1,019 research outputs found
Semi-algebraic colorings of complete graphs
We consider -colorings of the edges of a complete graph, where each color
class is defined semi-algebraically with bounded complexity. The case
was first studied by Alon et al., who applied this framework to obtain
surprisingly strong Ramsey-type results for intersection graphs of geometric
objects and for other graphs arising in computational geometry. Considering
larger values of is relevant, e.g., to problems concerning the number of
distinct distances determined by a point set.
For and , the classical Ramsey number is the
smallest positive integer such that any -coloring of the edges of ,
the complete graph on vertices, contains a monochromatic . It is a
longstanding open problem that goes back to Schur (1916) to decide whether
, for a fixed . We prove that this is true if each color
class is defined semi-algebraically with bounded complexity. The order of
magnitude of this bound is tight. Our proof is based on the Cutting Lemma of
Chazelle {\em et al.}, and on a Szemer\'edi-type regularity lemma for
multicolored semi-algebraic graphs, which is of independent interest. The same
technique is used to address the semi-algebraic variant of a more general
Ramsey-type problem of Erd\H{o}s and Shelah
Moduli spaces of colored graphs
We introduce moduli spaces of colored graphs, defined as spaces of
non-degenerate metrics on certain families of edge-colored graphs. Apart from
fixing the rank and number of legs these families are determined by various
conditions on the coloring of their graphs. The motivation for this is to study
Feynman integrals in quantum field theory using the combinatorial structure of
these moduli spaces. Here a family of graphs is specified by the allowed
Feynman diagrams in a particular quantum field theory such as (massive) scalar
fields or quantum electrodynamics. The resulting spaces are cell complexes with
a rich and interesting combinatorial structure. We treat some examples in
detail and discuss their topological properties, connectivity and homology
groups
Inside-Out Polytopes
We present a common generalization of counting lattice points in rational
polytopes and the enumeration of proper graph colorings, nowhere-zero flows on
graphs, magic squares and graphs, antimagic squares and graphs, compositions of
an integer whose parts are partially distinct, and generalized latin squares.
Our method is to generalize Ehrhart's theory of lattice-point counting to a
convex polytope dissected by a hyperplane arrangement. We particularly develop
the applications to graph and signed-graph coloring, compositions of an
integer, and antimagic labellings.Comment: 24 pages, 3 figures; to appear in Adv. Mat
Beck's Conjecture for Power Graphs
Beck's conjecture on coloring of graphs associated to various algebraic
objects has generated considerable interest in the community of discrete
mathematics and combinatorics since its inception in the year 1988. The version
of this conjecture for power-graphs of finite groups has been addressed and
partially settled by previous authors. In this paper we answer it in the
affirmative in complete generality, and, in effect, we establish a "nicer"
statement on a larger class of graphs. We also clear up certain ambiguities
present in the way the previous versions of the conjecture were posed
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