Frucht theorem for inverse semigroups

In the paper, the problem of representing a finite inverse semigroup by partial transformations of a graph is treated. The notions of weighted graph and its weighted partial isomorphisms are introduced. The main result is that any finite inverse semigroup is isomorphic to the semigroup of weighted partial isomorphisms of a weighted graph. This assertion is a natural generalization of the Frucht theorem for groups. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 246–251, February, 1997. This research was partially supported by the International Science Foundation under grant No. GSU 041049. Translated by A. I. Shtern     

Symmetrized Induced Ramsey Theory

We prove induced Ramsey theorems in which the monochromatic induced subgraph satisfies that all members of a prescribed set of its partial isomorphisms extend to automorphisms of the colored graph (without requirement of preservation of colors). We consider vertex and edge colorings, and extensions of partial isomorphisms in the set of all partial isomorphisms between singletons as considered by Babai and Sós (European J Combin 6(2):101–114, 1985), the set of all finite partial isomorphisms as considered by Hrushovski (Combinatorica 12(4):411–416, 1992), Herwig (Combinatorica 15:365–371, 1995) and Herwig-Lascar (Trans Amer Math Soc 5:1985–2021, 2000), and the set of all total isomorphisms. We observe that every finite graph embeds into a finite vertex transitive graph by a so called bi-embedding, an embedding that is compatible with a monomorphism between the corresponding automorphism groups. We also show that every countable graph bi-embeds into Rado’s universal countable graph Γ.     

Fixed points of posets and clique graphs

The fixed point property for partial orders has been the object of much attention in the past twenty years. Recently, M. Roddy ([7]) proved this famous conjecture of Rival (see [6]): the class of finite orders with the fixed point property is closed under finite products.In this article, we prove that a finite order has the fixed point property if the sequence of iterated clique graphs of its comparability graph tends to the trivial graph.     

On fixing boundary points of transitive hyperbolic graphs

We show that there is no one-ended, locally finite, planar, hyperbolic graph such that the stabilizer of one of its hyperbolic boundary points acts transitively on the vertices of the graph. This gives a partial answer to a question by Kaimanovich and Woess.     

Cut Vertices in Zero-Divisor Graphs of Finite Commutative Rings

A cut vertex of a connected graph is a vertex whose removal would result in a graph having two or more connected components. We examine the presence of cut vertices in zero-divisor graphs of finite commutative rings and provide a partial classification of the rings in which they appear.     

Catalan monoids,monoids of local endomorphisms,and their presentations

The Catalan monoid and partial Catalan monoid of a directed graph are introduced. Also introduced is the notion of a local endomorphism of a tree, and it is shown that the Catalan (resp. partial Catalan) monoid of a tree is simply its monoid of extensive local endomorphisms (resp. partial endomorphisms) of finite shift. The main results of this paper are presentations for the Catalan and partial Catalan monoids of a tree. Our presentation for the Catalan monoid of a tree is used to give an alternative proof for a result of Higgins. We also identify results of Aîzen?tat and Popova which give presentations for the Catalan monoid and partial Catalan monoid of a finite symmetric chain.     

Symmetries of the Partial Order of Traces

This paper deals with the automorphism group of the partial order of finite traces. We show that any group can arise as such an automorphism group if we allow arbitrary large dependence alphabets. Restricting to finite dependence alphabets, the automorphism groups are profinite and possess only finitely many simple decomposition factors. Finally, we show that the partial order associated with the Rado graph as dependence alphabet does not give rise to a homogeneous domain thereby answering an open question from Boldi, P., Cardone, F. and Sabadini, N. (1993).     

Update games and update networks

In this paper we model infinite processes with finite configurations as infinite games over finite graphs. We investigate those games, called update games, in which each configuration occurs an infinite number of times during a two-person play. We also present an efficient polynomial-time algorithm (and partial characterization) for deciding if a graph is an update network.     

Netlike partial cubes, V: Completion and netlike classes

We define a completion of a netlike partial cube G by replacing each convex 2n-cycle C of G with n≥3 by an n-cube admitting C as an isometric cycle. We prove that a completion of G is a median graph if and only if G has the Median Cycle Property (MCP) (see N. Polat, Netlike partial cubes III. The Median Cycle Property, Discrete Math.). In fact any completion of a netlike partial cube having the MCP is defined by a universal property and turns out to be a minimal median graph containing G as an isometric subgraph. We show that the completions of the netlike partial cubes having the MCP preserves the principal constructions of these graphs, such as: netlike subgraphs, gated amalgams and expansions. Conversely any netlike partial cube having the MCP can be obtained from a median graph by deleting some particular maximal finite hypercubes. We also show that, given a netlike partial cube G having the MCP, the class of all netlike partial cubes having the MCP whose completions are isomorphic to those of G share different properties, such as: depth, lattice dimension, semicube graph and crossing graph.     

Root Systems for Asymmetric Geometric Representations of Coxeter Groups

Results are obtained concerning root systems for asymmetric geometric representations of Coxeter groups. These representations were independently introduced by Vinberg and Eriksson, and generalize the standard geometric representation of a Coxeter group in such a way as to include certain restrictions of all Kac–Moody Weyl groups. In particular, a characterization of when a nontrivial multiple of a root may also be a root is given in the general context. Characterizations of when the number of such multiples of a root is finite and when the number of positive roots sent to negative roots by a group element is finite are also given. These characterizations are stated in terms of combinatorial conditions on a graph closely related to the Coxeter graph for the group. Other finiteness results for the symmetric case which are connected to the Tits cone and to a natural partial order on positive roots are extended to this asymmetric setting.     

