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- W2380201307 abstract "Let G=(V,E) be a graph with n vertices.A bicycle is a cycle that is also a cocycle. Let β(G) denote the dimension of the bicycle space of G. For H a subgraph of G, let (G,H) denote the number of spanning forests, where each tree in the forest contains exactly one edge from H. The H -extension X(G,H) of G is the graph obtained from G by adding a new vertex ν and joining ν to every vertex of G having odd degree in H. In this paper we show that (G,H) is even if, and only if, X(G,H) is either disconnected or contains a nonzero bicycle. For G an Eulerian graph, let λ(G) denote the minimum number of edges that must be contracted, so that some edge in the resulting graph G * lies in an odd number of perfect matchings.We also show that λ(G) = |β(G) - 1|. Further, the set of all edges of G * that lie in an odd number of perfect matchings forms a nonzero bicycle and the maximum size μ(G) of a matching in G satisfies that μ(G) ≥ (n - |β(G)- 1|)/2. For G a connected and non-Eulerian graph, let ψ(G) = β(X(G,G)) and let γ(G) denote the minimum number of edges that must be contracted so that the resulting graph contains an odd number of perfect matchings. We show that γ(G) = ψ(G) and then μ(G) ≥ (n - ψ(G))/2." @default.
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- W2380201307 date "1996-01-01" @default.
- W2380201307 modified "2023-09-25" @default.
- W2380201307 title "Bicycles, Matchings and Trees" @default.
- W2380201307 hasPublicationYear "1996" @default.
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