Strongly nice property and Schur positivity of graphs

TL;DR

This paper introduces the strongly nice property for graphs, linking it to Schur positivity of symmetric functions, and proves that claw-free graphs are exactly those with all induced subgraphs strongly nice, also resolving a conjecture on squid graphs.

Contribution

It defines the strongly nice property, characterizes claw-free graphs via this property, and applies it to resolve a conjecture on the Schur positivity of specific squid graphs.

Findings

Claw-free graphs are exactly those with all induced subgraphs strongly nice.

Squid graphs $Sq(2n-1;1^n)$ are not strongly nice for $n geq 3$.

The strongly nice property provides a new tool for studying Schur positivity of graphs.

Abstract

Motivated by the notion of nice graphs, we introduce the concept of strongly nice property, which can be used to study the Schur positivity of symmetric functions. We show that a graph and all its induced subgraphs are strongly nice if and only if it is claw-free, which strengthens a result of Stanley and provides further evidence for the well-known conjecture on the Schur positivity of claw-free graphs. As another application, we solve Wang and Wang's conjecture on the non-Schur positivity of squid graphs $Sq(2n-1;1^n)$ for $n \ge 3$ by proving that these graphs are not strongly nice.