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Classification of unmixed parity binomial edge ideals of cactus and chordal graphs

Deblina Dey, A. V. Jayanthan, Sarang Sane

Abstract

In this article, we characterize all unmixed and Cohen-Macaulay parity binomial edge ideals of cactus and chordal graphs in terms of the structural properties of the graph.

Classification of unmixed parity binomial edge ideals of cactus and chordal graphs

Abstract

In this article, we characterize all unmixed and Cohen-Macaulay parity binomial edge ideals of cactus and chordal graphs in terms of the structural properties of the graph.
Paper Structure (9 sections, 47 theorems, 24 equations, 8 figures)

This paper contains 9 sections, 47 theorems, 24 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Disconnector sets and unmixedness
  4. Cohen-Macaulay parity binomial edge ideal of cactus graphs
  5. The algorithm
  6. Algorithm:
  7. Conditions when non-bipartite chordal graphs are not unmixed
  8. Classification of unmixed chordal graphs
  9. Cohen-Macaulay parity binomial edge ideals

Key Result

Proposition 2.7

Let $S$ be a disconnector set of $G$ such that every element of $S$ is adjacent to vertices of a fixed non-bipartite connected component of $G\setminus S$. Then $S$ satisfies the sign-split property.

Figures (8)

  • Figure 1: Unmixed chordal graph classes: $\mathfrak{G}_1, \mathfrak{G_2}$ and $\mathfrak{G_3}$, where $P_1, P_2, P_3$ are path graphs of length at least one
  • Figure 2: A cactus graph. Note that $(C_1, v_1)$ and $(C_3, v_3)$ are pendant odd cycles of $G$ but $C_2$ is not.
  • Figure 3: The smallest graph $G$ in the class $\mathfrak{G}_2$
  • Figure 4: Step $0$: $H_0$
  • Figure 5: Step $1$: $H_1$
  • ...and 3 more figures

Theorems & Definitions (113)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Proposition 2.8
  • proof
  • Remark 2.9
  • ...and 103 more