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 3×33 \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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