Domino Exclusion Problem

. The classic domino exclusion problem consists of finding minimum number d ( n ) of dominoes on an n n  chessboard to prevent placement of another domino. This sequence of minimum numbers is discussed under A280984 at the On-Line Encyclopedia of Integer Sequences. With new theoretical insights and a specially designed computer program we were able to expand the sequence from n =18 to n = 33. New upper bounds of d ( n ) thought to be sharp have been obtained. The article also discusses the rectangle-free minimal domino packings. Small 3-dimensional grid squares up to n = 6 have been analysed.


Introduction
During the Covid-19 pandemic, the first author was writing a book on recreational mathematics.In one chapter, both old and new tasks about matchsticks were collected.In relation to the number 19, the following task was devised: "To create a vaccine that prevents Covid-19 virus from multiplying, it is necessary to colour a minimum number of unit edges of the 19 19  cell square so that each uncoloured edge has at least one point of contact with the coloured edge."As it turned out later this rather difficult task had far-reaching consequences.Solving this problem for small matchstick squares n n  the following sequence of minimum numbers was obtained, see Figure 1: 2, 3, 6, 9, 12, 17, ...Then, looking at The On-Line Encyclopedia of Integer Sequences (Shepard, 2017), it was understood that this matchstick problem is equivalent to the domino exclusion problem studied earlier in (Gyárfás et al., 1988).Domino exclusion problem consists of finding minimum number of dominoes on an n n  board to prevent placement of another domino, see Figure 2 as a transformation of Figure 1 in domino terminology.

Figure 1. Minimum number of coloured sticks
A third way to visualize the problem and not to draw unpainted square edges at all is to use graphs, see Figures 3 -4.
As far as we know, the first book in which one can find the tasks of excluding shapes (namely, pentominoes by monominoes on a chessboard 8 8 ) is the Golomb's classic book (Golomb, 1994).Exclusion problems in other areas (graph theory, statistical physics, percolation theory) may have to do with the following concepts: matching, minimum dominating sets, domination number, square grid graphs, an edge cover, dimmers, and others (Alanko et al., 2011), (Korte and Vygen, 2018).
The second author developed an efficient algorithm that allowed new progress in both the domino exclusion problem and its generalizations to n-dimensional grids.
The article uses generally accepted terms in mathematics: are the so-called ceiling and floor functions of x, respectively.be the minimum number of dominoes (edges isolating grid points) for which a packing exists.A domino packing is an arrangement of dominoes on a given board (here on a grid rectangle) to prevent placement of another domino.The following numbers are of particular importance in future estimates: ). , ( : A point is isolated if all its neighbours are connected by edges.Note that minimizing the number of edges or maximizing the number of isolated points or holes (1)

Estimates
Two important theorems are proved in this section.As a consequence of the first theorem, the following nice estimate is obtained: . 2 , , 3 Both old and new (updated) information one can found in (Kagey, 2019): "Fifteen terms are known, and a few folks have conjectured that (3) Walter Trump has just added the terms 19 -33 of the sequence (with with some examples of optimal solutions and announced an effective algorithm for finding the optimal solutions./Jun 14 at 20:30/" The sequence of the left hand side of (3) here means the numbers  (Gyárfás et al., 1988): is valid for strings (rectangles with one row or one column).
Equality can only exist if the number of grid points is 1 3   k n , see Figure 5 with H = 4 and D = 3.
For rectangles, a stronger estimate D H  can be proved (Theorem 1).The proof uses the same basic ideas as in the article (Gyárfás et al., 1988) but the presentation is shorter and quite different.the number of dominoes, respectively holes incident to the j-th side of the rectangle, see Figure 6 where follow immediately from the notations, but their essence, salt, is in their geometric interpretation.Let us interpret (4), e.g. for j = 1.The difference  Without loss of generality we can assume that 3 1 Assume R is the smallest rectangle that satisfies (5).The number of B points is an even number, so they can be grouped in pairs.Since at least one point in each pair is the domino endpoint, then Remark 2. With this proof technique, an estimate D H  can also be obtained for 3-dimensional grid rectangles. Corollary.
) , ( ) , ( Theorem 2. The following estimates are valid: Proof 1.The case n = 3k, estimate (6), is trivial in the sense that minimal arrangements (solutions) also for ] [ n m  rectangles one can obtain using an elementary pattern shown in Figure 10.Copies of ] 3 [  m can be added to each other as many times as needed.
Moreover, it is important that the additive property holds: rectangles are not adjacent, because in that case we would not get the minimal arrangement: . 3 8 3 .
For simplification purposes, let us use the following elementary calculations: The correctness of this inequality can be easily proved by taking r j k   5 and checking the five values of remainder r.
and instead of (10) we now have the inequality This inequality for arbitrary k is not correct at all, for example, k = 3.But here there is a subtle nuance, namely, the inequality does not have to be checked for all k, but only for those k for which n = 3k + 1 does not fit in the case 2a.We have previously found that the smallest square for which equality (12) does not hold is ] 19 19 [  .In this section we will look for the smallest rectangle for which the analogue of equality (12), i.e. ( 13) is no longer valid.
Proof.According to (2) it suffices to show solutions (arrangements) with the specified number ) , ( 0 m n D of dominoes.Such solutions are easy to find for small m.For m = 2, see Figure 13, and for m = 3k, see the periodic arrangement shown in Figure 10. The fact that the presented solutions contain the required number of dominoes can be easily verified using the property:  Now let's use a more advanced idea: in the role of elementary rectangles, let's take appropriately selected blocks that are periodically added.For m = 13 the minimal arrangements are shown in Figure 15.The key to the proof is now a periodically movable string (a block of red edges), which we move by three units to obtain all the required rectangles.From here, removing the first two rows, we easily get the minimal arrangements for rectangles with m = 11 rows.Similarly, removing the appropriate number of rows we will obtain minimal arrangements for the other required values for m (m = 10, 8, and 7).To avoid ambiguity, let us clarify that in the case n = 3k + 5, after removing the first two rows, the vertical domino is shifted down one unit, and after removing the first 5 rows, the vertical domino is replaced by a horizontal one.Theorem is proved.
Remark 4. As a further study, we propose the following hypothesis:

Rectangle-free packings
In the previous section, the partition of squares in rectangles was crucial to prove the theorems.Let us now consider the question of the existence of minimal packings which cannot be divided into smaller rectangles.Such packings will be called rectanglefree packings (arrangements, solutions).The smallest square for which a rectangle-free packing exists is the ] 5 5 [  square, the  20 obtained from the ] 5 5 [  square repeating the red fragment.The fact that there is no a rectangle-free packing for ] [ n n  square does not mean that there is no a rectangle-free packing for ] [ n m rectangle.See Figure 21 as an example.
Developing the idea of periodicity in two directions, we manage to find rectanglefree packings of squares with Equality ( 14) with precision to the notations is equivalent to equality (7) from (Gyárfás et al., 1988).Equality (15) immediately follows from (14 Equality ( 16) is very important.It shows that by minimizing B we minimize D.
With a backtracking algorithm we enumerate the domino packings of a ] [ n m  rectangle with a given number D of dominoes.We do this by placing dominoes row by row from left to right.In general there are 3 possibilities to continue in a grid cell: empty, horizontal or vertical domino.Therefore the number of paths is greater than 2 / 3 mn .Even rather small rectangles cannot be handled as the number of paths is too high.The new approach considers the known number B of bad domino constellations.As soon as (B + 1) such constellations are reached the current path can be abandoned.For small B this algorithm works very fast.Dependent on the used processor and programming language the enumeration (determination of the number of all packings for a number D of dominoes) for squares up to ] 20 20 [  can be done in less than a minute.The status of each cell is described in an oversized array sq(x,y) with 0  x  n + 1 and 0  y  m + 1, where x is the column and y the row, with the following numerical characteristics (Figure 24): The source code of the recursive procedure cpos is presented in an easy to read basic pseudo code.All variables are integers and all are public except of x, y and mBc.The main program askes for the values of m, n and D, calculates B, initializes the array sq() as shown above and calls the procedure by cpos(1,1).Bad domino constellations were count in Bc and compared with B. At the end the value of the variable Scnt is the number of different packings.Source code of the recursive procedure cpos(x,y)are presented in Appendix.Different cases for the planned domino as they occur in the procedure are shown in Figure 25.
The most important results obtained with a computer program are summarized in four tables.In the coloured cases it was proved by exhaustive computer search that packings with D 0 (m, n) dominoes do not exist.The results obtained by using a computer program are summarized in Table 4.

Conclusions
The article contains the theorems of pure mathematics, as well as computer-assisted proofs.New progress has been made in solving the domino exclusion problem, including a deeper understanding of the structure of minimal packings.Proof (or disproof) of the hypothesis formulated in the Remark 4 could be a natural continuation of this study.

Figure
Figure 4. Interpretation by graph the number of dominoes, holes that are incident (have contact, touch) to B, in the rectangle, except for the last row.Each such hole has a domino below it.It is important that each hole has its own corresponding domino. 1 T is the number of dominoes not yet counted. 1 T describes (redundant) dominoes exactly above which there are no holes and which do not belong to 1 D .Figure7shows several domino

Figure 5 .
Figure 5. Domino arrangement with H = D + 1 Figure 9 is permissible.Since the process is not complete in the column k m 3 , the theorem is proved.

Table 2 .
Smallest number D(m, n) of dominoes for which a packing exists In the coloured cases: D(m, n) = d 0 (m, n) + 1 Table 3. Number of minimal domino packings in ] [ n m  -rectangles with D 0 (m, n) dominoes (including reflections and rotations) used for a three-dimensional rectangles.As in two dimensions, the estimate (17) is sharp if any of the edge lengths is a multiple of 3. In this case, the minimum packing is obtained by repeating the minimum two-dimensional rectangles in layers.For illustration see Figure26with the minimal packing of

Figure 26 .
Figure 26.Minimal packing of in layers

Table 1 .
Number of domin packings in

Table 4 .
Number of minimal packings of cubeFrom Table4we see that