Partial cubes: structures, characterizations, and constructions

Partial cubes are isometric subgraphs of hypercubes. Structures on a graph defined by means of semicubes, and Djokovi?’s and Winkler’s relations play an important role in the theory of partial cubes. These structures are employed in the paper to characterize bipartite graphs and partial cubes of arbitrary dimension. New characterizations are established and new proofs of some known results are given.The operations of Cartesian product and pasting, and expansion and contraction processes are utilized in the paper to construct new partial cubes from old ones. In particular, the isometric and lattice dimensions of finite partial cubes obtained by means of these operations are calculated.     

On the coarse-geometric detection of subgroups

We generalize [20] to give sufficient conditions, primarily on coarse geometry, to ensure that a subset of a Cayley graph is a finite Hausdorff distance from a subgroup. Using this result, we prove a partial converse to the Flat Torus Theorem for CAT(0) groups. Also using this result, we give sufficient conditions for subgroups and splittings to be invariant under quasi-isometries.     

Partial Actions of Groups on Cell Complexes

 In this paper, we study partial group actions on 2-complexes. Our results include a characterization, in terms of generating sets, of when a partial group action on a connected 2-complex has a connected globalization. Using this result, we give a short combinatorial proof that a group acting without fixed points on a connected 2-complex, with finite quotient, is finitely generated. This result is then generalized to characterize finitely generated groups as precisely those groups having a partial action, without fixed points, on a finite tree, with a connected globalization. Finally, using Bass-Serre theory, we determine when a partial group action on a graph has a globalization which is a tree. The author was supported in part by NSF-NATO postdoctoral fellowship DGE-9972697, by Praxis XXI scholarship BPD 16306 98 and by FCT through Centro de Matemática da Universidade do Porto. Received September 20, 2001; in revised form June 25, 2002     

Ultragraphs and shift spaces over infinite alphabets

In this paper we further develop the theory of one-sided shift spaces over infinite alphabets, characterizing one-step shifts as edge shifts of ultragraphs and partially answering a conjecture regarding shifts of finite type (we show that there exist shifts of finite type that are not conjugate, via a conjugacy that is eventually finite periodic, to an edge shift of a graph). We also show that there exist edge shifts of ultragraphs that are shifts of finite type, but are not conjugate to a full shift, a result that is not true for edge shifts of graphs. One of the key results needed in the proofs of our conclusions is the realization of a class of ultragraph C*-algebras as partial crossed products, a result of interest on its own.     

On the hamiltonicity of line graphs of locally finite, 6‐edge‐connected graphs

The topological approach to the study of infinite graphs of Diestel and KÜhn has enabled several results on Hamilton cycles in finite graphs to be extended to locally finite graphs. We consider the result that the line graph of a finite 4‐edge‐connected graph is hamiltonian. We prove a weaker version of this result for infinite graphs: The line graph of locally finite, 6‐edge‐connected graph with a finite number of ends, each of which is thin, is hamiltonian.     

A Characterization of the Interval Function of a (Finite or Infinite) Connected Graph

By the interval function of a finite connected graph we mean the interval function in the sense of H. M. Mulder. This function is very important for studying properties of a finite connected graph which depend on the distance between vertices. The interval function of a finite connected graph was characterized by the present author. The interval function of an infinite connected graph can be defined similarly to that of a finite one. In the present paper we give a characterization of the interval function of each connected graph.     

Relations Between Connected and Self-Avoiding Hikes in Labelled Complete Digraphs

A walk in a directed graph is defined as a finite sequence of contiguous edges. Seeing the edges as indeterminates, walks are investigated as monomials and endowed with a partial order that extends to possibly unconnected objects called hikes. Analytical transformations of the weighted adjacency matrix reveal a relation between walks and self-avoiding hikes, giving rise to interesting combinatorial properties such as an expression of the number of ways to travel a walk in function of its self-avoiding divisors.     

Extending partial isomorphisms for the small index property of many ω-categorical structures

Theorem:Let A be a finite K _m -free graph, p ₁ , …, p _n partial isomorphisms on A. Then there exists a finite extension B, which is also a K _m -free graph, and automorphisms f _i of B extending the p _i . A paper by Hodges, Hodkinson, Lascar and Shelah shows how this theorem can be used to prove the small index property for the generic countable graph of this class. The same method also works for a certain class of continuum many non-isomorphic ω-categorical countable digraphs and more generally for structures in an arbitrary finite relational language, which are built in a similar fashion. Hrushovski proved this theorem for the class of all finite graphs [Hr]; the proof presented here stems from his proof. Supported by EC-grant ERBCHBGCT 920013.     

Algebraic Cayley graphs over finite local rings

In this work, we define and study the algebraic Cayley directed graph over a finite local ring. Its vertex set is the unit group of a finite extension of a finite local ring R and its adjacency condition is that the quotient is a monic primary polynomial. We investigate its connectedness and diameter bound, and we also show that our graph is an expander graph. In addition, if a local ring has nilpotency two, then we obtain a better view of our graph from the lifting of the graph over its residue field.     

An intersection problem for finite automata

A finite automaton with state space S and alphabet A can be thought of as a directed graph with vertex set S such that for each vertex t ϵ S the edges leaving t are in one-to-one correspondence with A. Motivated by a problem in logic, we propose a problem about intersections of languages accepted by finite automata and give some partial results.