The DP color function of clique-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 Dvok and Postle in 2015.As the analogue of the chromatic polynomial of a graph G, P (G, m), the DP color function of G, denoted by P DP (G, m), counts the minimum number of DP-colorings over all possible m-fold covers.Formulas for chromatic polynomials of cliquegluings of graphs, a fundamental graph operation, are well-known, but the effect of such gluings on the DP color function is not well understood.In this paper, we study the DP color function of K p -gluings of graphs.Recently, Becker et.al. asked whether P DP (G, m) (whenever m p, where the expression on the right is the DP-coloring analogue of the corresponding chromatic polynomial formula for a K p -gluing, G, of G 1 , . . ., G n .Becker et.al. showed this inequality holds when p = 1.In this paper, we show this inequality holds for edge-gluings (p = 2).On the other hand, we show it does not hold for triangle-gluings (p = 3), which also answers a question of Dong and Yang (2021).Finally, we show a relaxed version, based on a class of m-fold covers that we conjecture would yield the fewest DP-colorings for a given graph, of the inequality holds when p 3.