Sudoku Graph

Sudoku can be seen as a graph coloring problem, where the squares of the grid are vertices and the numbers are colors that must be different if in the same row, column, or \(3 \times 3\) grid (such vertices in the graph are connected by an edge). The sudoku is then a graph of 81 vertices and chromatic number 9. How many edges does this 81-vertex graph have?

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