The distinguishing index of the Cartesian product of finite graphs

Abstract

The distinguishing index D (G) of a graph G is the least natural number d such that G has an edge colouring with d colours that is only preserved by the identity automorphism. In this paper we investigate the distinguishing index of the Cartesian product of connected finite graphs. We prove that for every k ≥ 2, the k-th Cartesian power of a connected graph G has distinguishing index equal 2, with the only exception D (K 2 2) = 3. We also prove that if G and H are connected graphs that satisfy the relation 2 ≤ |G| ≤ |H| ≤ 2 |G| 2 G − 1 − |G| + 2, then D (G2H) ≤ 2 unless G2H = K 2 2 .

Edge motion and the distinguishing index

“…Another open problem is a lower bound for the sizes of graphs G and H that implies that D (G × H) ≥ d (or D (G H) ≥ d, respectively) for a given integer d. A corresponding result for D (G2H) was proved in [4].…”

“…As the distinguishing index of the Cartesian powers of graphs has been studied in [4], then higher powers of cycles, and of other graphs, with respect to the direct and strong product can be considered. Note that even cycles are bipartite, and the direct product of bipartite graphs is disconnected (if the number of isomorphic components of a graph G is large enough, then the distinguishing index of G is greater than that of a single component).…”

“…However, this is certainly not the case for edge colourings because there are graphs G of order n with D (G) > D (K n ). For results on the distinguishing index of the Cartesian powers of arbitrary connected graphs we refer the reader to [4]. In this paper, we only investigate the edge motion and its applications.…”

The chromatic distinguishing index of certain graphs

“…We call every C 4 in the book graph B n , a page of B n . The distinguishing index of Cartesian product of star K 1,n with path P m for m 2 and n 2 is D ′ (K 1,n P m ) = ⌈ 2m−1 √ n⌉, unless m = 2 and n = r 3 for some integer r. In the latter case D ′ (K 1,n P 2 ) = 3 √ n + 1, ( [6]). Since B n = K 1,n ✷P 2 , using this equality we obtain the distinguishing index of book graph B n .…”

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

“…Imrich and Klavzar in [6], and Gorzkowska et.al. in [2] showed that the distinguishing number and the distinguishing index of the square and higher powers of a connected graph G = K 2 , K 3 with respect to the Cartesian product is 2.…”