Cohen-Macaulayness of triangular graphs

TL;DR

This paper investigates which triangular graphs possess the Cohen-Macaulay property, providing a complete classification for certain small and large values of n, with results dependent on the field's characteristic.

Contribution

The paper determines the Cohen-Macaulayness of triangular graphs $T_n$ for specific values of n, including characteristic-dependent results for $T_7$ and $T_9$, advancing understanding in algebraic graph theory.

Findings

$T_2$, $T_3$, and $T_5$ are Cohen-Macaulay.

$T_4$, $T_6$, $T_8$, and $T_n$ for $n extgreater=10$ are not Cohen-Macaulay.

Over characteristic zero fields, $T_7$ and $T_9$ are Cohen-Macaulay.

Abstract

We study the Cohen-Macaulay property of triangular graphs $T_n$. We show that $T_2$, $T_3$ and $T_5$ are Cohen-Macaulay graphs, and that $T_4$, $T_6$, $T_8$ and $T_n$ are not Cohen-Macaulay graphs, for $n\geq 10$. Finally, we prove that over fields of characteristic zero $T_7$ and $T_9$ are Cohen-Macaulay.