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Problem of the Month (June 2017)

Consider graphs where each vertex is labeled with a number. We are interested in planar graphs where the sum of the vertices adjacent to n is 3n–1. (The case where the sum for every vertex is 3n is more natural, but then any 3-regular graph could be labeled with any one number.) What is the smallest such graph (by number of vertices, and sum of vertices as a tiebreaker) whose largest digit is n? Can you prove that such graphs always exist?

3n–1 Solutions
12345
678910

What if the sum of vertices is 3n+1, 4n–1, 4n+1, 5n–1 or 5n+1?


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