Graphs missing a connected partition

TL;DR

This paper investigates the structural properties of certain graphs with cut vertices and their chromatic symmetric functions, showing conditions under which these graphs lack an $e$-positive partition, advancing understanding of graph colorings.

Contribution

It establishes new conditions under which graphs with specific cut vertices cannot have an $e$-positive chromatic symmetric function, including trees with high-degree vertices and spiders with four legs.

Findings

Graphs with a cut vertex and at least five components lack certain connected partitions.

Such graphs cannot have an $e$-positive chromatic symmetric function.

Spiders with four legs also lack $e$-positivity in their chromatic symmetric functions.

Abstract

We prove that a graph with a cut vertex whose deletion produces at least five connected components must be missing a connected partition of some type. We prove that this also holds if there are four connected components that each have at least two vertices. In particular, the chromatic symmetric function of such a graph cannot be $e$-positive. This brings us very close to the conjecture by Dahlberg, She, and van Willigenburg of non-$e$-positivity for all trees with a vertex of degree at least four. We also prove that spiders with four legs cannot have an $e$-positive chromatic symmetric function.