The distinguishing  number and distinguishing index of  the lexicographic product of two graphs

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 not $k$-distinguishable but in which every ball of finite radius is $k$-distinguishable.In the second half of this paper we show that a large distinguishing number for an imprimitive group $G$ is traceable to a high distinguishing number either of a block of imprimitivity or of the induced action by $G$ on the corresponding system of imprimitivity. An immediate application is to automorphism groups of infinite imprimitive graphs. These results are companion to the study of the distinguishing number of infinite primitive groups and graphs in a previous paper by the authors together with T. W. Tucker.

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Distinguishability of Locally Finite Trees

The Electronic Journal of Combinatorics ◽

10.37236/947 ◽

2007 ◽

Vol 14 (1) ◽

Cited By ~ 11

Author(s):

Mark E. Watkins ◽

Xiangqian Zhou

Keyword(s):

Positive Integer ◽

Chromatic Number ◽

Inverse Image ◽

Vertex Coloring ◽

Finite Tree ◽

Locally Finite ◽

Distinguishing Number ◽

Nonidentity Element ◽

Locally Countable ◽

Proper Vertex

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.

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Symmetry Breaking in Graphs

The Electronic Journal of Combinatorics ◽

10.37236/1242 ◽

1996 ◽

Vol 3 (1) ◽

Cited By ~ 76

Author(s):

Michael O. Albertson ◽

Karen L. Collins

Keyword(s):

Symmetry Breaking ◽

Complete Graph ◽

Subject Classification ◽

Distinguishing Number

A labeling of the vertices of a graph G, $\phi :V(G) \rightarrow \{1,\ldots,r\}$, is said to be $r$-distinguishing provided no automorphism of the graph preserves all of the vertex labels. The distinguishing number of a graph G, denoted by $D(G)$, is the minimum $r$ such that $G$ has an $r$-distinguishing labeling. The distinguishing number of the complete graph on $t$ vertices is $t$. In contrast, we prove (i) given any group $\Gamma$, there is a graph $G$ such that $Aut(G) \cong \Gamma$ and $D(G)= 2$; (ii) $D(G) = O(log(|Aut(G)|))$; (iii) if $Aut(G)$ is abelian, then $D(G) \leq 2$; (iv) if $Aut(G)$ is dihedral, then $D(G) \leq 3$; and (v) If $Aut(G) \cong S_4$, then either $D(G) = 2$ or $D(G) = 4$. Mathematics Subject Classification 05C,20B,20F,68R

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A note on the ordinal power and the lexicographic product of partially ordered sets

Kodai Mathematical Seminar Reports ◽

10.2996/kmj/1138843209 ◽

1952 ◽

Vol 4 (1) ◽

pp. 19-22

Author(s):

Tadashi Ohkuma

Keyword(s):

Partially Ordered Sets ◽

Lexicographic Product ◽

Ordered Sets ◽

Partially Ordered

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Another H-super magic decompositions of the lexicographic product of graphs

Indonesian Journal of Combinatorics ◽

10.19184/ijc.2018.2.2.2 ◽

2018 ◽

Vol 2 (2) ◽

pp. 72

Author(s):

H Hendy ◽

Kiki A. Sugeng ◽

A.N.M Salman ◽

Nisa Ayunda

Keyword(s):

Bipartite Graph ◽

Complete Bipartite Graph ◽

Graph Decomposition ◽

Graph Labeling ◽

Sufficient Condition ◽

Lexicographic Product ◽

Simple Graphs ◽

Product Of Graphs

Let H and G be two simple graphs. The concept of an H-magic decomposition of G arises from the combination between graph decomposition and graph labeling. A decomposition of a graph G into isomorphic copies of a graph H is H-magic if there is a bijection f : V(G) ∪ E(G) → {1, 2, ..., ∣V(G) ∪ E(G)∣} such that the sum of labels of edges and vertices of each copy of H in the decomposition is constant. A lexicographic product of two graphs G1 and G2, denoted by G1[G2], is a graph which arises from G1 by replacing each vertex of G1 by a copy of the G2 and each edge of G1 by all edges of the complete bipartite graph Kn, n where n is the order of G2. In this paper we provide a sufficient condition for $\overline{C_{n}}[\overline{K_{m}}]$ in order to have a $P_{t}[\overline{K_{m}}]$-magic decompositions, where n > 3, m > 1, and t = 3, 4, n − 2.

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The Lexicographic Product of Po-groups and n-Perfect Pseudo Effect Algebras

International Journal of Theoretical Physics ◽

10.1007/s10773-013-1568-5 ◽

2013 ◽

Vol 52 (8) ◽

pp. 2760-2772 ◽

Cited By ~ 7

Author(s):

Anatolij Dvurečenskij ◽

Jan Krňávek

Keyword(s):

Effect Algebras ◽

Lexicographic Product

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