Uncountable Graphs Research Articles

The list-chromatic number and the coloring number of uncountable graphs

The list-chromatic number and the coloring number of uncountable graphs

A note on minor antichains of uncountable graphs

AbstractA simplified construction is presented for Komjáth's result that for every uncountable cardinal , there are graphs of size none of them being a minor of another.

Extremal triangle-free and odd-cycle-free colourings of uncountable graphs

The optimality of the Erdős–Rado theorem for pairs is witnessed by the colouring $$\Delta_\kappa : [2^\kappa]^2 \rightarrow \kappa$$ recording the least point of disagreement between two functions. This colouring has no monochromatic triangles or, more generally, odd cycles. We investigate a number of questions investigating the extent to which $$\Delta_\kappa$$ is an extremal such triangle-free or odd-cycle-free colouring. We begin by introducing the notion of $$\Delta$$ -regressive and almost $$\Delta$$ -regressive colourings and studying the structures that must appear as monochromatic subgraphs for such colourings. We also consider the question as to whether $$\Delta_\kappa$$ has the minimal cardinality of any maximal triangle-free or odd-cycle-free colouring into $$\kappa$$ . We resolve the question positively for odd-cycle-free colourings.

Hedetniemi's conjecture for uncountable graphs

It is proved that in Gödel's constructible universe, for every infinite successor cardinal \kappa , there exist graphs \mathcal G and \mathcal H of size and chromatic number \kappa , for which the product graph \mathcal G \times \mathcal H is countably chromatic. In particular, this provides an affirmative answer to a question of Hajnal from 1985.

Non-universality of automorphism groups of uncountable ultrahomogeneous structures

In [13], Mbombo and Pestov prove that the group of isometries of the generalized Urysohn space of density κ, for uncountable κ such that κ<κ=κ, is not a universal topological group of weight κ. We investigate automorphism groups of other uncountable ultrahomogeneous structures and prove that they are rarely universal topological groups for the corresponding classes. Our list of uncountable ultrahomogeneous structures includes random uncountable graph, tournament, linear order, partial order, group. That is in contrast with similar results obtained for automorphism groups of countable (separable) ultrahomogeneous structures.We also provide a more direct and shorter proof of the Mbombo–Pestov result.

Uncountable graphs and invariant measures on the set of universal countable graphs

AbstractWe give new examples and describe the complete lists of all measures on the set of countable homogeneous universal graphs and Ks‐free homogeneous universal graphs (for s ≥ 3) that are invariant with respect to the group of all permutations of the vertices. Such measures can be regarded as random graphs (respectively, random Ks‐free graphs). The well‐known example of Erdös–Rényi (ER) of the random graph corresponds to the Bernoulli measure on the set of adjacency matrices. For the case of the universal Ks‐free graphs there were no previously known examples of the invariant measures on the space of such graphs.The main idea of our construction is based on the new notions of measurable universal, and topologically universal graphs, which are interesting themselves. The realization of the construction can be regarded as two‐step randomization for universal measurable graph : “randomization in vertices” and “randomization in edges.” For Ks‐free, s ≥ 3, there is only randomization in vertices of the measurable graphs. The completeness of our lists is proved using the important theorem by Aldous about S∞‐invariant matrices, which we reformulate in appropriate way. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010

Uncountable graphs with all their vertices in one face

In 1967, Chartrand and Harary characterized finite outer-planar graphs, and Wagner studied uncountable graphs admitting a planar embedding. We continue this research for those outer-S graphs in surfaces S by studying uncountable graphs admitting S-embeddings with all their vertices in the same face (namely, outer-S embeddings).

The Countable Character of Uncountable Graphs

We show that a graph can always be decomposed into edge-disjoint subgraphs of countable cardinality in which the edge-connectivities and edge-separations of the original graph are preserved up to countable cardinals. We also show that the vertex set of any graph can be endowed with a well-ordering which has a certain compactness property with respect to edge-separation.

On immersions of uncountable graphs

In his paper on well-quasi-ordering infinite trees (Proc. Cambridge Philos. Soc. 61 (1965) 697), Nash-Williams proposed the conjecture that the class of all graphs (finite or infinite) is well-quasi-ordered by the immersion relation (which is denoted here by ⩽ 1). In addition, in a subsequent paper, Nash-Williams discussed a weaker version of his original conjecture to the effect that the class of graphs is well-quasi-ordered with respect to a relation ⩽ 2 which, roughly speaking, is obtained by redefining H⩽ 1 G so that distinct vertices of H can be mapped into the same vertex of G. It is the purpose of the present note to disprove Nash-Williams’ two immersion conjectures.

