Distinguishing labellings of group action on vector spaces and graphs

We introduce the total distinguishing number D (G) of a graph G as the least number d such that G has a total colouring (not necessarily proper) with d colours that is only preserved by the trivial automorphism. This is an analog to the notion of the distinguishing number D(G), and the distinguishing index D (G), which are defined for colourings of vertices and edges, respectively. We obtain a general sharp upper bound:We also introduce the total distinguishing chromatic number χ D (G) similarly defined for proper total colourings of a graph G. We prove that χ D (G) ≤ χ (G) + 1 for every connected graph G with the total chromatic number χ (G). Moreover, if χ (G) ≥ ∆(G)+ 2, then χ D (G) = χ (G). We prove these results by setting sharp upper bounds for the minimal number of colours in a proper total colouring such that each vertex has a distinct set of colour walks emanating from it.

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“…This compares with a more general result of Collins and Trenk [3], and of Klavžar, Wong and Zhu [7]. Theorem 1.1.…”

Section: Introduction and Definitionssupporting

confidence: 66%

“…Theorem 1.1. [3], [7] If G is a connected graph with maximum degree ∆, then D(G) ≤ ∆ + 1. Furthermore, equality holds if and only if G is a K n , K p,p or C 5 .…”

Section: Introduction and Definitionsmentioning

confidence: 99%

Distinguishing graphs by total colourings

2015

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“…It is also shown in [16] that D(A, X) = 3 in all these remaining cases except (q, n) = (2, 3), when D(A, X) = 4. It should be noted that motion is not used in either [8] or [16], and that the remaining cases can be handled by computer.…”

Section: Vector Spaces and Groupsmentioning

confidence: 85%

“…With considerably more work, it is shown in [16] that D(A, X) = 2 for all the other cases except those where (q, n) = (3, 2), or q = 2 and n ≤ 4. It is also shown in [16] that D(A, X) = 3 in all these remaining cases except (q, n) = (2, 3), when D(A, X) = 4.…”

Section: Vector Spaces and Groupsmentioning

confidence: 99%

Motion and distinguishing number two

Conder

Tucker

2011

Ars Math. Contemp.

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A group A acting faithfully on a finite set X is said to have distinguishing number two if there is a proper subset Y whose (setwise) stabilizer is trivial. The motion of A acting on X is defined as the largest integer k such that all non-trivial elements of A move at least k elements of X. The Motion Lemma of Russell and Sundaram states that if the motion is at least 2 log 2 |A|, then the action has distinguishing number two. When X is a vector space, group, or map, the Motion Lemma and elementary estimates of the motion together show that in all but finitely many cases, the action of Aut(X) on X has distinguishing number two. A new lower bound for the motion of any transitive action gives similar results for transitive actions with restricted point-stabilizers. As an instance of what can happen with intransitive actions, it is shown that if X is a set of points on a closed surface of genus g, and |X| is sufficiently large with respect to g, then any action on X by a finite group of surface homeomorphisms has distinguishing number two.

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“…Over the past few years, there has been a great flurry of activity on distinguishing, see for example [1,3,4,5,6,7,8,9,13,14,15,16].…”

Section: φ(V) ∈ E(h) If and Only If Uv ∈ E(g)mentioning

confidence: 99%

Distinguishing geometric graphs

Albertson

Boutin

2006

Journal of Graph Theory

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Abstract:We begin the study of distinguishing geometric graphs. Let G be a geometric graph. An automorphism of the underlying graph that preserves both crossings and noncrossings is called a geometric automorphism. A labeling, f : V(G) → {1, 2, . . . , r}, is said to be r-distinguishing if no nontrivial geometric automorphism preserves the labels. The distinguishing number of G is the minimum r such that G has an r-distinguishing labeling. We show that when K n is not the nonconvex K 4 , it can be 3-distinguished. Furthermore, when n ≥ 6, there is a K n that can be 1-distinguished. For n ≥ 4, K 2,n can realize any distinguishing number between 1 and n inclusive. Finally, we show that every K 3,3 can be 2-distinguished. We also offer several open questions.

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Distinguishing labellings of group action on vector spaces and graphs

Cited by 60 publications

References 9 publications

Distinguishing graphs by total colourings

Distinguishing graphs by total colourings

Motion and distinguishing number two

Distinguishing geometric graphs

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