2026/07/31 by Saeid Alikhani, Max Griswold
Mathematics · #math.CO #msc:05C31 #msc:05C69
6 pages, 1 figure
arxiv created 2026/07/31 · arxiv updated 2026/08/04
The domination integer root conjecture asserted that 0 and -2 are the only integer roots of the domination polynomial D(G, x) for any graph G. In this paper, we document a counterexample of order n = 33 possessing an integer domination root at x = -4. We provide the complete structural description of the graph G33, present its exact domination polynomial D(G33, x), and demonstrate its exact rational factorization. Furthermore, we outline the structural gadget mechanism involving transfer matrices and S-unit branch cancellations that gives rise to non-trivial zero evaluation at x = -4.