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 Dvořák 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 PDP (G,m), counts the minimum number of DP-colorings over all possible m-fold covers. Formulas for chromatic polynomials of clique-gluings of graphs 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 Kp-gluings of graphs. Recently, Becker et. al. asked whether PDP (G,m) ≤ ( ∏n i=1 PDP (Gi,m))/ ( ∏p−1 i=0 (m− i) )n−1 whenever m ≥ p, where the expression on the right is the DP-coloring analogue of the corresponding chromatic polynomial formula for a Kp-gluing of G1, . . . , Gn. 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.

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