We further study sets of labeled dice in which the relation "is a better die than" is non-transitive. Focusing on sets with an additional symmetry we call "balance," we prove that sets of n such m-sided dice exist for all n, m ≥ 3. We then show how to construct a set of n dice such that the relation behaves according to the direction of the arrows of any tournament (complete directed graph) on n vertices.
A signed graph is a graph Γ where the edges are assigned sign labels, either "+" or "−". The sign of a cycle is the product of the signs of its edges. Let SpecC(Γ) denote the list of lengths of cycles in Γ. We equip each signed graph with a vector whose entries are the numbers of negative k-cycles for k ∈ SpecC(Γ). These vectors generate a subspace of R SpecC (Γ) . Using matchings with a strong permutability property, we provide lower bounds on the dimension of this space; in particular, we show for complete graphs, complete bipartite graphs, and a few other graphs that this space is all of R SpecC(Γ) .
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