POINT VISIBILITY GRAPHS AND ${\mathcal O}$-CONVEX COVER

A visibility relation can be viewed as a graph: the uncountable graph of a visibility relationship between points in a polygon P is called the point visibility graph (PVG) of P. In this paper we explore the use of perfect graphs to characterize tractable subproblems of visibility problems. Our main result is a characterization of which polygons are guaranteed to have weakly triangulated PVGs, under a generalized notion of visibility called [Formula: see text]-visibility. Let [Formula: see text] denote a set of line orientations. A connected point set P is called [Formula: see text]-convex if the intersection of P with any line with orientation in [Formula: see text] is connected. Two points in a polygon are said to be [Formula: see text]-visible if there is an [Formula: see text]-convex path between them inside the polygon. Let [Formula: see text] denote the set of orientations perpendicular to orientations in [Formula: see text]. Let [Formula: see text] be the set of orientations θ such that a "reflex" local maximum in the boundary of P exists with respect to θ. Our characterization of which polygons have weakly-triangulated PVGs is based on restricting the cardinality and span of [Formula: see text]. This characterization allows us to exhibit a class of polygons admitting a polynomial algorithm for [Formula: see text]-convex cover.

Representing Embeddability as Set Inclusion

Abstract: few steps are made towards representation theory of embeddability among uncountable graphs. A monotone class of graphs is defined by forbidding countable subgraphs, related to the graphs [sic] end-structure. Using a combinatorial theorem of Shelah it is proved: - The complexity of the class in every regular uncountable [lambda] > NΓeu is at least [lambda superscript plus] + sup[mu superscript NΓeC] : [mu superscript plus] NΓeu there are 2[superscript lambda] pairwise non embeddable graphs in the class having strong homogeneity properties. - It is characterized when some invariants of a graph G [element of] G[subscript lambda] have to be inherited by one of fewer than [lambda] subgraphs whose union covers G. All three results are obtained as corollaries of a representation theorem (Theorem 1.10 below), that asserts the existence of a surjective homomorphism from the relation of embeddability over isomorphism types of regular cardinality [lambda] > NΓeu onto set inclusion over all subsets of reals or cardinality [lambda] or less. Continuity properties of the homomorphism are used to extend the first result to all singular cardinals below the first cardinal fixed point of second order. The first result shows that, unlike what Shelah showed in the class of all graphs, the relations of embeddability in this class is not independent of negations of the GCH.

A strongly non-Ramsey uncountable graph

It is consistent that there exists a graph X of cardinality א1 such that every graph has an edge coloring with א1 colors in which the induced copies of X (if there are any) are totally multicolored (get all possible colors). An important chapter of combinatorial set theory is partition calculus which concerns the infinite generalizations of Ramsey’s theorem. One of its corner stones is the Erdős–Rado theorem which says that if κ, μ are arbitrary cardinals then the complete graph on a large enough set has the property that whenever the edges are colored with μ colors then there is a monocolored subset of cardinal κ (see Chapter 17 of [1]). For κ = א1, μ = א0 the underlying set must have cardinality (2א0)+ and this is sharp. If the cardinal of the ground set is not enough large then—as discovered by Erdős, Hajnal, and Rado—very strong counterexamples exist, examples of μ-colorings, in which every κ-subset of the underlying set has edges in every color. An important example in this direction is the Todorcevic coloring, i.e., a function h : [ω1] → ω1 with the property that if X ⊆ ω1 is uncountable then h assumes every value on [X]2 (see [4]). This is usually denoted by ω1 6→ [ω1]ω1 . Another direction in Ramsey theory is when we want not necessarily complete monochromatic graphs but to exclude trivialities we require that the required monocolored copy of the predetermined graph be induced. If the graph and the number of colors are both finite then there always exists a graph with the required property. It was shown, however, by A. Hajnal and the author that at least consistently there may be a negative answer. In [2] we proved that if a Cohen real is added to a model of set theory then in the resulting model there is a graph X such that every graph has 1991 Mathematics Subject Classification: 05D10, 03E35, 05C55. Partially supported by the European Communities (Cooperation in Science and Technology with Central and Eastern European Countries) contract no. ERBCIPACT930113.

A note on minors of uncountable graphs

AbstractIf λ &gt; ω then there are 2λ graphs of cardinal λ none of them being a minor of another.

Open-interval graphs versus closed-interval graphs

A graph G = ( V, E) is said to be represented by a family F of nonempty sets if there is a bijection f: V → F such that uv ϵ E if and only if f( u)∩ f( v) ≠ Ø. It is proved that if G is a Countable graph then G can be represented by open intervals on the real line if and only if G can be represented by closed intervals on the real line, however, this is no longer true when G is an uncountable graph. Similar results are also proved when intervals are required to have unit length.