Subfield: Combinatorics


Question: The columns and the rows of a $3 n \times 3 n$ square board are numbered $1,2, \ldots, 3 n$. Every square $(x, y)$ with $1 \leq x, y \leq 3 n$ is colored asparagus, byzantium or citrine according as the modulo 3 remainder of $x+y$ is 0,1 or 2 respectively. One token colored asparagus, byzantium or citrine is placed on each square, so that there are $3 n^{2}$ tokens of each color.

Suppose that one can permute the tokens so that each token is moved to a distance of at most $d$ from its original position, each asparagus token replaces a byzantium token, each byzantium token replaces a citrine token, and each citrine token replaces an asparagus token. Prove that it is possible to permute the tokens so that each token is moved to a distance of at most $d+2$ from its original position, and each square contains a token with the same color as the square.

Solution:
Without loss of generality it suffices to prove that the A-tokens can be moved to distinct $\mathrm{A}$-squares in such a way that each $\mathrm{A}$-token is moved to a distance at most $d+2$ from its original place. This means we need a perfect matching between the $3 n^{2} \mathrm{~A}$-squares and the $3 n^{2} \mathrm{~A}$-tokens such that the distance in each pair of the matching is at most $d+2$.

To find the matching, we construct a bipartite graph. The A-squares will be the vertices in one class of the graph; the vertices in the other class will be the A-tokens.

Split the board into $3 \times 1$ horizontal triminos; then each trimino contains exactly one Asquare. Take a permutation $\pi$ of the tokens which moves A-tokens to B-tokens, B-tokens to C-tokens, and C-tokens to A-tokens, in each case to a distance at most $d$. For each A-square $S$, and for each A-token $T$, connect $S$ and $T$ by an edge if $T, \pi(T)$ or $\pi^{-1}(T)$ is on the trimino containing $S$. We allow multiple edges; it is even possible that the same square and the same token are connected with three edges. Obviously the lengths of the edges in the graph do not exceed $d+2$. By length of an edge we mean the distance between the A-square and the A-token it connects.

Each A-token $T$ is connected with the three A-squares whose triminos contain $T, \pi(T)$ and $\pi^{-1}(T)$. Therefore in the graph all tokens are of degree 3 . We show that the same is true for the A-squares. Let $S$ be an arbitrary A-square, and let $T_{1}, T_{2}, T_{3}$ be the three tokens on the trimino containing $S$. For $i=1,2,3$, if $T_{i}$ is an A-token, then $S$ is connected with $T_{i}$; if $T_{i}$ is a B-token then $S$ is connected with $\pi^{-1}\left(T_{i}\right)$; finally, if $T_{i}$ is a C-token then $S$ is connected with $\pi\left(T_{i}\right)$. Hence in the graph the A-squares also are of degree 3.

Since the A-squares are of degree 3 , from every set $\mathcal{S}$ of A-squares exactly $3|\mathcal{S}|$ edges start. These edges end in at least $|\mathcal{S}|$ tokens because the A-tokens also are of degree 3. Hence every set $\mathcal{S}$ of A-squares has at least $|\mathcal{S}|$ neighbors among the A-tokens.

Therefore, by HALL's marriage theorem, the graph contains a perfect matching between the two vertex classes. So there is a perfect matching between the A-squares and A-tokens with edges no longer than $d+2$. It follows that the tokens can be permuted as specified in the problem statement.