HOMOLOGICAL LINEAR QUOTIENTS AND EDGE IDEALS OF GRAPHS

Abstract

It is well known that the edge ideal $I(G)$ of a simple graph G has linear quotients if and only if $G^c$ is chordal. We investigate when the property of having linear quotients is inherited by homological shift ideals of an edge ideal. We will see that adding a cluster to the graph $G^c$ when $I(G)$ has homological linear quotients results in a graph with the same property. In particular, $I(G)$ has homological linear quotients when $G^c$ is a block graph. We also show that adding pinnacles to trees preserves the property of having homological linear quotients for the edge ideal of their complements. Furthermore, $I(G)$ has homological linear quotients for every graph G such that $G^c$ is a $\lambda $-minimal chordal graph.

1 Introduction

Let $S=K[x_1,\ldots , x_n]$ be the polynomial ring in the variables $x_1,\ldots , x_n$ over a field K with its natural multigrading. Throughout, a monomial and its multidegree will be used interchangeably and $S(\boldsymbol {x}^{\boldsymbol {a}})$ denotes the free S-module with one generator of multidegree $\boldsymbol {x}^{\boldsymbol {a}}$ . A monomial ideal $I\subseteq S$ has a (unique up to isomorphism) minimal multigraded resolution

$$ \begin{align*} \mathbf{F}: 0 \rightarrow F_p \rightarrow \cdots \rightarrow F_1 \rightarrow F_0 \end{align*} $$

with

$$ \begin{align*}F_k=\bigoplus_{{\boldsymbol{a}}\in \mathbb{Z}^n}S(\boldsymbol{x}^{\boldsymbol{a}})^{\,\beta_{k,\boldsymbol{a}}}.\end{align*} $$

The kth homological shift ideal of I denoted by $\mathrm {HS}_k(I)$ is the ideal generated by the kth multigraded shifts of I, that is,

$$ \begin{align*} \mathrm{HS}_k(I)=(\{{\boldsymbol{x}}^{\boldsymbol{a}}\mid \beta_{k,\boldsymbol{a}}\neq 0 \, \}). \end{align*} $$

Recently, properties of monomial ideals which are inherited by their homological shift ideals have attracted attention. It is shown in [1, Theorem 3.2] that if I is a matroidal ideal, then so are its homological shift ideals. It is still an open question whether a similar statement holds if one replaces matroidal by polymatroidal. However, there are some partial positive answers for some classes of polymatroidal ideals including polymatroidal ideals satisfying the strong exchange property [13, Corollary 3.6], Veronese-type ideals [13, Theorem 3.3], polymatroidal ideals generated in degree two [7, Theorem 4.5] and for the first homological shift ideal of any polymatroidal ideal [6, Theorem 2.2]. In [3, Proposition 3.1], analogues of these results for the property of being equigenerated squarefree Borel are presented and in [2], a quasi-additive property of homological shift ideals is studied.

Having linear quotients is another property that has received considerable attention. Following [7], we say that a monomial ideal I has homological linear quotients when I has linear quotients and $\mathrm {HS}_k(I)$ inherits this property for every k. It is shown in [3, Theorem 2.4] and [3, Theorems 2.4 and 3.3] that principal Borel ideals as well as squarefree Borel ideals have homological linear quotients (see also [14]). It is shown in [13, Theorem 2.2] that even $\mathbf {c}$ -bounded principal Borel ideals have homological linear quotients. It is also proved in [7, Theorem 1.3] that if a monomial ideal I has linear quotients, then $\mathrm {HS}_1(I)$ has the same property.

