19,584 research outputs found
Combinatorial problems of (quasi-)crystallography
Several combinatorial problems of (quasi-)crystallography are reviewed with
special emphasis on a unified approach, valid for both crystals and
quasicrystals. In particular, we consider planar sublattices, similarity
sublattices, coincidence sublattices, their module counterparts, and central
and averaged shelling. The corresponding counting functions are encapsulated in
Dirichlet series generating functions, with explicit results for the triangular
lattice and the twelvefold symmetric shield tiling. Other combinatorial
properties are briefly summarised.Comment: 12 pages, 2 PostScript figures, LaTeX using vch-book.cl
Combinatorial Problems on -graphs
Bir\'{o}, Hujter, and Tuza introduced the concept of -graphs (1992),
intersection graphs of connected subgraphs of a subdivision of a graph .
They naturally generalize many important classes of graphs, e.g., interval
graphs and circular-arc graphs. We continue the study of these graph classes by
considering coloring, clique, and isomorphism problems on -graphs.
We show that for any fixed containing a certain 3-node, 6-edge multigraph
as a minor that the clique problem is APX-hard on -graphs and the
isomorphism problem is isomorphism-complete. We also provide positive results
on -graphs. Namely, when is a cactus the clique problem can be solved in
polynomial time. Also, when a graph has a Helly -representation, the
clique problem can be solved in polynomial time. Finally, we observe that one
can use treewidth techniques to show that both the -clique and list
-coloring problems are FPT on -graphs. These FPT results apply more
generally to treewidth-bounded graph classes where treewidth is bounded by a
function of the clique number
Some combinatorial problems
AbstractThere are many interesting and sophisticated problems posed in the IMO, Putnam and domestic Olympiads. Some of these problems have deep mathematical background, nice generalizations, and lead to new areas of research in combinatorics. We investigate several topics in this category and mention some results and open problems
Combinatorial problems in finite fields and Sidon sets
We use Sidon sets to present an elementary method to study some combinatorial
problems in finite fields, such as sum product estimates, solubility of some
equations and distribution of sequences in small intervals. We obtain classic
and more recent results avoiding the use of exponential sums, the usual tool to
deal with these problems.Comment: 13 page
Recognizing Graph Theoretic Properties with Polynomial Ideals
Many hard combinatorial problems can be modeled by a system of polynomial
equations. N. Alon coined the term polynomial method to describe the use of
nonlinear polynomials when solving combinatorial problems. We continue the
exploration of the polynomial method and show how the algorithmic theory of
polynomial ideals can be used to detect k-colorability, unique Hamiltonicity,
and automorphism rigidity of graphs. Our techniques are diverse and involve
Nullstellensatz certificates, linear algebra over finite fields, Groebner
bases, toric algebra, convex programming, and real algebraic geometry.Comment: 20 pages, 3 figure
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