On distinguishing coloring and some variants of proper coloring of graphs derived from subdivision operations

Abstract: Let G be a simple, finite, connected, and undirected graph, and T be a finite tree. The middle graph M(G) of G is obtained from the subdivision graph S(G) after joining pairs of subdivided vertices that lie on adjacent edges of G and the central graph C(G) of G is obtained from S(G) after joining all non-adjacent vertices of G. We show that if the order of G is at least 4, then Aut(G), Aut(C(G)), and Aut(M(G)) are isomorphic (as abstract groups) and apply these results to obtain new sharp upper bounds of the distinguishing number and the distinguishing index of C(G) and M(G) inspired by an algorithm due to Kalinowski, Pilsniak, and Wozniak from 2016. Furthermore, we study the total distinguishing chromatic number of C(G) and S(G), use Latin squares to verify the AVD-total coloring conjecture for central graphs of regular graphs and some other classes of graphs (which is a partial progress towards answering an open question of Panda, Verma, and Keerti from 2020), and obtain new bounds of the total dominator chromatic number of C(G) and C(T).