scholarly journals Bounds on the Distinguishing Chromatic Number

10.37236/177 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Karen L. Collins ◽  
Mark Hovey ◽  
Ann N. Trenk
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Automorphism Group ◽  
Chromatic Number ◽  
Automorphism Groups ◽  
Upper Bounds ◽  
Underlying Graph ◽  
Minimum Number

Collins and Trenk define the distinguishing chromatic number $\chi_D(G)$ of a graph $G$ to be the minimum number of colors needed to properly color the vertices of $G$ so that the only automorphism of $G$ that preserves colors is the identity. They prove results about $\chi_D(G)$ based on the underlying graph $G$. In this paper we prove results that relate $\chi_D(G)$ to the automorphism group of $G$. We prove two upper bounds for $\chi_D(G)$ in terms of the chromatic number $\chi(G)$ and show that each result is tight: (1) if Aut$(G)$ is any finite group of order $p_1^{i_1} p_2^{i_2} \cdots p_k^{i_k}$ then $\chi_D(G) \le \chi(G) + i_1 + i_2 \cdots + i_k$, and (2) if Aut$(G)$ is a finite and abelian group written Aut$(G) = {\Bbb Z}_{p_{1}^{i_{1}}}\times \cdots \times {\Bbb Z}_{p_{k}^{i_{k}}}$ then we get the improved bound $\chi_D(G) \le \chi(G) + k$. In addition, we characterize automorphism groups of graphs with $\chi_D(G) = 2$ and discuss similar results for graphs with $\chi_D(G)=3$.

scholarly journals The Chromatic Number of Finite Group Cayley Tables

10.37236/7874 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Luis Goddyn ◽  
Kevin Halasz ◽  
E. S. Mahmoodian
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Chromatic Number ◽  
Finite Groups ◽  
Upper Bound ◽  
Latin Square ◽  
Finite Abelian Groups ◽  
Cayley Table ◽  
Minimum Number ◽  
Cayley Tables

The chromatic number of a latin square $L$, denoted $\chi(L)$, is the minimum number of partial transversals needed to cover all of its cells. It has been conjectured that every latin square satisfies $\chi(L) \leq |L|+2$. If true, this would resolve a longstanding conjecture—commonly attributed to Brualdi—that every latin square has a partial transversal of size $|L|-1$. Restricting our attention to Cayley tables of finite groups, we prove two results. First, we resolve the chromatic number question for Cayley tables of finite Abelian groups: the Cayley table of an Abelian group $G$ has chromatic number $|G|$ or $|G|+2$, with the latter case occurring if and only if $G$ has nontrivial cyclic Sylow 2-subgroups. Second, we give an upper bound for the chromatic number of Cayley tables of arbitrary finite groups. For $|G|\geq 3$, this improves the best-known general upper bound from $2|G|$ to $\frac{3}{2}|G|$, while yielding an even stronger result in infinitely many cases.

Conflict-Free Colourings of Uniform Hypergraphs With Few Edges

2012 ◽  
Vol 21 (4) ◽  
pp. 611-622 ◽  
Author(s):  
A. KOSTOCHKA ◽  
M. KUMBHAT ◽  
T. ŁUCZAK
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Upper Bounds ◽  
Uniform Hypergraph ◽  
Lower And Upper Bounds ◽  
Uniform Hypergraphs ◽  
Minimum Number

A colouring of the vertices of a hypergraph is called conflict-free if each edge e of contains a vertex whose colour does not repeat in e. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of , and is denoted by χCF(). Pach and Tardos proved that for an (2r − 1)-uniform hypergraph with m edges, χCF() is at most of the order of rm1/r log m, for fixed r and large m. They also raised the question whether a similar upper bound holds for r-uniform hypergraphs. In this paper we show that this is not necessarily the case. Furthermore, we provide lower and upper bounds on the minimum number of edges of an r-uniform simple hypergraph that is not conflict-free k-colourable.

