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Scarf complexes of graphs and their powers

Sara Faridi, Tài Huy Hà, Takayuki Hibi, Susan Morey

Abstract

Every multigraded free resolution of a monomial ideal I contains the Scarf multidegrees of I. We say I has a Scarf resolution if the Scarf multidegrees are sufficient to describe a minimal free resolution of I. The main question of this paper is which graphs G have edge ideal I(G) with a Scarf resolution? We show that I(G) has a Scarf resolution if and only if G is a gap-free forest. We also classify connected graphs for which all powers of I(G) have Scarf resolutions. Along the way, we give a concrete description of the Scarf complex of any forest. For a general graph, we give a recursive construction for its Scarf complex based on Scarf complexes of induced subgraphs.

Scarf complexes of graphs and their powers

Abstract

Every multigraded free resolution of a monomial ideal I contains the Scarf multidegrees of I. We say I has a Scarf resolution if the Scarf multidegrees are sufficient to describe a minimal free resolution of I. The main question of this paper is which graphs G have edge ideal I(G) with a Scarf resolution? We show that I(G) has a Scarf resolution if and only if G is a gap-free forest. We also classify connected graphs for which all powers of I(G) have Scarf resolutions. Along the way, we give a concrete description of the Scarf complex of any forest. For a general graph, we give a recursive construction for its Scarf complex based on Scarf complexes of induced subgraphs.
Paper Structure (8 sections, 21 theorems, 53 equations, 3 figures)

This paper contains 8 sections, 21 theorems, 53 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Simplicial Resolutions
  3. The Scarf complex of edge ideals of graphs
  4. The Scarf complex of a subgraph
  5. The Scarf complex of an induced subgraph
  6. The Scarf complex of a forest
  7. Graphs which are Scarf
  8. The Scarf complex of powers of edge ideals

Key Result

Theorem 1

Let $G$ be a graph with edge ideal $I = I(G)$. If $G$ is connected and $t>1$, then

Figures (3)

  • Figure 1: The Scarf complex of powers of a path $I=(ab,bc,cd)$
  • Figure 2: $1$-Skeleton of the Scarf complex of powers of a claw $I=(ab,ac,ad)$
  • Figure 3: The Scarf complex of powers of a square $I=(ab,bc,cd,da)$

Theorems & Definitions (54)

  • Theorem : \ref{['t:beautiful']}: The "Beautiful Oberwolfach Theorem"
  • Example 2.1
  • Theorem 2.2: Supporting a Free Resolution BPS
  • Example 2.3
  • Definition 2.4: Scarf Ideals
  • Lemma 2.5
  • proof
  • Definition 3.1: Edge Ideal of Graphs
  • Example 3.2
  • Example 3.3
  • ...and 44 more