Sudoku in the Context of Graph Theory

Sudoku is a popular puzzle, typically using a 9x9 grid. Within the 9x9 grid, there are 9 3x3 boxes. Each box, row, and column must contain the digits 1-9 in its cells with no repetition. A certain number of digits are already given to the solver, which “force” them to put each digit in a particular cell. In other words, there is only one solution with the given digits. We are particularly interested in a generalization of sudoku to graph theory. Vertices in graphs can be colored, or labeled, so that no two adjacent vertices share the same color. A sudoku coloring of a graph is a partial coloring in which certain vertices are colored in order to “force” the remainder of the vertices in the graph to be colored one specific way. This is similar to the given numbers in the sudoku puzzle “forcing” a solution. The sudoku number of a graph G, denoted sn(G), is the minimum number of vertices that must be colored in the graph to guarantee a unique solution. Lau et al. introduced the idea of the sudoku number of a graph, denoted sn(G), and proved that sn(G)=1 if and only if G is bipartite. Pokrovskiy proved that sn(G)=n-1 if and only if G is complete. We investigate graphs with sn(G)=2. We have found some broad classes of graphs with sn(G)=2 as well as some more specific constructions.