Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Simon Y. M. Gong, David G. L. Wang, K. Zhang

TL;DR

The paper addresses the problem of identifying $e$-positive graphs within the Stanley--Stembridge framework by focusing on trinacria graphs $T_{abc}$ and proving $e$-positivity for the family $T_{(b+2)b2}$. The authors employ the composition method, decomposing $X_{T_{(b+2)b2}}$ into $X_G=Y_2e_1^2+Y_1e_1+Y_0$ and proving $e$-positivity of each $Y_i$ via tailored combinatorial techniques: a positive $e_I$-expansion for $Y_2$ using a term-pairing bijection; a charging argument for $Y_1$ with a structured partition of index sets; and a progressive repair scheme for $Y_0$ based on an involution on a negative set. These results yield a concrete, constructive verification of $e$-positivity for this trinacria family and endorse the sharpness of earlier threshold results by MS25. The work broadens the catalog of $e$-positive graphs beyond unit interval graphs and illustrates the utility of the composition method in complex graph families.

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.

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

TL;DR

The paper addresses the problem of identifying -positive graphs within the Stanley--Stembridge framework by focusing on trinacria graphs and proving -positivity for the family . The authors employ the composition method, decomposing into and proving -positivity of each via tailored combinatorial techniques: a positive -expansion for using a term-pairing bijection; a charging argument for with a structured partition of index sets; and a progressive repair scheme for based on an involution on a negative set. These results yield a concrete, constructive verification of -positivity for this trinacria family and endorse the sharpness of earlier threshold results by MS25. The work broadens the catalog of -positive graphs beyond unit interval graphs and illustrates the utility of the composition method in complex graph families.

Abstract

In this paper, we identify a new family of -positive graphs, called the trinacria graphs , thereby providing a partial answer to Stanley's question on which graphs are -positive. The trinacria graph is the graph on vertices obtained by attaching paths , and~ 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.
Paper Structure (6 sections, 23 theorems, 80 equations, 1 figure, 1 table)

This paper contains 6 sections, 23 theorems, 80 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the $e$-positivity of $T_{(b+2)b2}$
  4. $Y_2$ is $e$-positive
  5. $Y_1$ is $e$-positive
  6. $Y_0$ is $e$-positive

Key Result

Theorem 1.1

Every trinacria graph of the form $T_{(b+2)b2}$ is $e$-positive.

Figures (1)

  • Figure 1: The trinacria graph $T_{abc}$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Proposition 2.1: OS14
  • Proposition 2.2: SW16
  • Proposition 2.3: Zhe22
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 33 more