Infinite Graphs

If $X$ is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\delta$-hyperbolic $($in the Gromov sense$)$ if any side of $T$ is contained in a $\delta$-neighborhood of the union of the other two sides, for every geodesic triangle $T$ in $X$. We denote by $\delta(X)$ the sharp hyperbolicity constant of $X$, i.e., $\delta(X):=\inf\{\delta\ge 0: \, X \, \text{ is $\delta$-hyperbolic}\,\}$. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. One of the main aims of this paper is to obtain quantitative information about the distortion of the hyperbolicity constant of the graph $G\setminus e$ obtained from the graph $G$ by deleting an arbitrary edge $e$ from it. These inequalities allow to obtain the other main result of this paper, which ch...

New compact representations of infinite graphs are investigated. Finite automata are used to represent labelled hypergraphs which can be also multi-graphs. Our approach consists of a general framework where vertices are represented by a regular prefixfree language and edges are represented by a regular language and a function over tuples. We consider three different functions over tuples: given a tuple the first function returns its first difference, the second one returns its suffix and the last one returns its infixes. The first-difference function is substantially a direct generalization to infinite multi-hypergraphs of the representation introduced by Ehrenfeucht et al. for finite graphs. This representation, though very interesting for finite graphs, turns out to be quite unsatisfactory for infinite graphs. The other two functions we consider while preserving some interesting features of their representation also achieves a high expressive power. As a matter of fact, our formalism either with the suffix or infix function results to be more powerful than the equational graphs introduced by Courcelle and the simple graphs defined by Caucal. The monadic second order theories of these two classes of graphs are undecidable, but still many interesting graph properties are decidable. The use of a regular prefixfree language to represent the vertices allows (fixed the language of the edges) to express a graph by a labelled tree, moreover, the use of finite automata to represent the edges allows the verification of graph properties.

New compact representations of infinite graphs are investigated. Finite automata are used to represent labelled hypergraphs which can be also multi-graphs. Our approach consists of a general framework where vertices are represented by a regular prefixfree language and edges are represented by a regular language and a function over tuples. We consider three different functions over tuples: given a tuple the first function returns its first difference, the second one returns its suffix and the last one returns its infixes. The first-difference function is substantially a direct generalization to infinite multi-hypergraphs of the representation introduced by Ehrenfeucht et al. for finite graphs. This representation, though very interesting for finite graphs, turns out to be quite unsatisfactory for infinite graphs. The other two functions we consider while preserving some interesting features of their representation also achieves a high expressive power. As a matter of fact, our formalism either with the suffix or infix function results to be more powerful than the equational graphs introduced by Courcelle and the simple graphs defined by Caucal. The monadic second order theories of these two classes of graphs are undecidable, but still many interesting graph properties are decidable. The use of a regular prefixfree language to represent the vertices allows (fixed the language of the edges) to express a graph by a labelled tree, moreover, the use of finite automata to represent the edges allows the verification of graph properties.

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Linearly bounded Turing machines have been mainly studied as acceptors for context-sensitive languages. We define a natural class of infinite automata representing their observable computational behavior, called linearly bounded graphs. These automata naturally accept the same languages as the linearly bounded machines defining them. We present some of their structural properties as well as alternative characterizations in terms of rewriting systems and context-sensitive transductions. Finally, we compare these graphs to rational graphs, which are another class of automata accepting the context-sensitive languages, and prove that in the bounded-degree case, rational graphs are a strict sub-class of linearly bounded graphs.

We introduce the notion of an edge-end and characterize those countable graphs which have edge-end-faithful spanning trees. We also prove that for a natural class of graphs, there always exists a tree which is faithful on the undominated ends and rayless over the dominated does.

Investigating whether a finite graph is Hamiltonian is a prominent task in graph theory, which has been transferred to infinite graphs. In order to overcome the problem of what a Hamiltonian cycle should be for infinite graphs, we follow the topological approach introduced by Diestel and Kühn [1, 2]. Especially for locally finite graphs, i.e., graphs where each vertex has finite degree, this topological viewpoint has been quite successful. For a locally finite connected graph G we define its infinite cycles via its Freudenthal compactification |G| as follows: We call a homeomorphic image of the unit circle S 1 ⊆ R 2 in |G| an infinite cycle of G. Consequently, we call an infinite cycle of G containing all vertices of G a Hamilton cycle of G. In this talk I will first give an overview about Hamiltonicity in locally finite graphs. Then I will present recent results about extensions to locally finite graphs of sufficient conditions, formulated in terms of forbidden induced subgraphs, forcing finite graphs to be Hamiltonian. Our results include claw-and net-free graphs, claw-and bull-free graphs, but also further graph classes which are structurally richer.

