Papers by Wilfried Imrich
European Journal of Combinatorics
Aldous and Fill conjectured that the maximum relaxation time for the random walk on a connected r... more Aldous and Fill conjectured that the maximum relaxation time for the random walk on a connected regular graph with n vertices is (1 + o(1)) 3n 2 2π 2. This conjecture can be rephrased in terms of the spectral gap as follows: the spectral gap (algebraic connectivity) of a connected k-regular graph on n vertices is at least (1 + o(1)) 2kπ 2 3n 2 , and the bound is attained for at least one value of k. Based upon previous work of Brand, Guiduli, and Imrich, we prove this conjecture for cubic graphs. We also investigate the structure of quartic (i.e. 4-regular) graphs with the minimum spectral gap among all connected quartic graphs. We show that they must have a path-like structure built from specific blocks.
The Electronic Journal of Combinatorics, 2021
A vertex colouring of a graph is asymmetric if it is preserved only by the identity automorphism.... more A vertex colouring of a graph is asymmetric if it is preserved only by the identity automorphism. The minimum number of colours needed for an asymmetric colouring of a graph $G$ is called the asymmetric colouring number or distinguishing number $D(G)$ of $G$. It is well known that $D(G)$ is closely related to the least number of vertices moved by any non-identity automorphism, the so-called motion $m(G)$ of $G$. Large motion is usually correlated with small $D(G)$. Recently, Babai posed the question whether there exists a function $f(d)$ such that every connected, countable graph $G$ with maximum degree $\Delta(G)\leq d$ and motion $m(G)>f(d)$ has an asymmetric $2$-colouring, with at most finitely many exceptions for every degree. We prove the following result: if $G$ is a connected, countable graph of maximum degree at most 4, without an induced claw $K_{1,3}$, then $D(G)= 2$ whenever $m(G)>2$, with three exceptional small graphs. This answers the question of Babai for $d=4$ ...
European Journal of Combinatorics, 2020
If a graph G has distinguishing number 2, then there exists a partition of its vertex set into tw... more If a graph G has distinguishing number 2, then there exists a partition of its vertex set into two parts, such that no nontrivial automorphism of G fixes setwise the two parts. Such a partition is called a 2-distinguishing coloring of G, and the parts are called its color classes. If G admits such a coloring, it is often possible to find another in which one of the color classes is sparse in a certain sense. In this case we say that G has 2-distinguishing density zero. An extreme example of this would be an infinite graph admitting a 2-distinguishing coloring in which one of the color classes is finite. If a graph G contains a vertex v such that, for all n ∈ N, any two distinct vertices equidistant from v have nonequal n-spheres, then we say that G satisfies the Distinct Spheres Condition. In this paper we prove a general result: any countable connected graph that satisfies the Distinct Spheres Condition is 2-distinguishable with density zero. We present two proofs of this, one that uses a deterministic coloring, and another (that applies only to locally finite graphs) using a random coloring. From this * Florian Lehner was supported by the Austrian Science Fund (FWF) Grant no. J 3850-N32 result, we deduce that several important families of countably infinite and connected graphs are 2-distinguishable with density zero, including those that are locally finite and primitive. Furthermore, we prove that any connected graph with infinite motion and subquadratic growth is 2distinguishable with density zero.
Lecture Notes in Mathematics, 1990
We derive a bound for the rank of the fixed point set Fp(0') of a monomorphism 0' and the fixed p... more We derive a bound for the rank of the fixed point set Fp(0') of a monomorphism 0' and the fixed point group Fix(0') of homomorphisms 0' of subgroups G of a free group F into F. We do not require the ranks of G and F to be finite; these conditions are replaced by a finiteness condition for 0'.
Monatshefte für Mathematik, 2020
Let T be a finite or infinite tree and m the minimum number of vertices moved by the non-identity... more 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 minimum number of colors needed to color the v... more The distinguishing number D(G) of a graph G is the minimum number of colors needed to color the vertices of G such that the coloring is preserved only by the trivial automorphism. In this paper we improve results about the distinguishing number of Cartesian products of finite and infinite graphs by removing restrictions to prime or relatively prime factors.