Regarding having homological linear quotients, we restrict our attention to edge ideals of graphs. Let G be a simple graph on n vertices and $I(G)\subseteq S$ be its edge ideal. From [10] and [12, Theorem 10.2.6], $I(G)$ has linear quotients if and only if $G^c$ is a chordal graph. It is shown in [7, Proposition 3.2] that if $I(G)$ has homological linear quotients, then adding a whisker to $G^c$ gives a graph such that the edge ideal of its complement also has homological linear quotients. As a result, $I(G)$ has homological linear quotients when $G^c$ is a tree. Generalising these two results, we show in Theorem 2.6 that when $I(G)$ has homological linear quotients, then adding clusters to $G^c$ leads to a graph such that the edge ideal of its complement has homological linear quotients. In particular, this implies that $I(G)$ has homological linear quotients when $G^c$ is a block graph (see Corollary 2.7).

Next, we consider another construction of adding pinnacles which preserves the property of having linear quotients for homological shift ideals (see Section 3 for the definition). We will see in Theorem 3.1 that if $G^c$ is obtained by adding pinnacles to a tree, then $I(G)$ has homological linear quotients. Finally, we see in Corollary 3.4 that $I(G)$ has homological linear quotients if $G^c$ is a $\lambda $ -minimal graph.

2 Block graphs

Throughout, $S=K[x_1,\ldots ,x_n]$ denotes a polynomial ring over a field K with its natural multigrading. If $u,v\in S$ are monomials, then $u:v$ denotes the monomial ${u}/{\gcd (u,v)}$ . For a monomial $u\in S$ , we set $\max u= \max \{k \mid x_k$ divides $u\}$ . When ${\ell = \max u}$ , we may sometimes write $x_\ell =\max u$ for ease of use.

Let $I\subseteq S$ be a monomial ideal. We denote its minimal set of monomial generators by $G(I)$ . A monomial ideal $I\subseteq S$ is said to have linear quotients if there exists an ordering $u_1,\ldots ,u_r$ of the elements of $G(I)$ , called an admissible order, such that for each $i=1,\ldots ,r-1$ , the colon ideal $(u_1,\ldots ,u_i):(u_{i+1})$ is generated by a subset of $\{x_1,\ldots , x_n\}$ . If I has linear quotients with respect to the ordering $u_1,\ldots ,u_r$ of $G(I)$ , we define

$$ \begin{align*} \mathrm{set}(u_{i+1}) = \{x_j \mid x_j\in (u_1,\ldots,u_i):(u_{i+1}) \}. \end{align*} $$

Remark 2.1. Let a monomial ideal $I\subseteq S$ have linear quotients. By [15, Lemma 1.5], a minimal multigraded free resolution $\mathbf {F}$ of I can be described as follows: the S-module $F_i$ in homological degree i of $\mathbf {F}$ is the multigraded free S-module whose basis is formed by the monomials $u x_{\ell _1}\ldots x_{\ell _i}$ for which $u\in \mathrm {G}(I)$ and $x_{\ell _1},\ldots ,x_{\ell _i}$ are distinct elements of $\mathrm {set}(u)$ .

Henceforth, all graphs considered in this paper are simple graphs. Let G be a graph on the vertex set $V(G)=\{x_1,\ldots ,x_n\}$ with edge set $E(G)$ . The ideal

$$ \begin{align*} I(G) = (x_{i}x_{j} \mid \{x_{i},x_{j}\} \in E(G)) \subseteq S \end{align*} $$

is called the edge ideal of G. The complement of G, denoted by $G^c$ , is the graph on the vertex set $V(G)$ whose edge set is

$$ \begin{align*} E(G^{c})=\{\{x_i,x_j\} \mid x_i\neq x_j~\mathrm{and}\ \{x_i,x_j\}\notin E(G)\}. \end{align*} $$

The set of all vertices adjacent to a vertex $x_i$ in G, denoted by $\mathrm {N}_G(x_i)$ , is called the neighbourhood of $x_i$ in G. The distance between vertices $x_i$ and $x_j$ of a connected graph G, denoted by , is the number of edges in the shortest path connecting them.

