Partial Petrial polynomials for complete graphs and paths
Qi Yan, Yuancheng Li
TL;DR
The paper studies the partial Petrial polynomial for ribbon graphs, focusing on bouquets and the role of intersection graphs. It shows that for bouquets, the partial Petrial polynomial is governed by the bouquet's intersection graph, and extends this to circle graphs by defining $P_G^{\times}(z)$ via a bouquet $B$ with $G=I(B)$. For complete graphs $K_n$, the authors derive explicit, parity-dependent formulas for $P_{K_n}^{\times}(z)$, proving nonzero coefficients on all degrees $1$ through $n$; for paths, they obtain a two-term closed form, establishing that the path case yields a binomial polynomial. These results illuminate how topological dualities interact with graph structure and invite further characterization of graphs with binomial partial Petrial polynomials.
Abstract
Recently, Gross, Mansour, and Tucker introduced the partial Petrial polynomial, which enumerates all partial Petrials of a ribbon graph by Euler genus. They provided formulas or recursions for various families of ribbon graphs, including ladder ribbon graphs. In this paper, we focus on the partial Petrial polynomial of bouquets, which are ribbon graphs with exactly one vertex. We prove that the partial Petrial polynomial of a bouquet primarily depends on its intersection graph, meaning that two bouquets with identical intersection graphs will have the same partial Petrial polynomial. Additionally, we introduce the concept of the partial Petrial polynomial for circle graphs and prove that for a connected graph with $n$ vertices ($n\geq 2$), the polynomial has non-zero coefficients for all terms of degrees from 1 to $n$ if and only if the graph is complete. Finally, we present the partial Petrial polynomials for paths.