Positive $e$-expansions of the chromatic symmetric functions of KPKPs, twinned lollipops, and kayak paddles

Abstract: We find a positive $e_I$-expansion for the chromatic symmetric function of KPKP graphs, which are graphs obtained by connecting a vertex in a complete graph with a vertex in the maximal clique of a lollipop graph by a path. This generalizes the positive $e_I$-expansion for the chromatic symmetric function of lollipops obtained by Tom, for that of KPK graphs obtained by Wang and Zhou, and as well for those of KKP graphs and PKP graphs obtained by Qi, Tang and Wang. As an application, we confirm the $e$-positivity of twinned lollipops. We also discover the first positive $e_I$-expansion for the chromatic symmetric function of kayak paddle graphs which are formed by connecting a vertex on a cycle and a vertex on another cycle with a path. This refines the $e$-positivity of kayak paddle graphs which was obtained by Aliniaeifard, Wang, and van Willigenburg.