Abstract The adjacent vertex-distinguishing total chromatic number of a graph 𝐺, denoted by χ𝑎𝑡 (𝐺), is the smallest denoted by 𝑘 for which 𝐺 has a proper total 𝑘-coloring such that any two ...

For vertex-reinforced random walks, we extend results previously obtained on ℤ by Volkov [Ann. Probab. 29 (2001) 66–91] and by Basdevant, Schapira and Singh [Ann. Probab. 42 (2014) 527–558], and on ...