On toric ideals arising from signed graphs

• Published in: Journal of algebraic combinatorics Vol. 53; no. 4; pp. 1265 - 1298
• Main Authors: Huh, JiSun, Kim, Sangwook, Park, Boram
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-06-2021; Springer Nature B.V
• Subjects: Binomials; Combinatorics; Computer Science; Convex and Discrete Geometry; Graph theory; Graphs; Group Theory and Generalizations; Lattices; Mathematics; Mathematics and Statistics; Order; Ordered Algebraic Structures; Signed graph; 05C22; Toric ideal; Complete intersection; 14M25; Primitive element; 05C25
• ISSN: 0925-9899; 1572-9192
• DOI: 10.1007/s10801-020-00962-3

A
signed graph
is a pair
(
G
,
τ
)
of a graph
G
and its sign
τ
, where a
sign
τ
is a function from
{
(
e
,
v
)
∣
e
∈
E
(
G
)
,
v
∈
V
(
G
)
,
v
∈
e
}
to
{
1
,
-
1
}
. Note that graphs or digraphs are special cases of signed graphs. In this paper, we study the toric ideal
I
(
G
,
τ
)
associated with a signed graph
(
G
,
τ
)
, and the results of the paper give a unified idea to explain some known results on the toric ideals of a graph or a digraph. We characterize all primitive binomials of
I
(
G
,
τ
)
and then focus on the complete intersection property. More precisely, we find a complete list of graphs
G
such that
I
(
G
,
τ
)
is a complete intersection for every sign
τ
.