scholarly journals On automorphism groups of low complexity subshifts

2015 ◽  
Vol 36 (1) ◽  
pp. 64-95 ◽  
Author(s):  
SEBASTIÁN DONOSO ◽  
FABIEN DURAND ◽  
ALEJANDRO MAASS ◽  
SAMUEL PETITE
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Polynomial Complexity ◽  
Automorphism Groups ◽  
Low Complexity ◽  
Topological Dynamics ◽  
Main Technique ◽  
Shift Map ◽  
The One ◽  
A Finite Group

In this article, we study the automorphism group$\text{Aut}(X,{\it\sigma})$of subshifts$(X,{\it\sigma})$of low word complexity. In particular, we prove that$\text{Aut}(X,{\it\sigma})$is virtually$\mathbb{Z}$for aperiodic minimal subshifts and certain transitive subshifts with non-superlinear complexity. More precisely, the quotient of this group relative to the one generated by the shift map is a finite group. In addition, we show that any finite group can be obtained in this way. The class considered includes minimal subshifts induced by substitutions, linearly recurrent subshifts and even some subshifts which simultaneously exhibit non-superlinear and superpolynomial complexity along different subsequences. The main technique in this article relies on the study of classical relations among points used in topological dynamics, in particular, asymptotic pairs. Various examples that illustrate the technique developed in this article are provided. In particular, we prove that the group of automorphisms of a$d$-step nilsystem is nilpotent of order$d$and from there we produce minimal subshifts of arbitrarily large polynomial complexity whose automorphism groups are also virtually$\mathbb{Z}$.

The Coherence Number of 2-Groups

1990 ◽  
Vol 33 (4) ◽  
pp. 503-508 ◽  
Author(s):  
James McCool
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Dihedral Group ◽  
Upper Bounds ◽  
Distinguished Element ◽  
A Finite Group

AbstractLet G be a finite group. A natural invariant c(G) of G has been defined by W.J. Ralph, as the order (possibly infinite) of a distinguished element of a certain abelian group associated to G. Ralph has shown that c(Zn) = 1 and c(Z2 ⴲ Z2) = 2. In the present paper we show that c(G) is finite whenever G is a dihedral group or a 2-group, and obtain upper bounds for c(G) in these cases.

Diameters and automorphism groups of inclusion graphs over nilpotent groups

2019 ◽  
Vol 19 (05) ◽  
pp. 2050097
Author(s):  
Shikun Ou ◽  
Dein Wong ◽  
Zhijun Wang
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Automorphism Groups ◽  
Undirected Graph ◽  
Nilpotent Groups ◽  
Dominating Sets ◽  
A Finite Group

The inclusion graph of a finite group [Formula: see text], written as [Formula: see text], is defined to be an undirected graph that its vertices are all nontrivial subgroups of [Formula: see text], and in which two distinct subgroups [Formula: see text], [Formula: see text] are adjacent if and only if either [Formula: see text] or [Formula: see text]. In this paper, we determine the diameter of [Formula: see text] when [Formula: see text] is nilpotent, and characterize the independent dominating sets as well as the automorphism group of [Formula: see text].

Finite Groups Which are Automorphism Groups of Infinite Groups Only

1985 ◽  
Vol 28 (1) ◽  
pp. 84-90
Author(s):  
Jay Zimmerman
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Field ◽  
Finite Groups ◽  
Finite Simple Group ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Schur Multiplier ◽  
Infinite Groups ◽  
Infinite Set

AbstractThe object of this paper is to exhibit an infinite set of finite semisimple groups H, each of which is the automorphism group of some infinite group, but of no finite group. We begin the construction by choosing a finite simple group S whose outer automorphism group and Schur multiplier possess certain specified properties. The group H is a certain subgroup of Aut S which contains S. For example, most of the PSL's over a non-prime finite field are candidates for S, and in this case, H is generated by all of the inner, diagonal and graph automorphisms of S.

