18,525 research outputs found
Generalisation : graphs and colourings
The interaction between practice and theory in mathematics is a central theme. Many mathematical structures and theories result from the formalisation of a real problem. Graph Theory is rich with such examples. The graph structure itself was formalised by Leonard Euler in the quest to solve the problem of the Bridges of Königsberg. Once a structure is formalised, and results are proven, the mathematician seeks to generalise. This can be considered as one of the main praxis in mathematics. The idea of generalisation will be illustrated through graph colouring. This idea also results from a classic problem, in which it was well known by topographers that four colours suffice to colour any map such that no countries sharing a border receive the same colour. The proof of this theorem eluded mathematicians for centuries and was proven in 1976. Generalisation of graphs to hypergraphs, and variations on the colouring theme will be discussed, as well as applications in other disciplines.peer-reviewe
On the expected number of perfect matchings in cubic planar graphs
A well-known conjecture by Lov\'asz and Plummer from the 1970s asserted that
a bridgeless cubic graph has exponentially many perfect matchings. It was
solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other
hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the
special case of cubic planar graphs. In our work we consider random bridgeless
cubic planar graphs with the uniform distribution on graphs with vertices.
Under this model we show that the expected number of perfect matchings in
labeled bridgeless cubic planar graphs is asymptotically , where
and is an explicit algebraic number. We also
compute the expected number of perfect matchings in (non necessarily
bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs.
Our starting point is a correspondence between counting perfect matchings in
rooted cubic planar maps and the partition function of the Ising model in
rooted triangulations.Comment: 19 pages, 4 figure
Combinatorics and Geometry of Transportation Polytopes: An Update
A transportation polytope consists of all multidimensional arrays or tables
of non-negative real numbers that satisfy certain sum conditions on subsets of
the entries. They arise naturally in optimization and statistics, and also have
interest for discrete mathematics because permutation matrices, latin squares,
and magic squares appear naturally as lattice points of these polytopes.
In this paper we survey advances on the understanding of the combinatorics
and geometry of these polyhedra and include some recent unpublished results on
the diameter of graphs of these polytopes. In particular, this is a thirty-year
update on the status of a list of open questions last visited in the 1984 book
by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure
Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs
The chromatic polynomial of a graph G counts the number of proper colorings
of G. We give an affirmative answer to the conjecture of Read and
Rota-Heron-Welsh that the absolute values of the coefficients of the chromatic
polynomial form a log-concave sequence. We define a sequence of numerical
invariants of projective hypersurfaces analogous to the Milnor number of local
analytic hypersurfaces. Then we give a characterization of correspondences
between projective spaces up to a positive integer multiple which includes the
conjecture on the chromatic polynomial as a special case. As a byproduct of our
approach, we obtain an analogue of Kouchnirenko's theorem relating the Milnor
number with the Newton polytope.Comment: Improved readability. Final version, to appear in J. Amer. Math. So
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