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SemiGroups Acting on ContextFree Graphs
"... Let \Gamma be some contextfree graph. We give sufficient conditions on a semigroup of bisimulations H ensuring that the quotient Hn\Gamma is contextfree. Using these sufficient conditions we show that the quotient Aut(\Gamma )n\Gamma of \Gamma by its full group of automorphisms is always context ..."
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Let \Gamma be some contextfree graph. We give sufficient conditions on a semigroup of bisimulations H ensuring that the quotient Hn\Gamma is contextfree. Using these sufficient conditions we show that the quotient Aut(\Gamma )n\Gamma of \Gamma by its full group of automorphisms is always contextfree. We then give examples showing optimality (in some sense) of the above result.
Distinguishability of locally finite trees
 Electron. J. Combin
"... The distinguishing number ∆(X) of a graph X is the least positive integer n for which there exists a function f: V (X) → {0, 1, 2, · · · , n−1} such that no nonidentity element of Aut(X) fixes (setwise) every inverse image f −1 (k), k ∈ {0, 1, 2, · · · , n − 1}. All infinite, locally finite t ..."
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The distinguishing number ∆(X) of a graph X is the least positive integer n for which there exists a function f: V (X) → {0, 1, 2, · · · , n−1} such that no nonidentity element of Aut(X) fixes (setwise) every inverse image f −1 (k), k ∈ {0, 1, 2, · · · , n − 1}. All infinite, locally finite trees without pendant vertices are shown to be 2distinguishable. A proof is indicated that extends 2distinguishability 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 ∆(J) = ∆(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 vertexcoloring. 1
A Note on the Growth Rate of Planar Graphs
, 1996
"... If \Gamma is a planar, locally finite, vertex transitive, 1ended graph, then there is a particular `niceness' about the arrangement of the regions incident to a vertex in \Gamma. Using this feature, it can be shown that \Gamma can be embedded in either the Euclidean plane or the hyperbolic plane in ..."
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If \Gamma is a planar, locally finite, vertex transitive, 1ended graph, then there is a particular `niceness' about the arrangement of the regions incident to a vertex in \Gamma. Using this feature, it can be shown that \Gamma can be embedded in either the Euclidean plane or the hyperbolic plane in such a way that every edge has the same length and every angle in an ncycle bounding a region has the same measure. Moreover, there is a simple condition which tells whether \Gamma is embedded in the Euclidean plane or the hyperbolic plane. The geometry of these two planes is then exploited to show that \Gamma must have either quadratic growth or exponential growth depending on which plane it is embedded in. 1. Introduction. The graphs considered in this paper are simple, infinite, and planar. The symbols V (\Gamma), E (\Gamma) and aut(\Gamma) will denote respectively, the vertex set, the edge set, and the automorphism group of \Gamma. All graphs will be assumed to be locally finite; th...
Automorphisms of graphs
, 2001
"... This chapter surveys automorphisms of finite graphs, concentrating on the asymmetry of typical graphs, prescribing automorphism groups (as either permutation groups or abstract groups), and special properties of vertextransitive graphs and related classes. There are short digressions on infinite gr ..."
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This chapter surveys automorphisms of finite graphs, concentrating on the asymmetry of typical graphs, prescribing automorphism groups (as either permutation groups or abstract groups), and special properties of vertextransitive graphs and related classes. There are short digressions on infinite graphs and graph homomorphisms.
Mutually embeddable graphs and the Tree Alternative conjecture
, 2004
"... We prove that if a rayless tree T is mutually embeddable and nonisomorphic with another rayless tree, then T is mutually embeddable and nonisomorphic with infinitely many rayless trees. The proof relies on a fixed element theorem of Halin, which states that every rayless tree has either a vertex o ..."
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We prove that if a rayless tree T is mutually embeddable and nonisomorphic with another rayless tree, then T is mutually embeddable and nonisomorphic with infinitely many rayless trees. The proof relies on a fixed element theorem of Halin, which states that every rayless tree has either a vertex or an edge that is fixed by every selfembedding. We state a conjecture that proposes an extension of our result to all trees. Key words: Rayless tree, mutually embeddable, selfembedding 1991 MSC: 05C05, 20M20 Email addresses: abonato@rogers.com (Anthony Bonato), Claude.Tardif@rmc.ca (Claude Tardif).
