On the Workday Number for Finite Multigraphs in a Variation of Cops and Robbers

Eric Schneider

Oftentimes, there are “bad guys” such as diseases, wildfires, or thieves that the “good guys” such as the CDC, firemen, or police wish to control or capture. However, the good guys only have a limited quantity of assets such as money, people, and time, so it is important for them to use the least amount of resources. One well-known way of analyzing such problems is known as “Cops and Robbers on a Graph”. I analyzed a different version of this model to find out how to minimize the cost (called the Workday Number) to catch the bad guys. I discovered how to compute a way to catch the bad guys in two days while still minimizing the cost … How did I come up with my research topic? During the proof-based power round of the national American Regions Mathematics League Competition (ARML), there was one problem which introduced and asked questions about the Workday Number. In answering it, I realized that I could combine the idea of flow networks from computer science with monovariants from my math experience to give bounds on the Workday number. Unknown to me at the time, the panel of judges, all mathematicians, had deliberated for over thirty minutes over the correctness of my solution. Although it appeared correct, and they could not find any holes in it, it simply did not match any of the official proofs that they had.