The distinguishing number of Cartesian products of complete graphs

Imrich, Wilfried; Jerebic, Janja; Klavar, Sandi

doi:10.1016/j.ejc.2007.11.018

Cited by 58 publications

(49 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For a 2-connected planar graph G, the distinguishing index may attain 1 + ∆(G) as it is shown by the complete bipartite graph K 2,q with q = r 2 for a positive integer r. In this case, D (K 2,q ) = r + 1 as it follows from the result obtained independently by Fisher and Isaak [11] and by Imrich, Jerebic and Klavžar [14]. They proved the following theorem.…”

Section: -Connected Graphsmentioning

confidence: 50%

Improving upper bounds for the distinguishing index

Pilśniak

2017

Ars Math. Contemp.

View full text Add to dashboard Cite

The distinguishing index of a graph G, denoted by D (G), is the least number of colours in an edge colouring of G not preserved by any non-trivial automorphism. We characterize all connected graphs G with D (G) ≥ ∆(G). We show that D (G) ≤ 2 if G is a traceable graph of order at least seven, and D (G) ≤ 3 if G is either claw-free or 3-connected and planar. We also investigate the Nordhaus-Gaddum type relation: 2 ≤ D (G) + D (G) ≤ max{∆(G), ∆(G)} + 2 and we confirm it for some classes of graphs.

show abstract

Section: -Connected Graphsmentioning

confidence: 50%

Improving upper bounds for the distinguishing index

Pilśniak

2017

Ars Math. Contemp.

View full text Add to dashboard Cite

show abstract

“…In fact, the rows of A are distinct, and since the number of label t + i − 1(mod s) in the ith column is one and in the jth column, j = i, is zero, so the columns have distinct degrees. Hence χ ′ D (K s,t ) = s. Before we prove the next result, we need the following preliminaries: By the result obtained by Fisher and Isaak [4] and independently by Imrich, Jerebic and Klavžar [7] the distinguishing index of complete bipartite graphs is as follows.…”

Section: The Distinguishing Chromatic Index Of Certain Graphsmentioning

confidence: 98%

The chromatic distinguishing index of certain graphs

Alikhani

Soltani

2020

AKCE International Journal of Graphs and Combinatorics

View full text Add to dashboard Cite

The distinguishing index of a graph G, denoted by D ′ (G), is the least number of labels in an edge labeling of G not preserved by any non-trivial automorphism. The distinguishing chromatic index χ ′ D (G) of a graph G is the least number d such that G has a proper edge labeling with d labels that is preserved only by the identity automorphism of G. In this paper we compute the distinguishing chromatic index for some specific graphs. Also we study the distinguishing chromatic index of corona product and join of two graphs.

show abstract

“…In general, the structure of Cartesian products makes matrices particularly useful for studying questions involving their automorphisms. See [7,16], and [13] for further examples of matrix techniques used to find determining sets or distinguishing labelings. …”

Section: Characteristic Matricesmentioning

confidence: 99%

The determining number of a Cartesian product

Boutin

2009

Journal of Graph Theory

View full text Add to dashboard Cite

A set S of vertices is a determining set for a graph G if every automorphism of G is uniquely determined by its action on S. The determining number of G, denoted Det(G), is the size of a smallest determining set. This paper begins by proving that ifm is the prime factor decomposition of a connected graph then Det(G) = max{Det(G k i i )}. It then provides upper and lower bounds for the determining number of a Cartesian power of a prime connected graph. Further, this paper shows that Det(Q n ) = log 2 n +1 which matches the lower bound, and that Det(K n 3 ) = log 3 (2n+1) +1 which for all n is within one of the upper bound. The paper concludes by proving that if H is prime and connected, Det(H n ) = (log n).᭧

show abstract

The distinguishing number of Cartesian products of complete graphs

Cited by 58 publications

References 13 publications

Improving upper bounds for the distinguishing index

Improving upper bounds for the distinguishing index

The chromatic distinguishing index of certain graphs

The determining number of a Cartesian product

Contact Info

Product

Resources

About