On $m$-Closed Graphs

TL;DR

This paper introduces and studies the concept of m-closed graphs, generalizing closed graphs, and provides classifications and primary decompositions for specific cases like 3-closed trees.

Contribution

It generalizes the notion of closed graphs to m-closed graphs and classifies 3-closed trees, offering new insights into their algebraic and combinatorial properties.

Findings

Equivalent condition for 3-closed property of trees

Classification of a specific class of 3-closed trees

Analysis of primary decomposition for these graphs

Abstract

A graph is closed when its vertices have a labeling by $[n]$ such that the binomial edge ideal $J_G$ has a quadratic Gr\"{o}bner basis with respect to the lexicographic order induced by $x_1 > \cdots > x_n > y_1> \cdots > y_n$. In this paper, we generalize this notion and study the so called $m-$closed graphs. We find equivalent condition to $3-$closed property of an arbitrary tree $T$. Using it, we classify a class of $3-$closed trees. The primary decomposition of this class of graphs is also studied.