“Distinguishing colorings and determining sets for graphs” by Lauren Keough PhD, Grand Valley State University
 
The distinguishing number of a graph G is the fewest number of colors needed so that the only color-preserving automorphism is trivial. A determining set is a subset of the vertices such that any automorphism that fixes these vertices pointwise is trivial. We will discuss determining and distinguishing numbers for some families of hypercubes and for the (generalized) Mycielskian of a graph.
  
The Mycielskian of a graph G, μ(G), is constructed by adding a shadow vertex ui for each vertex vi in G, one additional vertex w, and edges so that N(ui) = N(vi)∪{w}.  In 2019, Alikhani and Soltani showed that the distinguishing number of μ(G) is at most one more than the distinguishing number of G when G is twin-free and conjecture that the distinguishing number of μ(G),  is at most the distinguishing number of G for most graphs. We prove a stronger version of their conjecture, as well as results for generalized Mycielskians, and determining numbers. 
 
This is joint work with Debra Boutin, Sally Cockburn, Sarah Loeb, Kat Perry, and Puck Rombach.