Abstract
The distinguishing index $D'(G)$ of a graph $G$ is the least number $k$ such that $G$ has an edge colouring with $k$ colours that is only preserved by the trivial automorphism. Pilśniak proved that a connected, claw-free graph has the distingushing index at most three. In this paper, we show that the distingushing index of a connected, claw-free graph with at least six vertices is bounded from above by two. We also consider more general graphs in this sense. Namely, we prove that if $G$ is a connected, $K_{1,s}$-free graph of order at least six, then $D'(G) \leq s-1$.
Highlights
The notation follows the standard terminology as in [2]
We say that an edge colouring c breaks an automorphism φ of the graph G if there exists an edge such that its colour is different from the colour of its image under the automorphism’s action
We say that the automorphism φ preserves the colouring c
Summary
The notation follows the standard terminology as in [2]. We consider edge colourings, not necessarily proper, of simple, connected graphs. Kalinowski and Pilsniak in their paper proved a general upper bound on the distinguishing index of connected graphs. They proved that if G is a connected graph with at least three vertices and maximal degree ∆(G), D (G) ∆(G), with the exception of three cycles C3, C4 and C5. They proved that this bound is best possible in a general case. The goal of our research was to obtain a sharp bound for the distinguishing index of claw-free graphs, proving the conjecture posed by Pilsniak.
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