416 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
Contractors for flows
We answer a question raised by Lov\'asz and B. Szegedy [Contractors and
connectors in graph algebras, J. Graph Theory 60:1 (2009)] asking for a
contractor for the graph parameter counting the number of B-flows of a graph,
where B is a subset of a finite Abelian group closed under inverses. We prove
our main result using the duality between flows and tensions and finite Fourier
analysis. We exhibit several examples of contractors for B-flows, which are of
interest in relation to the family of B-flow conjectures formulated by Tutte,
Fulkerson, Jaeger, and others.Comment: 22 pages, 1 figur
Distinguishing graphs by their left and right homomorphism profiles
We introduce a new property of graphs called ‘q-state Potts unique-ness’ and relate it to chromatic and Tutte
uniqueness, and also to ‘chromatic–flow uniqueness’, recently studied by Duan, Wu and Yu.
We establish for which edge-weighted graphs H homomor-phism functions from multigraphs G to H are
specializations of the Tutte polynomial of G, in particular answering a question of Freed-man, Lovász and
Schrijver. We also determine for which edge-weighted graphs H homomorphism functions from
multigraphs G to H are specializations of the ‘edge elimination polynomial’ of Averbouch, Godlin and
Makowsky and the ‘induced subgraph poly-nomial’ of Tittmann, Averbouch and Makowsky.
Unifying the study of these and related problems is the notion of the left and right homomorphism profiles
of a graph.Ministerio de Educación y Ciencia MTM2008-05866-C03-01Junta de Andalucía FQM- 0164Junta de Andalucía P06-FQM-0164
Graph homomorphisms, the Tutte polynomial and “q-state Potts uniqueness”
We establish for which weighted graphs H homomorphism functions from multigraphs
G to H are specializations of the Tutte polynomial of G, answering a question
of Freedman, Lov´asz and Schrijver.
We introduce a new property of graphs called “q-state Potts uniqueness” and relate
it to chromatic and Tutte uniqueness, and also to “chromatic–flow uniqueness”,
recently studied by Duan, Wu and Yu.Ministerio de Educación y Ciencia MTM2005-08441-C02-0
Tutte's dichromate for signed graphs
We introduce the ``trivariate Tutte polynomial" of a signed graph as an
invariant of signed graphs up to vertex switching that contains among its
evaluations the number of proper colorings and the number of nowhere-zero
flows. In this, it parallels the Tutte polynomial of a graph, which contains
the chromatic polynomial and flow polynomial as specializations. The number of
nowhere-zero tensions (for signed graphs they are not simply related to proper
colorings as they are for graphs) is given in terms of evaluations of the
trivariate Tutte polynomial at two distinct points. Interestingly, the
bivariate dichromatic polynomial of a biased graph, shown by Zaslavsky to share
many similar properties with the Tutte polynomial of a graph, does not in
general yield the number of nowhere-zero flows of a signed graph. Therefore the
``dichromate" for signed graphs (our trivariate Tutte polynomial) differs from
the dichromatic polynomial (the rank-size generating function).
The trivariate Tutte polynomial of a signed graph can be extended to an
invariant of ordered pairs of matroids on a common ground set -- for a signed
graph, the cycle matroid of its underlying graph and its frame matroid form the
relevant pair of matroids. This invariant is the canonically defined Tutte
polynomial of matroid pairs on a common ground set in the sense of a recent
paper of Krajewski, Moffatt and Tanasa, and was first studied by Welsh and
Kayibi as a four-variable linking polynomial of a matroid pair on a common
ground set.Comment: 53 pp. 9 figure
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