On endfaithful spanning trees in infinite graphs
, 1990
"... Let G be an infinite connected graph. A ray (from v) inGis a 1way infinite path in G (with initial vertex v). An infinite connected subgraph ofa ray R ⊂ G is called a tail of R. If X ⊂ G is finite, the infinite component of R\X will be called the tail of R in G\X. The following assertions are equiv ..."
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Let G be an infinite connected graph. A ray (from v) inGis a 1way infinite path in G (with initial vertex v). An infinite connected subgraph ofa ray R ⊂ G is called a tail of R. If X ⊂ G is finite, the infinite component of R\X will be called the tail of R in G\X. The following assertions are equivalent for rays P, Q ⊂ G:
Between Ends and Fibers
, 2006
"... Let Γ be an infinite, locally finite, connected graph with distance function δ. Given a ray P in Γ and a constant C ≥ 1, a vertexsequence {xn} ∞ n=0 ⊆ V P is said to be regulated by C if, for all n ∈ N, xn+1 never precedes xn on P, each vertex of P appears at most C times in the sequence, and δP ( ..."
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Let Γ be an infinite, locally finite, connected graph with distance function δ. Given a ray P in Γ and a constant C ≥ 1, a vertexsequence {xn} ∞ n=0 ⊆ V P is said to be regulated by C if, for all n ∈ N, xn+1 never precedes xn on P, each vertex of P appears at most C times in the sequence, and δP (xn, xn+1) ≤ C. R. Halin (1964) defined two rays to be endequivalent if they are joined by infinitely many pairwisedisjoint paths; the resulting equivalence classes are called ends. More recently H.A. Jung (1993) defined rays P and Q to be bequivalent if there exist sequences {xn} ∞ n=0 ⊆ V P and {yn} ∞ n=0 ⊆ V Q regulated by some constant C ≥ 1 such that δ(xn, yn) ≤ C for all n ∈ N; he named the resulting equivalence classes bfibers. Let F0 denote the set of nondecreasing functions from N into the set of positive real numbers. The relation P ∼f Q (called fequivalence) generalizes Jung’s condition to δ(xn, yn) ≤ Cf(n). As f runs through F0, uncountably many equivalence relations are
A proof of the rooted tree alternative conjecture
, 2008
"... In [2] Bonato and Tardif conjectured that the number of isomorphism classes of trees mutually embeddable with a given tree T is either 1 or ∞. We prove the analogue of their conjecture for rooted trees. We also discuss the original conjecture for locally finite trees and state some new conjectures. ..."
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In [2] Bonato and Tardif conjectured that the number of isomorphism classes of trees mutually embeddable with a given tree T is either 1 or ∞. We prove the analogue of their conjecture for rooted trees. We also discuss the original conjecture for locally finite trees and state some new conjectures. 1
ZEOLITES: GEOMETRY AND COMBINATORICS ∗
"... We study the background associated with phenomena observed in zeolites using combinatorial and geometric techniques. We define combinatorial ddimensional zeolites and show that not all combinatorial zeolites have a unit distance realization in R d, and of those that have a unit distance realization ..."
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We study the background associated with phenomena observed in zeolites using combinatorial and geometric techniques. We define combinatorial ddimensional zeolites and show that not all combinatorial zeolites have a unit distance realization in R d, and of those that have a unit distance realization, not all have nonoverlapping unitdistance realizations. Only few classes of finite 2d zeolites are known and a long standing conjecture of Harborth suggests that there is only one type of nonoverlapping unit distance realizable finite 2d zeolites. However, for all d ≥ 2, we prove that there are uncountably many distinct infinite nonoverlapping unit distance realizable zeolites. Infinite 3d zeolites are the objects of interest to Chemists and Physicists and have important industrial applications. 1.