinate has exactly two $0$s and two $1$s.

This reformulation may be a better starting point for a structural attack than thinking directly in matroid language.

Current Status

I did not find a counterexample.

The strongest completed evidence from this turn is:

• exhaustive verification for all rank-r matroids on 2r elements with r \le 4;
• exhaustive verification for simple graphic rank-4 instances on 8 edges;
• no sampled failure among several hundred rank-5 linear matroids over GF(2), GF(3), GF(4), GF(5).