The Distinguishing Index of Infinite Graphs

Izak Broere, Monika Pilśniak

Abstract


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.

 


Keywords


distinguishing index, automorphism, infinite graph, countable graph, edge colouring, Infinite Motion Lemma

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