2025/05/09 by Alexander Clow, Clow, Alexander, Davies, Alfie +2
Computer Science · #91A46 (Primary) 05C20 (Secondary) #Advanced Graph Theory Research #Artificial Intelligence in Games #Combinatorics (math.CO) #Complexity and Algorithms in Graphs #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2505.06206
openalex publication_date 2025/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Recently, Clow and McKay proved that the Digraph Placement ruleset is universal for normal play: for all normal play combinatorial games X, there is a Digraph Placement game G with G=X. Clow and McKay also showed that the 22 game values born by day 2 correspond to Digraph Placement games with at most 4 vertices. This bound is best possible. We extend this work using a combination of exhaustive and random searches to demonstrate all 1474 values born by day 3 correspond to Digraph Placement games on at most 8 vertices. We provide a combinatorial proof that this bound is best possible. We conclude by giving improved bounds on the number of vertices required to construct all game values born by days 4 and 5.