The DP color function of joins and vertex-gluings of graphs

DP-coloring (also called correspondence coloring) is a generalization of list coloring that has been widely studied in recent years after its introduction by Dvořák and Postle 2015. As the analogue chromatic polynomial P(G,m), DP color function graph G, denoted PDP(G,m), counts minimum number DP-colorings over all possible m-fold covers. Chromatic polynomials for joins vertex-gluings graphs are well understood, but effect these operations on not known. In this paper we make progress understanding join with complete certain graphs. We also develop tools to study under operations, threshold (smallest m) beyond which constructed equals polynomial.