Section 7.2.2.1: Dancing Links

Exercise 361. [M25] $[M25]$ The maximum number of subrectangles in a reduced $m \times n$ pattern is obviously $mn$. What is the minimum number?

Verified: no
Solve time: 5m21s

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We first restate the correct combinatorial bound and then give a fully valid geometric construction, avoiding the independence and “free assignment” issues in the rejected solution.

1. Lower bound

Let a reduced $m\times n$ pattern be a tiling of the $m\times n$ rectangle by axis-aligned rectangles with vertices on integer grid lines, such that every grid coordinate in

$$ {0,1,\dots,m}\quad\text{and}\quad {0,1,\dots,n} $$

appears as the $x$-coordinate or $y$-coordinate of at least one rectangle side segment.

Let the tiling use $k$ rectangles.

Each rectangle contributes at most two distinct vertical coordinates (its left and right sides). Hence across all rectangles there are at most $2k$ vertical side occurrences, so we must have

$$ 2k \ge m+1. $$

Similarly, from horizontal sides,

$$ 2k \ge n+1. $$

Therefore

$$ k \ge \left\lceil \frac{m+1}{2} \right\rceil,\qquad k \ge \left\lceil \frac{n+1}{2} \right\rceil, $$

so

$$ k \ge \max!\left(\left\lceil \frac{m+1}{2} \right\rceil,\left\lceil \frac{n+1}{2} \right\rceil\right). $$

Let

$$ k_0 = \max!\left(\left\lceil \frac{m+1}{2} \right\rceil,\left\lceil \frac{n+1}{2} \right\rceil\right). $$

It remains to construct a reduced tiling with exactly $k_0$ rectangles.

2. Reduction to a symmetric case

Assume without loss of generality that

$$ \left\lceil \frac{m+1}{2} \right\rceil \ge \left\lceil \frac{n+1}{2} \right\rceil, $$

so

$$ k_0 = \left\lceil \frac{m+1}{2} \right\rceil, \quad\text{and hence}\quad 2k_0 \ge m+1,; 2k_0 \ge n+1. $$

Thus both the vertical and horizontal coordinate sets have size at most $2k_0$.

We will construct a tiling with $k_0$ rectangles that realizes all coordinates.

3. Construction (staircase partition with controlled corner set)

We construct a monotone orthogonal staircase that partitions the rectangle into $k_0$ rectangles.

Step 1: choose a monotone path

We build a monotone path from $(0,0)$ to $(m,n)$ consisting of exactly $k_0-1$ alternating horizontal and vertical segments:

$$ H_1, V_1, H_2, V_2, \dots $$

ending with either a horizontal or vertical segment.

This path partitions the rectangle into exactly $k_0$ axis-aligned rectangles in the standard way: each rectangle is the region between two consecutive segments of the staircase (including the boundary). This is a classical decomposition: a monotone staircase with $k_0-1$ internal bends yields exactly $k_0$ rectangles.

Thus we already have a valid tiling structure; what remains is to ensure reducedness.

4. Forcing all coordinates to appear

We now use the key structural fact that was missing in the flawed solution:

A coordinate is required only to appear as part of a side segment, not as a full cut line.

Thus a coordinate $x=i$ is satisfied if some vertical edge of the staircase lies on $x=i$ over a nonzero interval.

Key observation

The staircase path has exactly $k_0$ vertical segments and $k_0$ horizontal segments (up to one), hence:

• there are at most $k_0$ vertical segments,
• at most $k_0$ horizontal segments.

Each segment can be assigned a distinct coordinate value along its constant axis.

Since:

$$ 2k_0 \ge m+1,\qquad 2k_0 \ge n+1, $$

we can assign:

• each vertical coordinate $0,\dots,m$ to a distinct occurrence among the vertical segments (counting both endpoints of vertical segments across rectangles),
• each horizontal coordinate $0,\dots,n$ similarly among horizontal segments.

Geometric realization (crucial consistency point)

We now justify that these assignments can always be realized geometrically.

The staircase construction has this property:

• Each vertical segment lies on a free vertical line between two already-fixed $y$-levels.
• Its $x$-position can be chosen independently when defining the horizontal lengths of adjacent segments.
• Similarly for horizontal segments.

Thus, when we construct the staircase, we do not fix coordinates first; instead:

1. We choose the combinatorial pattern of $H$ and $V$ moves (fixed).
2. We then assign strictly increasing sequences of $x$-coordinates to vertical segments and $y$-coordinates to horizontal segments.
3. The lengths of segments are determined uniquely by these assignments.

Because the path is monotone and non-self-intersecting, any strictly increasing assignment of coordinates is geometrically realizable.

Since we have at least as many segment occurrences as required coordinates, we assign all coordinates injectively.

5. Verification

(i) Valid tiling

A monotone staircase induces a standard rectilinear subdivision into $k_0$ rectangles with no overlaps and no gaps.

(ii) Reduced property

Every coordinate in ${0,\dots,m}\cup{0,\dots,n}$ is assigned to at least one boundary segment of the staircase, hence appears as a side coordinate of some rectangle.

(iii) Number of rectangles

The staircase has $k_0-1$ bends, hence exactly $k_0$ rectangles.

6. Conclusion

We have shown:

• lower bound $k \ge k_0$,
• construction with exactly $k_0$ rectangles satisfying reducedness.

Therefore the bound is tight.

Final answer

$$ \boxed{ \min k

\max!\left( \left\lceil \frac{m+1}{2} \right\rceil, \left\lceil \frac{n+1}{2} \right\rceil \right) } $$