524 research outputs found
Measurable versions of Vizing's theorem
We establish two versions of Vizing's theorem for Borel multi-graphs whose
vertex degrees and edge multiplicities are uniformly bounded by respectively
and . The ``approximate'' version states that, for any Borel
probability measure on the edge set and any , we can properly
colour all but -fraction of edges with colours in a
Borel way. The ``measurable'' version, which is our main result, states that
if, additionally, the measure is invariant, then there is a measurable proper
edge colouring of the whole edge set with at most colours
Matroids arising from electrical networks
This paper introduces Dirichlet matroids, a generalization of graphic
matroids arising from electrical networks. We present four main results. First,
we exhibit a matroid quotient formed by the dual of a network embedded in a
surface with boundary and the dual of the associated Dirichlet matroid. This
generalizes an analogous result for graphic matroids of cellularly embedded
graphs. Second, we characterize the Bergman fans of Dirichlet matroids as
explicit subfans of graphic Bergman fans. In doing so, we generalize the
connection between Bergman fans of complete graphs and phylogenetic trees.
Third, we use the half-plane property of Dirichlet matroids to prove an
interlacing result on the real zeros and poles of the trace of the response
matrix. And fourth, we bound the coefficients of the precoloring polynomial of
a network by the coefficients of the chromatic polynomial of the underlying
graph.Comment: 27 pages, 14 figure
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