The Distinguishing Index of Infinite Graphs

  • Izak Broere
  • Monika Pilśniak
Keywords: distinguishing index, automorphism, infinite graph, countable graph, edge colouring, Infinite Motion Lemma


The  distinguishing index $D^\prime(G)$ of a graph $G$ is the least cardinal $d$ such that $G$ has an edge colouring with $d$ colours that is only preserved by the trivial automorphism. This is similar to the notion of the distinguishing number $D(G)$ of a graph $G$, which is defined with respect to vertex colourings.

We derive several bounds for infinite graphs, in particular, we prove the general bound $D^\prime(G)\leq\Delta(G)$ for an arbitrary infinite graph. Nonetheless,  the distinguishing index is at most two for many countable graphs, also for the infinite random graph and for uncountable tree-like graphs.

We also investigate the concept of the motion of edges and its relationship with the Infinite Motion Lemma.


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