The trinacria graphs $T_{(b+2)b2}$ are $e$-positive

TL;DR

This paper introduces a new family of graphs called trinacria graphs and proves they are e-positive, advancing understanding of graph positivity properties with combinatorial methods.

Contribution

The paper identifies and proves the e-positivity of the trinacria graphs, partially answering Stanley's question on e-positive graphs using novel combinatorial techniques.

Findings

Trinacria graphs are e-positive.

The proof employs divide-and-conquer and charging arguments.

Provides partial answer to Stanley's e-positivity question.

Abstract

In this paper, we identify a new family of $e$-positive graphs, called the trinacria graphs $T_{(b+2)b2}$, thereby providing a partial answer to Stanley's question on which graphs are $e$-positive. The trinacria graph $T_{abc}$ is the graph on $a+b+c+3$ vertices obtained by attaching paths $P_a$, $P_b$ and~$P_c$ to the vertices of a triangle, respectively. Our proof relies on several ad hoc combinatorial ideas, and employs divide-and-conquer techniques, charging arguments, and progressive repair methods.