We present a distance metric based upon the notion of minimum-cost injective mappings between sets. Our function satisfies metric properties as long as the cost of the minimum mappings is derived from a semimetric, for which the triangle inequality is not necessarily satisfied. We show that the Jaccard distance (alternatively biotope, Tanimoto, or Marczewski-Steinhaus distance) may be considered the special case for finite sets where costs are derived from the discrete metric. Extensions that allow premetrics (not necessarily symmetric), multisets (generalized to include probability distributions), and asymmetric mappings are given that expand the versatility of the metric without sacrificing metric properties. The function has potential applications in pattern recognition, machine learning, and information retrieval.

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Let G be a simple graph, and let ∆(G) and χ ′ (G) denote the maximum degree and chromatic index of G, respectively. Vizing proved that χ ′ (G) = ∆(G) or ∆(G) + 1. We say G is ∆-critical if χ ′ (G) = ∆ + 1 and χ ′ (H) < χ ′ (G) for every proper subgraph H of G. In 1968, Vizing conjectured that if G is a ∆-critical graph, then G has a 2-factor. Let G be an n-vertex ∆-critical graph. It was proved that if ∆(G) ≥ n/2, then G has a 2-factor; and that if ∆(G) ≥ 2n/3 + 12, then G has a hamiltonian cycle, and thus a 2-factor. It is well known that every 2-tough graph with at least three vertices has a 2-factor. We investigate the existence of a 2-factor in a ∆-critical graph under "moderate" given toughness and maximum degree conditions. In particular, we show that if G is an n-vertex ∆-critical graph with toughness at least 3/2 and with maximum degree at least n/3, then G has a 2-factor. In addition, we develop new techniques in proving the existence of 2-factors in graphs.

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