Let T be a finite or infinite tree and m the minimum number of vertices moved by the non-identity automorphisms of T. We give bounds on the maximum valence d of T that assure the existence of a vertex coloring of T with two colors that is preserved only by the identity automorphism. For finite m we obtain the bound $$d\le 2^{m/2}$$d≤2m/2 when T is finite, and $$d\le 2^{(m-2)/2}+2$$d≤2(m-2)/2+2 when T is infinite. For countably infinite m the bound is $$d\le 2^m.$$d≤2m. This relates to a question of Babai, who asked whether there existed a function f(d) such that every connected, locally finite graph G with maximum valence d has a 2-coloring of its vertices that is only preserved by the identity automorphism if the minimum number m of vertices moved by each non-identity automorphisms of G is at least $$m\ge f(d)$$m≥f(d). Our results give a positive answer for trees. The trees need not be locally finite, their maximal valence can be $$2^{\aleph _0}$$2ℵ0. For finite m we also extend ou...

The distinguishing number D(G) of a graph G is the least cardinal d such that G has a labeling with d labels which is only preserved by the trivial automorphism. We show that the distinguishing number of infinite, locally finite, connected graphs G with infinite motion and growth o(n 2 / log 2 n) is either 1 or 2, which proves the Infinite Motion Conjecture of Tom Tucker for this type of graphs. The same holds true for graphs with countably many ends that do not grow too fast. We also show that graphs G of arbitrary cardinality are 2-distinguishable if every nontrivial automorphism moves at least uncountably many vertices m(G), where m(G) ≥ |Aut(G)|. This extends a result of Imrich et al. to graphs with automorphism groups of arbitrary cardinality.

We introduce the endomorphism distinguishing number $D_e(G)$ of a graph $G$ as the least cardinal $d$ such that $G$ has a vertex coloring with $d$ colors that is only preserved by the trivial endomorphism. This generalizes the notion of the distinguishing number $D(G)$ of a graph $G$, which is defined for automorphisms instead of endomorphisms.As the number of endomorphisms can vastly exceed the number of automorphisms, the new concept opens challenging problems, several of which are presented here. In particular, we investigate relationships between $D_e(G)$ and the endomorphism motion of a graph $G$, that is, the least possible number of vertices moved by a nontrivial endomorphism of $G$. Moreover, we extend numerous results about the distinguishing number of finite and infinite graphs to the endomorphism distinguishing number.

I have added at the top of the document a Figure of a Binary Tree as Data Structure to Help us comprehend the nature of the Cardinality of $\Sigma^*$
There are two plausible arguments for this presented here. One that would imply its Cardinality is $\aleph0$ and another that it is $2^{\aleph0}$. I ask the reader to examine both and check if both or only one argument is correct.

It is consistent that there is a set mapping from the four-tuples of omega_n into the finite subsets with no free subsets of size t_n for some natural number t_n. For any n< omega it is consistent that there is a set mapping from the pairs of omega_n into the finite subsets with no infinite free sets.

It is consistent that there is a set mapping from the four-tuples of ωninto the finite subsets with no free subsets of sizetnfor some natural numbertn. For anyn< ω it is consistent that there is a set mapping from the pairs of ωninto the finite subsets with no infinite free sets. For anyn< ω it is consistent that there is a set mapping from the pairs of ωninto ωnwith no uncountable free sets.

The distinguishing number $D(G)$ of a graph $G$ is the least cardinal number $\aleph$ such that $G$ has a labeling with $\aleph$ labels that is only preserved by the trivial automorphism. We show that the distinguishing number of the countable random graph is two, that tree-like graphs with not more than continuum many vertices have distinguishing number two, and determine the distinguishing number of many classes of infinite Cartesian products. For instance, $D(Q_{n}) = 2$, where $Q_{n}$ is the infinite hypercube of dimension ${n}$.

The distinguishing number D(G) of a graph G is the least cardinal d such that G has a labeling with d labels which is only preserved by the trivial automorphism. We show that the distinguishing number of infinite, locally finite, connected graphs G with infinite motion and growth o(n 2 / log 2 n) is either 1 or 2, which proves the Infinite Motion Conjecture of Tom Tucker for this type of graphs. The same holds true for graphs with countably many ends that do not grow too fast. We also show that graphs G of arbitrary cardinality are 2-distinguishable if every nontrivial automorphism moves at least uncountably many vertices m(G), where m(G) ≥ |Aut(G)|. This extends a result of Imrich et al. to graphs with automorphism groups of arbitrary cardinality.

