Down-left graphs and a connection to toric ideals of graphs

Abstract

We introduce a family of graphs, which we call down-left graphs, and study their combinatorial and algebraic properties. We show that members of this family are well-covered, $C_5$-free, and vertex decomposable. By applying a result of Hà-Woodroofe and Moradi--Khosh-Ahang, the (Castelnuovo-Mumford) regularity of the associated edge ideals is the induced matching number of the graph. As an application, we give a combinatorial interpretation for the regularity of the toric ideals of chordal bipartite graphs that are $(K_{3,3} \setminus e)$-free.

Figures (9)

• Figure 1: The down-left graph $G(3,4)$
• Figure 2: The graphs $C_5$ and $K_{3,3} \setminus e$
• Figure 3: The graph $G(5,6,\vec{a},\vec{b})$ with $\vec{a} = (0,0,1,2,2)$ and $\vec{b} = (5,5,6,6,7)$. The black vertices belong to $G(5,6)$, but not $G(5,6,\vec{a},\vec{b})$. The graph $G(5,6,\vec{a},\vec{b})^\circ$ is obtained by removing all the isolated vertices.
• Figure 4: Three possible ways to place three vertices without an induced triangle. Grey areas represent places that are adjacent to either $v_1$ or $v_2$.
• Figure 5: A chordal bipartite graph

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2: Theorem \ref{['thm:reg-downleft']}
• Definition 2.1
• Remark 2.2
• Lemma 2.3: W
• Definition 2.4
• Theorem 2.5: MV
• Theorem 2.6: K
• Theorem 2.7
• Theorem 2.8: CV