Consider an m -by-n chessboard with m and n both odd. To fix...

Consider an $m$ -by-n chessboard with $m$ and $n$ both odd. To fix the notation, suppose that the square in the upper left-hand corner is colored white. Show that if a white square is cut out anywhere on the board, the resulting pruned board has a perfect cover by dominoes.

Key Concepts

Domino Tiling

Domino tiling involves covering a region completely with domino pieces, where each domino covers exactly two adjacent cells. This concept is widely used in combinatorics and recreational mathematics to investigate how shapes can be partitioned or covered without overlaps or gaps.

Parity and Checkerboard Coloring

The parity argument uses a checkerboard coloring scheme to alternate two colors on a grid. In tiling problems, this coloring helps reveal imbalances, ensuring that the numbers of cells of each color are balanced (or purposely imbalanced) to satisfy conditions necessary for a tiling, as each domino will cover one cell of each color.

Bipartite Graphs

A bipartite graph is a type of graph whose vertices can be divided into two disjoint sets such that no two vertices within the same set are adjacent. In domino tiling, the grid can be represented as a bipartite graph where cells of one color form one vertex set and cells of the other color form the other set, facilitating the analysis of possible coverings.

Perfect Matching

A perfect matching in a graph is a set of edges where every vertex in the graph is incident to exactly one edge from the set. In the context of domino tiling, finding a perfect matching in the corresponding bipartite graph ensures that every cell can be paired with exactly one adjacent cell, providing a valid, complete covering for the board.

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