Let Σ = {0, 1}, herein in the discussion below: Theorem 0.1 Here we show that Cantor's Diagonalization is wrong in his proof of the Uncountable Cardinality of the Reals or that the Cardinality of the reals is Countable Infinity. Proof: 1 Let us partition the integers into the N∪{0} and the the negative integers. Let T = (0, 1) be a subset of the reals. Let the set T be listable or have a bijection with the countably infinite set of natural numbers. Map the number 0 to the first real r 0 and the second number 1 to r 1 the second real, the number 2 to r 2 and so on in binary till countable infinity. Let all r i be different. Now I assume the reader here is familiar with Cantor's Argument in his Diagonalization Method. We simply choose a diagonal starting from the 0 th line such that we flip the bits of the diagonal. Now this real number of the flipped diagonal is different from the ones listed so far. However instead of claiming we have a contradiction, we claim that we just have not listed the reals from set T completely. So we put this diagnol say t −1 on the top of the list at the −1th position above the number 0 that is on top of the r 0 th real.Then we rediagonalize from the −1th and the t −1 th real, position and flip bits of our diagonal and again place it on the −2th position above the −1th number for the real now denoted as t −2 noting that this number t −2 is different from all those below that is t −1 , r 0 , r 1 , r 2 , r 3 • • •. Or we could form another different seperate list and so on to place these new reals of diagonals in. We could carry this method of Cantor's Diagonalization forever and come to the conclusion that we have not listed the reals completely. This could imply that there is no contradiction as Cantor claimed in listing the reals from the set T .

For an ordered set W = {w 1 , w 2 … w k } V (G) of vertices, we refer to the ordered k-tuple r(v W) = (d(v, w1), d(v, w 2) … d(v, w k)) as the (metric) representation of v with respect to W. A set W of a connected graph G is called a resolving set of G if distinct vertices of G have distinct representations with respect to W. A resolving set with minimum cardinality is called a minimum resolving set or a basis. The dimension, dim(G), is the number of vertices in a basis for G. By imposing additional constraints on the resolving set, many resolving parameters are formed. In this paper, we introduce cyclic resolving set and find the cyclic resolving number for a grid graph and augmented grid graph.

In this expository paper we revise some extensions of Kuratowski planarity criterion, providing a link between the embeddings of infinite graphs without accumulation points and the embeddings of finite graphs with some distinguished vertices in only one face. This link is valid for any surface and for some pseudosurfaces. On the one hand, we present some key ideas that are not easily accessible. On the other hand, we state the relevance of infinite, locally finite graphs in practice and suggest some ideas for future research

Cardinality of Sigma* After I submitted the first version of this paper to http : //academia.edu, I read Dr. Carl Startuvant's map from * to N. I tried to put his version of the map on a mathematical footing. This is a counting problem and cannot be done by finitely described Algorithms as there are, in retrospect, infinities involved. So it has to be proved by a bijection that is mathematically acceptable in reason. The first version of my proof of the cardinality of * which shows that the cardinality of * is uncountably infinite below and on the next page is the second version of the proof due to Dr. Carl Startuant's response on http : //academia.edu, which disproves the map he suggests is one to one. I have asked for a suitable response on http : //academia.edu on my first posting on what the error in my proof of the uncountably infinite cardinality of * was but I did not get a any response on those grounds. I also think that if both proofs are correct then we may have to conclude that the countable infinity is as large as uncountable infinity in size. But I believe the chances of this being so are remote. I would rather believe that I have erred somewhere.

A coloring is distinguishing (or symmetry breaking) if no non-identity automorphism preserves it. The distinguishing threshold of a graph G, denoted by θ(G), is the minimum number of colors k so that every k-coloring of G is distinguishing. We generalize this concept to edge-coloring by defining an alternative index θ(G). We consider θ for some families of graphs and find its connection with edge-cycles of the automorphism group. Then we show that θ(G) = 2 if and only if G ≃ K1,2 and θ (G) = 3 if and only if G ≃ P4,K1,3 or K3. Moreover, we prove some auxiliary results for graphs whose distinguishing threshold is 3 and show that although there are infinitely many such graphs, but they are not line graphs. Finally, we compute θ(G) when G is the Cartesian product of simple prime graphs.

