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Chromatic symmetric functions of conjoined graphs

E. Y. J. Qi, D. Q. B. Tang, D. G. L. Wang

TL;DR

This work analyzes chromatic symmetric functions of conjoined graphs formed by joining rooted graphs with paths, focusing on path-, spider-, and chain-conjoined constructions. It employs the composition method to obtain neat positive $e_I$-expansions for several graph families, including KPCs, PKPs, and KKPs, and establishes $e$-positivity for these classes. A key contribution is a unified framework that translates graph-joining operations into explicit symmetric-function identities, enabling systematic positivity proofs and revealing connections to noncommutative symmetric function techniques. The paper also addresses hat-chains, presenting an $e$-positivity conjecture and a recent involution-based confirmation, thereby broadening the landscape of $e$-positive graphs.

Abstract

We introduce path-conjoined graphs defined for two rooted graphs by joining their roots with a path, and investigate the chromatic symmetric functions of its two generalizations: spider-conjoined graphs and chain-conjoined graphs. By using the composition method developed by Zhou and the third author recently, we obtain neat positive $e_I$-expansions for the chromatic symmetric functions of clique-path-cycle graphs, path-clique-path graphs, and clique-clique-path graphs. We pose the $e$-positivity conjecture for hat-chains.

Chromatic symmetric functions of conjoined graphs

TL;DR

This work analyzes chromatic symmetric functions of conjoined graphs formed by joining rooted graphs with paths, focusing on path-, spider-, and chain-conjoined constructions. It employs the composition method to obtain neat positive -expansions for several graph families, including KPCs, PKPs, and KKPs, and establishes -positivity for these classes. A key contribution is a unified framework that translates graph-joining operations into explicit symmetric-function identities, enabling systematic positivity proofs and revealing connections to noncommutative symmetric function techniques. The paper also addresses hat-chains, presenting an -positivity conjecture and a recent involution-based confirmation, thereby broadening the landscape of -positive graphs.

Abstract

We introduce path-conjoined graphs defined for two rooted graphs by joining their roots with a path, and investigate the chromatic symmetric functions of its two generalizations: spider-conjoined graphs and chain-conjoined graphs. By using the composition method developed by Zhou and the third author recently, we obtain neat positive -expansions for the chromatic symmetric functions of clique-path-cycle graphs, path-clique-path graphs, and clique-clique-path graphs. We pose the -positivity conjecture for hat-chains.
Paper Structure (5 sections, 27 theorems, 102 equations, 5 figures)

This paper contains 5 sections, 27 theorems, 102 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Path-conjoined graphs
  4. Spider-conjoined graphs
  5. Chain-conjoined graphs

Key Result

Proposition 2.1

Let $G$ be a graph with a stable set $T$ of order $3$. Denote the edges linking the vertices in $T$ by $e_1$, $e_2$ and $e_3$. For any set $S\subseteq \{1,2,3\}$, denote by $G_S$ the graph with vertex set $V(G)$ and edge set $E(G)\cup\{e_j\colon j\in S\}$. Then

Figures (5)

  • Figure 1: The lollipop $K_m^l$, the tadpole $C_m^l$, and the $3$-spider $S(abc)$.
  • Figure 2: The path-conjoined graph $P^k(G,H)$.
  • Figure 3: The spider-conjoined graph $S_j^{gh}(G,H)$.
  • Figure 4: The chain-conjoined graph $C(G_1, k_1, G_2, k_2, \dots ,G_{l-1}, k_{l-1}, G_l)$.
  • Figure 5: The hat-chain $C^{\tau_0\tau_1\dotsm\tau_l}( K_1,\, C_{m_1},\, C_{m_2},\, \dots,\, C_{m_l},\, K_1)$.

Theorems & Definitions (47)

  • Proposition 2.1: OS14, the triple-deletion properties
  • Proposition 2.2: AWv24, the arithmetic progression property
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • proof
  • Proposition 2.5: SW16
  • Proposition 2.6: Lollipops, Tom23X
  • Proposition 2.7: Tadpoles, WZ24X
  • Proposition 2.8: $3$-Spiders, Zhe22
  • ...and 37 more