2,626 research outputs found
Spanning trees of 3-uniform hypergraphs
Masbaum and Vaintrob's "Pfaffian matrix tree theorem" implies that counting
spanning trees of a 3-uniform hypergraph (abbreviated to 3-graph) can be done
in polynomial time for a class of "3-Pfaffian" 3-graphs, comparable to and
related to the class of Pfaffian graphs. We prove a complexity result for
recognizing a 3-Pfaffian 3-graph and describe two large classes of 3-Pfaffian
3-graphs -- one of these is given by a forbidden subgraph characterization
analogous to Little's for bipartite Pfaffian graphs, and the other consists of
a class of partial Steiner triple systems for which the property of being
3-Pfaffian can be reduced to the property of an associated graph being
Pfaffian. We exhibit an infinite set of partial Steiner triple systems that are
not 3-Pfaffian, none of which can be reduced to any other by deletion or
contraction of triples.
We also find some necessary or sufficient conditions for the existence of a
spanning tree of a 3-graph (much more succinct than can be obtained by the
currently fastest polynomial-time algorithm of Gabow and Stallmann for finding
a spanning tree) and a superexponential lower bound on the number of spanning
trees of a Steiner triple system.Comment: 34 pages, 9 figure
Renormalization: an advanced overview
We present several approaches to renormalization in QFT: the multi-scale
analysis in perturbative renormalization, the functional methods \`a la
Wetterich equation, and the loop-vertex expansion in non-perturbative
renormalization. While each of these is quite well-established, they go beyond
standard QFT textbook material, and may be little-known to specialists of each
other approach. This review is aimed at bridging this gap.Comment: Review, 130 pages, 33 figures; v2: misprints corrected, refs. added,
minor improvements; v3: some changes to sect. 5, refs. adde
Generic rigidity with forced symmetry and sparse colored graphs
We review some recent results in the generic rigidity theory of planar
frameworks with forced symmetry, giving a uniform treatment to the topic. We
also give new combinatorial characterizations of minimally rigid periodic
frameworks with fixed-area fundamental domain and fixed-angle fundamental
domain.Comment: 21 pages, 2 figure
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