Distinguishing graphs by edge-colourings

Distinguishing graphs by total colourings

“…It was also proved in [5] that D (G) ≤ D(G) + 1 for any connected graph of order n ≥ 3, and this bound is sharp for each n. Actually, quite frequently D (G) < D(G). For a complete graph D (K n ) = 2 for any n ≥ 6, and also for a complete bipartite graph D (K p,p ) = 2 for p ≥ 4, whereas D(K n ) and D(K p,p ) are equal to ∆ + 1.…”

“…Edge colourings breaking automorphisms were investigated by the first two authors in [5]. If a graph G does not contain K 2 as a connected component, then the distinguishing index D (G) of a graph G as the least number d such that G admits an edge colouring with d colours that is only preserved by the trivial automorphism.…”

“…Theorem 1.3. [5] If G is a connected graph of order n ≥ 3 with maximum degree ∆, then D (G) ≤ ∆ unless G is C 3 , C 4 or C 5 .…”

Improving upper bounds for the distinguishing index

“…Kalinowski and Pilśniak [17] also introduced a distinguishing chromatic index χ D (G) of a graph G as the least number of colours in a proper edge colouring that breaks all non-trivial automorphisms of G. They proved the following somewhat unexpected result. The following Nordhaus-Gaddum type inequalities for the distinguishing chromatic index are the same as in Theorem 1.2 but we have to be more careful in the proof.…”

“…Let P and Q be the two sets of bipartition of K p,q with |P | = p and |Q| = q. If p = q, then p ≥ 4, and there exists a spanning asymmetric tree of K p,p (see [17]). If p < q ≤ 2 p − p + 1, then for the proof of Theorem 3.9, Imrich and Klavžar in [15] constructed a distinguishing vertex 2-colouring of K p 2K q that corresponds to a distinguishing edge 2-colouring f of K p,q , where a colouring of vertices in a K q -layer can be represented by a sequence from {1, 2} q and it corresponds to a colouring of edges incident to a vertex in P (the same is true for K p -layers and vertices in Q).…”