Partial Petrial Polynomials of Bouquets

Abstract

Gross, Mansour, and Tucker [European J. Combin., 95 (2021): 103329] introduced the \emph{partial Petrial polynomial} of a ribbon graph $G$, denoted by ${}^{\partial }{\epsilon }_G^{\times }(z)$. For a prime bouquet $B_n$, Yan and Li [Discrete Appl. Math., 375 (2025): 281-289] determined ${}^{\partial }{\epsilon }_{B_n}^{\times }(z)$ when the intersection graph $I(B_n)$ is either the complete graph or a path, and provided an equivalent condition under which the lowest degree of the nonzero coefficient in ${}^{\partial }{\epsilon }_{B_n}^{\times }(z)$ is $1$. In this paper, we determine ${}^{\partial }{\epsilon }_{B_n}^{\times }(z)$ when the intersection graph is a cycle. Moreover, we present a complete characterization of the prime bouquets whose lowest nonzero term in ${}^{\partial }{\epsilon }_{B_n}^{\times }(z)$ is of degree $2$ and determine the partial Petrial polynomial for the prime bouquets. As corollaries, we determine ${}^{\partial }{\epsilon }_{B_n}^{\times }(z)$ when $I(B_n)$ is the complete bipartite and tripartite graph.