DISTINGUISHING INDEX OF THE CARTESIAN
PRODUCT OF FINITE GRAPHS
Aleksandra Gorzkowska, Rafal Kalinowski, Monika Pilśniak
Faculty of Applied Mathematics
AGH University of Science and Technology
e-mail: [email protected], [email protected], [email protected]
The distinguishing index D0 (G) of a graph G is the least number d such that
G has an edge colouring with d colours that is only preserved by the identity
automorphism. The Cartesian product of graphs G and H is a graph denoted
GH whose verte set is V (G) × V (H). Two vertices (g, h) and (g 0 , h0 ) are
adjacent if either g = g 0 and hh0 ∈ E(H), or gg 0 ∈ E(G) and h = h0 .
In the talk 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 the distinguishing index equal
to two with the exception D0 (K22 ) = 3. We also prove that if connected finite
graphs G and H satisfy the relation |G| ≤ |H| ≤ 2|G| (2kGk − 1) − |G| + 2,
then D0 (GH) ≤ 2 unless GH = K22 .
Keywords: edge colourings, distinguishing index, graph automorphisms,
Cartesian producs, Cartesian powers.
AMS Subject Classification: 05C25, 05C80, 03E10.
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