2008/04/24 by Claude Laflamme, Laflamme, C., Lionel Nguyen Van Thé +3
Computer Science · #03C13 #03C15 #05C25 #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.0804.4019
openalex publication_date 2008/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The distinguishing number of a graph G is the smallest positive integer r such that G has a labeling of its vertices with r labels for which there is no non-trivial automorphism of G preserving these labels. Albertson and Collins computed the distinguishing number for various finite graphs, and Imrich, Klavžar and Trofimov computed the distinguishing number of some infinite graphs, showing in particular that the Random Graph has distinguishing number 2. We compute the distinguishing number of various other finite and countable homogeneous structures, including undirected and directed graphs, and posets. We show that this number is in most cases two or infinite, and besides a few exceptions conjecture that this is so for all primitive homogeneous countable structures.