Explicit Formulas and Unimodality Phenomena for General Position Polynomials

TL;DR

This paper investigates the properties of the general position polynomial in graphs, providing explicit formulas for certain classes, analyzing unimodality and log-concavity, and exploring conditions under which these properties hold or fail.

Contribution

It offers explicit formulas for general position polynomials of complete multipartite graphs and analyzes their unimodality and log-concavity, advancing understanding of these properties in graph polynomials.

Findings

For balanced complete multipartite graphs with part size r ≤ 4, the polynomial is log-concave and unimodal.

Counterexamples show that for larger r, unimodality and log-concavity can fail.

Unimodality is retained for coronas G∘K₁ in several natural graph classes.

Abstract

The general position problem in graphs seeks the largest set of vertices such that no three vertices lie on a common geodesic. Its counting refinement, the general position polynomial $\psi(G)$, asks for all such possible sets. In this paper, We describe general position sets for several classes of graphs and provide explicit formulas for the general position polynomials of complete multipartite graphs. We specialize to balanced complete multipartite graphs and show that for part size $r\le 4$, the polynomial $\psi(K_{r,\dots,r})$ is log-concave and unimodal for all numbers of parts, while for larger $r$, counterexamples show that these properties fail. Finally, we analyze the corona $G\circ K_1$ and prove that unimodality of $\psi(G)$ is retained for numerous natural classes (paths, edgeless graphs, combs). This contributes to an open problem, but the general case remains unknown. Our findings support the parallel between general position polynomials and classical position-type parameters, and identify balanced multipartite graphs and coronas as promising testbeds for additional research.