A vertex coloring of a graph G is distinguishing if non-identity automorphisms do not preserve it. The distinguishing number, D(G), is the minimum number of colors required for such a coloring and the distinguishing threshold, θ(G), is the minimum number of colors k such that any arbitrary k-coloring is distinguishing. Moreover, Φk(G) is the number of distinguishing coloring of G using at most k colors. In this paper, for some graph operations, namely, vertex-sum, rooted product, corona product and lexicographic product, we find formulae of the distinguishing number and threshold using Φk(G).

There are different definitions of ends in non-locally-finite graphs which are all equivalent in the locally finite case. We prove the compactness of the end-topology that is based on the principle of removing finite sets of vertices and give a proof of the compactness of the end-topology that is constructed by the principle of removing finite sets of edges. For the latter case there exists already a proof in [1], which only works on graphs with countably infinite vertex sets and in contrast to which we do not use the Theorem of Tychonoff. We also construct a new topology of ends that arises from the principle of removing sets of vertices with finite diameter and give applications that underline the advantages of this new definition.

We consider maps which are constructed from plane trees by connecting each pair of consecutive leaves on their contour by a single edge. When every non-leaf vertex of the underlying tree has exactly one child which is a leaf, these maps are bijectively related to plane trees with marked corners. We study the scaling limits of such maps where Boltzmann weights are assigned to each face. The main result is that when the degree of a typical face is in the domain of attraction of an α-stable law, with α ∈ (1, 2), the maps, properly rescaled, converge to the α-stable looptrees of Curien and Kortchemski.

Résumé: Cette thèse s' inscrit dans l'étude de familles de graphes infinis de présentation finie, de leurs propriétés structurelles, ainsi que des comparaisons entre ces familles. Étant donné un alphabet fini Σ, un graphe infini étiqueté par Σ peut être caractérisé par un ensemble fini de relations binaires (Ra) a∈ Σ sur un domaine dénombrable V quelconque. De multiples caractérisations finies de tels ensembles de relations existent, soit de façon explicite grâce à des systèmes de réécriture ou à divers formalismes de la théorie des ...

A group A acting faithfully on a set X is 2-distinguishable if there is a 2-coloring of X that is not preserved by any nonidentity element of A, equivalently, if there is a proper subset of X with trivial setwise stabilizer. The motion of an element a in A is the number of points of X that are moved by a, and the motion of the group A is the minimal motion of its nonidentity elements. For finite A, the Motion Lemma says that if the motion of A is large enough (specifically at least 2 log_2 |A|), then the action is 2-distinguishable. For many situations where X has a combinatorial or algebraic structure, the Motion Lemma implies the action of Aut(X) on X is 2-distinguishable in all but finitely many instances. We prove an infinitary version of the Motion Lemma for countably infinite permutation groups, which states that infinite motion is large enough to guarantee 2-distinguishability. From this we deduce a number of results, including the fact that every locally finite, connected gr...

A group of permutations G of a set V is k-distinguishable if there exists a partition of V into k parts such that only the identity permutation in G fixes setwise all of the cells of the partition. The least cardinal number k such that (G,V) is k-distinguishable is its distinguishing number. In particular, a graph X is k-distinguishable if its automorphism group Aut(X) has distinguishing number at most k in its action on the vertices of X. Various results in the literature demonstrate that when an infinite graph fails to have some property, then often some finite subgraph is similarly deficient. In this paper we show that whenever an infinite connected graph X is not k-distinguishable (for a given cardinal k), then it contains a ball B of finite radius whose distinguishing number is at least k. Moreover, this lower bound cannot be sharpened, since for any integer k greater than 3 there exists an infinite, locally finite, connected graph X that is not k-distinguishable but in which e...

A group of permutations $G$ of a set $V$ is $k$-distinguishable if there exists a partition of $V$ into $k$ cells such that only the identity permutation in $G$ fixes setwise all of the cells of the partition. The least cardinal number $k$ such that $(G,V)$ is $k$-distinguishable is its distinguishing number $D(G,V)$. In particular, a graph $\Gamma$ is $k$-distinguishable if its automorphism group $\rm{Aut}(\Gamma)$ satisfies $D(\rm{Aut}(\Gamma),V\Gamma)\leq k$.Various results in the literature demonstrate that when an infinite graph fails to have some property, then often some finite subgraph is similarly deficient. In this paper we show that whenever an infinite connected graph $\Gamma$ is not $k$-distinguishable (for a given cardinal $k$), then it contains a ball of finite radius whose distinguishing number is at least $k$. Moreover, this lower bound cannot be sharpened, since for any integer $k \geq 3$ there exists an infinite, locally finite, connected graph $\Gamma$ that is no...

