The Polynomial Profile of Distance Games on Paths and Cycles

Distance games are games played on graphs in which the players alternately colour vertices, and which vertices can be coloured only depends on the distance to previously coloured vertices. The polynomial profile encodes the number of positions with a fixed number of vertices from each player. We extend previous work on finding the polynomial profile of several distance games (Col, Snort, and Cis) played on paths. We give recursions and generating functions for the polynomial profiles of generalizations of these three games when played on paths. We also find the polynomial profile of Cis played on cycles and the total number of positions of Col and Snort on cycles, as well as pose a conjecture about the number of positions when playing Col and Snort on complete bipartite graphs.