The dimension of the negative cycle vectors of a signed graph

Alex Schaefer, Thomas Zaslavsky


A signed graph is a graph Γ with edges labeled “+” and “−”. The sign of a cycle is the product of its edge signs. 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)|.


Signed graph, negative cycle vector, permutable matching

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ISSN: 1855-3974

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