A vertex coloring is called distinguishing if the identity is the only automorphism that can preserve it. The distinguishing number of a graph is the minimum number of colors required for such a coloring. The distinguishing threshold of a graph G is the minimum number of colors k required that any arbitrary k-coloring of G is distinguishing. We prove a statement that gives a necessary and sufficient condition for a vertex coloring of the Cartesian product to be distinguishing. Then we use it to calculate the distinguishing threshold of a Cartesian product graph. Moreover, we calculate the number of non-equivalent distinguishing colorings of grids.

A graph G is k-ordered if for every ordered sequence of k vertices, there is a cycle in G that encounters the vertices of the sequence in the given order. We prove that if G is a connected graph distinct from a path, then there is a number t G such that for every t ≥ t G the t-iterated line graph of G, L t (G), is (δ(L t (G))+1)-ordered. Since there is no graph H which is (δ(H)+2)-ordered, the result is best possible.

In this short paper, we describe another class of forcing notions which preserve measurability of a large cardinal κ from the optimal hypothesis, while adding new unbounded subsets to κ. In some ways these forcings are closer to the Cohen-type forcings-e.g. we show that they are not minimalhowever, they share some properties with tree-like forcings. We show that they admit fusion-type arguments which allow for a uniform lifting argument.

The distinguishing number $\Delta(X)$ of a graph $X$ is the least positive integer $n$ for which there exists a function $f:V(X)\to\{0,1,2,\cdots,n-1\}$ such that no nonidentity element of $\hbox{Aut}(X)$ fixes (setwise) every inverse image $f^{-1}(k)$, $k\in\{0,1,2,\cdots,n-1\}$. All infinite, locally finite trees without pendant vertices are shown to be 2-distinguishable. A proof is indicated that extends 2-distinguishability to locally countable trees without pendant vertices. It is shown that every infinite, locally finite tree $T$ with finite distinguishing number contains a finite subtree $J$ such that $\Delta(J)=\Delta(T)$. Analogous results are obtained for the distinguishing chromatic number, namely the least positive integer $n$ such that the function $f$ is also a proper vertex-coloring.

In this paper we are interested in the automorphism group of the poset B m,n. B m,n constitutes the words obtained from the cyclic word of length n on an alphabet of m letters in by deleting on all possible ways and their natural order. We prove: Résumé: Le but de ce papier est la détermination du groupe d'automorphismes des ordres B m,n. Il s'agit des mots obtenus à partir du mot cyclique de longeur n sur un alphabet de m lettres par suppression successive de lettres et ordonnés naturellement. On prouve: Aut B m,n =    S n for 1 ≤ n ≤ m, S 2 ⊗ S 2m−n for m + 1 ≤ n ≤ 2m − 1, S 2 for 2m ≤ n.

One of the natural topologies for infinite graphs with edge-ends is ETop. Also ETop is the coarest topology among other topologies for infinite graphs.
In this note, we characterize this topology with different methods and we show that it is always compact.

A problem by Diestel is to extend algebraic flow theory of finite graphs to infinite graphs
with ends. In order to pursue this problem, we define an A-flow and non-elusive H-flow for
arbitrary graphs and for abelian topological Hausdorff groups H and compact subsets A ⊆ H.
We use these new definitions to extend several well-known theorems of flows in finite graphs
to infinite graphs

Let M be the countably infinite metric fan. We show that C k (M, 2) is sequential and contains a closed copy of Arens space S 2 . It follows that if X is metrizable but not locally compact, then C k (X) contains a closed copy of S 2 , and hence does not have the property AP.

This note presents a new, elementary proof of a generalization of a theorem of Halin to graphs with unbounded degrees, which is then applied to show that every connected, countably infinite graph G with a subdegree-finite, infinite automorphism group whose cardinality is strictly less than continuum, has a finite set F of vertices that is setwise stabilized only by the identity automorphism. A bound on the size of such sets, which are called distinguishing, is also provided.
To put this theorem of Halin and its generalization into perspective, we also discuss several related non-elementary, independent results and their methods of proof.

We introduce the notion of an edge-end and characterize those countable graphs which have edge-end-faithful spanning trees. We also prove that for a natural class of graphs, there always exists a tree which is faithful on the undominated ends and rayless over the dominated does. 1997 Academic Press article no. TB961734 225

We investigate the following problem: What countable graphs must a graph of uncountable chromatic number contain? We define two graphsΓ andΔ which are very similar and we show thatΓ is contained in every graph of uncountable chromatic number, whileΔ is (consistently) not.