107 research outputs found
Ramified rectilinear polygons: coordinatization by dendrons
Simple rectilinear polygons (i.e. rectilinear polygons without holes or
cutpoints) can be regarded as finite rectangular cell complexes coordinatized
by two finite dendrons. The intrinsic -metric is thus inherited from the
product of the two finite dendrons via an isometric embedding. The rectangular
cell complexes that share this same embedding property are called ramified
rectilinear polygons. The links of vertices in these cell complexes may be
arbitrary bipartite graphs, in contrast to simple rectilinear polygons where
the links of points are either 4-cycles or paths of length at most 3. Ramified
rectilinear polygons are particular instances of rectangular complexes obtained
from cube-free median graphs, or equivalently simply connected rectangular
complexes with triangle-free links. The underlying graphs of finite ramified
rectilinear polygons can be recognized among graphs in linear time by a
Lexicographic Breadth-First-Search. Whereas the symmetry of a simple
rectilinear polygon is very restricted (with automorphism group being a
subgroup of the dihedral group ), ramified rectilinear polygons are
universal: every finite group is the automorphism group of some ramified
rectilinear polygon.Comment: 27 pages, 6 figure
A sharp threshold for random graphs with a monochromatic triangle in every edge coloring
Let be the set of all finite graphs with the Ramsey property that
every coloring of the edges of by two colors yields a monochromatic
triangle. In this paper we establish a sharp threshold for random graphs with
this property. Let be the random graph on vertices with edge
probability . We prove that there exists a function with
, as tends to infinity
Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0 and Pr [ G(n,(1+\eps)\hat
c/\sqrt{n}) \in \R ] \to 1. A crucial tool that is used in the proof and is
of independent interest is a generalization of Szemer\'edi's Regularity Lemma
to a certain hypergraph setting.Comment: 101 pages, Final version - to appear in Memoirs of the A.M.
Generalized Colorings of Graphs
A graph coloring is an assignment of labels called “colors” to certain elements of a graph subject to certain constraints. The proper vertex coloring is the most common type of graph coloring, where each vertex of a graph is assigned one color such that no two adjacent vertices share the same color, with the objective of minimizing the number of colors used. One can obtain various generalizations of the proper vertex coloring problem, by strengthening or relaxing the constraints or changing the objective. We study several types of such generalizations in this thesis. Series-parallel graphs are multigraphs that have no K4-minor. We provide bounds on their fractional and circular chromatic numbers and the defective version of these pa-rameters. In particular we show that the fractional chromatic number of any series-parallel graph of odd girth k is exactly 2k/(k − 1), confirming a conjecture by Wang and Yu. We introduce a generalization of defective coloring: each vertex of a graph is assigned a fraction of each color, with the total amount of colors at each vertex summing to 1. We define the fractional defect of a vertex v to be the sum of the overlaps with each neighbor of v, and the fractional defect of the graph to be the maximum of the defects over all vertices. We provide results on the minimum fractional defect of 2-colorings of some graphs. We also propose some open questions and conjectures. Given a (not necessarily proper) vertex coloring of a graph, a subgraph is called rainbow if all its vertices receive different colors, and monochromatic if all its vertices receive the same color. We consider several types of coloring here: a no-rainbow-F coloring of G is a coloring of the vertices of G without rainbow subgraph isomorphic to F ; an F -WORM coloring of G is a coloring of the vertices of G without rainbow or monochromatic subgraph isomorphic to F ; an (M, R)-WORM coloring of G is a coloring of the vertices of G with neither a monochromatic subgraph isomorphic to M nor a rainbow subgraph isomorphic to R. We present some results on these concepts especially with regards to the existence of colorings, complexity, and optimization within certain graph classes. Our focus is on the case that F , M or R is a path, cycle, star, or clique
Some Multicolor Ramsey Numbers Involving Cycles
Establishing the values of Ramsey numbers is, in general, a difficult task from the computational point of view. Over the years, researchers have developed methods to tackle this problem exhaustively in ways that require intensive computations. These methods are often backed by theoretical results that allow us to cut the search space down to a size that is within the limits of current computing capacity.
This thesis focuses on developing algorithms and applying them to generate Ramsey colorings avoiding cycles. It adds to a recent trend of interest in this particular area of finite Ramsey theory. Our main contributions are the enumeration of all (C_5,C_5,C_5;n) Ramsey colorings and the study of the Ramsey numbers R(C_4,C_4,K_4) and R4(C_5)
Balanced-chromatic number and Hadwiger-like conjectures
Motivated by different characterizations of planar graphs and the 4-Color
Theorem, several structural results concerning graphs of high chromatic number
have been obtained. Toward strengthening some of these results, we consider the
\emph{balanced chromatic number}, , of a signed graph
. This is the minimum number of parts into which the vertices of a
signed graph can be partitioned so that none of the parts induces a negative
cycle. This extends the notion of the chromatic number of a graph since
, where denotes the signed graph
obtained from~ by replacing each edge with a pair of (parallel) positive and
negative edges. We introduce a signed version of Hadwiger's conjecture as
follows.
Conjecture: If a signed graph has no negative loop and no
-minor, then its balanced chromatic number is at most .
We prove that this conjecture is, in fact, equivalent to Hadwiger's
conjecture and show its relation to the Odd Hadwiger Conjecture.
Motivated by these results, we also consider the relation between
subdivisions and balanced chromatic number. We prove that if has
no negative loop and no -subdivision, then it admits a balanced
-coloring. This qualitatively generalizes a result of
Kawarabayashi (2013) on totally odd subdivisions
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