Partial Petrial polynomials for complete graphs and paths

TL;DR

This paper investigates the partial Petrial polynomial for specific classes of ribbon graphs, establishing its dependence on intersection graphs, characterizing it for complete graphs, and computing it for paths, advancing understanding in topological graph invariants.

Contribution

It introduces the dependence of the partial Petrial polynomial on intersection graphs of bouquets and characterizes it for complete graphs and paths.

Findings

Partial Petrial polynomial depends on the intersection graph of bouquets.

Complete graphs have non-zero coefficients for all degrees in their polynomial.

Explicit partial Petrial polynomials are computed for paths.

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.