Ars Mathematica Contemporanea, 2020
We consider edge colourings, not necessarily proper. The distinguishing index D (G) of a graph G ... more We consider edge colourings, not necessarily proper. The distinguishing index D (G) of a graph G is the least number of colours in an edge colouring that is preserved only by the identity automorphism. It is known that D (G) ≤ ∆ for every countable, connected graph G with finite maximum degree ∆ except for three small cycles. We prove that D (G) ≤ √ ∆ + 1 if additionally G does not have pendant edges.
Ars Mathematica Contemporanea, 2014
This note describes fast algorithms for computing the prime factors of connected, nonbipartite gr... more This note describes fast algorithms for computing the prime factors of connected, nonbipartite graphs with respect to the direct product, and of connected graphs with respect to the strong product. The complexities are O(m min(n 2 , ∆ 3)) for the direct product, and O(m a(G)∆) for the strong, where n is the order of the graph G to be factored, m its size, a(G) its arboricity, and ∆ its maximum degree. That is, the complexities are linear in m for fixed ∆.
Ars Mathematica Contemporanea, 2014
This paper is concerned with the fast computation of a relation d on the edge set of connected gr... more This paper is concerned with the fast computation of a relation d on the edge set of connected graphs that plays a decisive role in the recognition of approximate Cartesian products, the weak reconstruction of Cartesian products, and the recognition of Cartesian graph bundles with a triangle free basis. A special case of d is the relation δ * , whose convex closure yields the product relation σ that induces the prime factor decomposition of connected graphs with respect to the Cartesian product. For the construction of d so-called Partial Star Products are of particular interest. Several special data structures are used that allow to compute Partial Star Products in constant time. These computations are tuned to the recognition of approximate graph products, but also lead to a linear time algorithm for the computation of δ * for graphs with maximum bounded degree. Furthermore, we define quasi Cartesian products as graphs with non-trivial δ *. We provide several examples, and show that quasi Cartesian products can be recognized in linear time for graphs with bounded maximum degree. Finally, we note that quasi products can be recognized in sublinear time with a parallelized algorithm.
Applicable Analysis and Discrete Mathematics, 2020
Let H be an arbitrary graph with vertex set V (H) = [nH] = {l,?, nH}. The generalized Sierpi?ski ... more Let H be an arbitrary graph with vertex set V (H) = [nH] = {l,?, nH}. The generalized Sierpi?ski graph SnH , n ? N, is defined on the vertex set [nH]n, two different vertices u = un ?u1 and v = vn ? v1 being adjacent if there exists an h? [n] such that (a) ut = vt, for t > h, (b) uh ? vh and uhvh ? E(H), and (c) ut = vh and vt = uh for t < h. If H is the complete graph Kk, then we speak of the Sierpi?ski graph Sn k . We present an algorithm that recognizes Sierpi?ski graphs Sn k in O(|V (Sn k )|1+1=n) = O(|E(Sn k )|) time. For generalized Sierpi?ski graphs SnH we present a polynomial time algorithm for the case when H belong to a certain well defined class of graphs. We also describe how to derive the base graph H from an arbitrarily given SnH .
The Electronic Journal of Combinatorics, 2014
We introduce the endomorphism distinguishing number $D_e(G)$ of a graph $G$ as the least cardinal... more 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.
The Electronic Journal of Combinatorics, 2007
The distinguishing number $D(G)$ of a graph $G$ is the least cardinal number $\aleph$ such that $... more 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 Electronic Journal of Combinatorics, 2017
We consider infinite graphs. The distinguishing number $D(G)$ of a graph $G$ is the minimum numbe... more We consider infinite graphs. The distinguishing number $D(G)$ of a graph $G$ is the minimum number of colours in a vertex colouring of $G$ that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is called the distinguishing index, denoted by $D'(G)$. We prove that $D'(G)\leq D(G)+1$. For proper colourings, we study relevant invariants called the distinguishing chromatic number $\chi_D(G)$, and the distinguishing chromatic index $\chi'_D(G)$, for vertex and edge colourings, respectively. We show that $\chi_D(G)\leq 2\Delta(G)-1$ for graphs with a finite maximum degree $\Delta(G)$, and we obtain substantially lower bounds for some classes of graphs with infinite motion. We also show that $\chi'_D(G)\leq \chi'(G)+1$, where $\chi'(G)$ is the chromatic index of $G$, and we prove a similar result $\chi''_D(G)\leq \chi''(G)+1$ for proper total colourings. A number of conjectures are formulated.