The automorphism group of a finitely generated virtually abelian group

2016 ◽  
Vol 8 (1) ◽  
Author(s):  
Bettina Eick
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Automorphism Groups ◽  
Finitely Generated ◽  
Crystallographic Groups ◽  
Practical Algorithm

AbstractWe describe a practical algorithm to compute the automorphism group of a finitely generated virtually abelian group. As application, we describe the automorphism groups of some small-dimensional crystallographic groups.

Representations of holomorphs of group extensions with Abelian kernels

1975 ◽  
Vol 78 (3) ◽  
pp. 357-368 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Finite Rank ◽  
Automorphism Groups ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Soluble Groups ◽  
Group Extensions

This paper is devoted to the construction of faithful representations of the automorphism group and the holomorph of an extension of an abelian group by some other group, the representations here being homomorphisms into certain restricted parts of the automorphism groups of smallish abelian groups. We apply these to two very specific cases, namely to finitely generated metabelian groups and to certain soluble groups of finite rank. We describe the applications first.

scholarly journals The Distinguishing Chromatic Number of Kneser Graphs

10.37236/3066 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Zhongyuan Che ◽  
Karen L. Collins
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Chromatic Number ◽  
Finite Abelian Group ◽  
Proper Coloring ◽  
Kneser Graph ◽  
Nontrivial Automorphism ◽  
Vertex Set ◽  
Kneser Graphs ◽  
Element Set

A labeling $f: V(G) \rightarrow \{1, 2, \ldots, d\}$ of the vertex set of a graph $G$ is said to be proper $d$-distinguishing if it is a proper coloring of $G$ and any nontrivial automorphism of $G$ maps at least one vertex to a vertex with a different label. The distinguishing chromatic number of $G$, denoted by $\chi_D(G)$, is the minimum $d$ such that $G$ has a proper $d$-distinguishing labeling. Let $\chi(G)$ be the chromatic number of $G$ and $D(G)$ be the distinguishing number of $G$. Clearly, $\chi_D(G) \ge \chi(G)$ and $\chi_D(G) \ge D(G)$. Collins, Hovey and Trenk have given a tight upper bound on $\chi_D(G)-\chi(G)$ in terms of the order of the automorphism group of $G$, improved when the automorphism group of $G$ is a finite abelian group. The Kneser graph $K(n,r)$ is a graph whose vertices are the $r$-subsets of an $n$ element set, and two vertices of $K(n,r)$ are adjacent if their corresponding two $r$-subsets are disjoint. In this paper, we provide a class of graphs $G$, namely Kneser graphs $K(n,r)$, whose automorphism group is the symmetric group, $S_n$, such that $\chi_D(G) - \chi(G) \le 1$. In particular, we prove that $\chi_D(K(n,2))=\chi(K(n,2))+1$ for $n\ge 5$. In addition, we show that $\chi_D(K(n,r))=\chi(K(n,r))$ for $n \ge 2r+1$ and $r\ge 3$.

scholarly journals Regular dessins uniquely determined by a nilpotent automorphism group

2018 ◽  
Vol 21 (3) ◽  
pp. 397-415 ◽  
Author(s):  
Na-Er Wang ◽  
Roman Nedela ◽  
Kan Hu
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Upper Bound ◽  
Automorphism Groups ◽  
Nilpotency Class ◽  
Isomorphism Classes ◽  
One To One

Abstract It is well known that the automorphism group of a regular dessin is a two-generator finite group, and the isomorphism classes of regular dessins with automorphism groups isomorphic to a given finite group G are in one-to-one correspondence with the orbits of the action of {{\mathrm{Aut}}(G)} on the ordered generating pairs of G. If there is only one orbit, then up to isomorphism the regular dessin is uniquely determined by the group G and it is called uniquely regular. In this paper we investigate the classification of uniquely regular dessins with a nilpotent automorphism group. The problem is reduced to the classification of finite maximally automorphic p-groups G, i.e., the order of the automorphism group of G attains Hall’s upper bound. Maximally automorphic p-groups of nilpotency class three are classified.

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