The Electronic Journal of Combinatorics, 2018
It is known that for graphs $A$ and $B$ with odd cycles, the direct product $A\times B$ is vertex... more It is known that for graphs $A$ and $B$ with odd cycles, the direct product $A\times B$ is vertex-transitive if and only if both $A$ and $B$ are vertex-transitive. But this is not necessarily true if one of $A$ or $B$ is bipartite, and until now there has been no characterization of such vertex-transitive direct products. We prove that if $A$ and $B$ are both bipartite, or both non-bipartite, then $A\times B$ is vertex-transitive if and only if both $A$ and $B$ are vertex-transitive. Also, if $A$ has an odd cycle and $B$ is bipartite, then $A\times B$ is vertex-transitive if and only if both $A\times K_2$ and $B$ are vertex-transitive.
The Electronic Journal of Combinatorics, 2019
The distinguishing number $D(G)$ of a graph $G$ is the smallest number of colors that is needed t... more The distinguishing number $D(G)$ of a graph $G$ is the smallest number of colors that is needed to color the vertices such that the only color-preserving automorphism fixes all vertices. We give a complete classification for all connected graphs $G$ of maximum valence $\Delta(G) = 3$ and distinguishing number $D(G) = 3$. As one of the consequences we show that all infinite connected graphs with $\Delta(G) = 3$ are $2$-distinguishable.
The Electronic Journal of Combinatorics, 2014
A graph $G$ is said to be 2-distinguishable if there is a labeling of the vertices with two label... more A graph $G$ is said to be 2-distinguishable if there is a labeling of the vertices with two labels such that only the trivial automorphism preserves the labels. Call the minimum size of a label class in such a labeling of $G$ the cost of 2-distinguishing $G$.We show that the connected, locally finite, infinite graphs with finite 2-distinguishing cost are exactly the graphs with countable automorphism group. Further we show that in such graphs the cost is less than three times the size of a smallest determining set. We also another, sharper bound on the 2-distinguishing cost, in particular for graphs of linear growth.
Ars Mathematica Contemporanea, 2013
The distinguishing number D(G) of a graph G is the least cardinal d such that G has a labeling wi... more 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.
Discrete Applied Mathematics, 2019
We extend results about asymmetric colorings of finite Cartesian products of graphs to strong and... more We extend results about asymmetric colorings of finite Cartesian products of graphs to strong and direct products of graphs and digraphs. On the way we shorten proofs for the existence of prime factorizations of finite digraphs and characterize the structure of the automorphism groups of strong and direct products. The paper ends with results on asymmetric colorings of Cartesian products of finite and infinite digraphs.
Ars Mathematica Contemporanea, 2015
We extend the definition of the Cartesian product to graphs with loops and show that the Sabiduss... more We extend the definition of the Cartesian product to graphs with loops and show that the Sabidussi-Vizing unique factorization theorem for connected finite simple graphs still holds in this context for all connected finite graphs with at least one unlooped vertex. We also prove that this factorization can be computed in O(m) time, where m is the number of edges of the given graph.
Ars Mathematica Contemporanea, 2009
Under suitable conditions of connectivity or non-bipartiteness, each of the three standard graph ... more Under suitable conditions of connectivity or non-bipartiteness, each of the three standard graph products (the Cartesian product, the direct product and the strong product) satisfies the unique prime factorization property, and there are polynomial algorithms to determine the prime factors. This is most easily proved for the Cartesian product. For the other products, current proofs involve a notion of a Cartesian skeleton which transfers their multiplication properties to the Cartesian product. The present article introduces simplified definitions of Cartesian skeletons for the direct and strong products, and provides new, fast and transparent algorithms for their construction. Since the complexity of the prime factorization of the direct and the strong product is determined by the complexity of the construction of the Cartesian skeleton, the new algorithms also improve the complexity of the prime factorizations of graphs with respect to the direct and the strong product. We indicate how these simplifications fit into the existing literature.
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Papers by Wilfried Imrich