A graph G is called a chordal graph if it has no induced cycle of length greater than three. An ordering $x_1>x_2>\cdots >x_n$ of vertices of a graph G is called a perfect elimination ordering if whenever a vertex $x_i$ is adjacent to vertices $x_j$ and $x_k$ with ${i<j<k}$ , then $x_j$ and $x_k$ are also adjacent. Chordal graphs are characterised in [5, 11] as those graphs whose vertices admit a perfect elimination ordering.

Remark 2.2. While it is known by [10] and [12, Theorem 10.2.6] that the edge ideal $I(G)$ of a graph G has linear quotients if and only if $G^c$ is chordal, this property is not inherited by homological shift ideals. For example, consider the graph G presented in Figure 1. Here, the labelling of vertices gives a perfect elimination ordering of vertices with respect to $x_1>\cdots >x_6$ and even more with respect to $x_6>\cdots >x_1$ . One has

$$ \begin{align*} I(G^c)=(x_1x_4,x_1 x_5,x_1 x_6,x_2 x_6,x_3 x_6), \end{align*} $$

and

$$ \begin{align*} \mathrm{HS}_2(I(G^c))=(x_1 x_2x_3 x_6,x_1 x_4x_5 x_6), \end{align*} $$

which does not have linear quotients with respect to any ordering of its generators.

Figure 1 A chordal graph such that $\mathrm{HS}_2(I(G^c))$ does not have linear quotients.

Let $u= x_{i_1}\cdots x_{i_m}\in S$ be a squarefree monomial with $i_1<\cdots <i_m$ . We say that $x_{i_t}$ is a source variable of u with respect to a graph G, or shortly a source of u when the graph is clear from the context, if the following conditions hold:

• • $1\leq i_t< \max u$ ;

• • $x_{i_t}$ is adjacent to $x_{i_s}$ in G for $t<s\leq \max u$ .

Theorem 2.3 [13, Theorem 4.1].

Let G be a chordal graph. Suppose that $x_1> x_2 > \cdots > x_n$ is a perfect elimination ordering of $V(G)$ . Then, for each k,

$$ \begin{align*} \mathrm{HS}_k(I(G^c))= \bigg(u \,\Big|\!\! \begin{array}{l} u\ \mathrm{is\ a\ squarefree\ monomial\ of\ degree}\ k+2 \\ \mathrm{which\ has\ a\ source\ with\ respect\ to}\ G^c \end{array}\!\!\bigg). \end{align*} $$

A graph G is said to be a biconnected graph if it is connected and nonseparable, that is, if we remove any of its vertices, the graph remains connected. A biconnected component is a maximal biconnected subgraph. A graph G is called a block graph if every biconnected component is a clique.

Let G be a graph and $v\in V(G)$ . We say that the graph H is obtained from G by adding a t-cluster or simply a cluster via v when we add $t-1$ new vertices $y_1,\ldots ,y_{t-1}$ to $V(G)$ , and add all edges $\{y_iy_j \mid 1\leq i<j\leq t \}$ to $E(G)$ (note that we set $v=y_t$ ).

The first statement of the following lemma is a special case of [13, Proposition 1.7].

Lemma 2.4. Let $I\subset K[{\boldsymbol {x}}]=K[x_1,\ldots ,x_n]$ be a monomial ideal that has homological linear quotients and consider the ideal $\mathfrak {m}=(y_1,\ldots ,y_m)$ in $K[{\boldsymbol {y}}]=K[y_1,\ldots ,y_m] $ with m new variables. Then the kth homological shift ideal of $\mathfrak {m} I\subseteq K[{\boldsymbol {x}},{\boldsymbol {y}}]$ is

$$ \begin{align*} \mathrm{HS}_k(\mathfrak{m} I)&= (y_1,\ldots,y_m)\mathrm{HS}_{k}(I)+(y_iy_j\mid 1\leq i<j\leq m)\mathrm{HS}_{k-1}(I) \\ &\quad + (y_iy_jy_k\mid 1\leq i<j<k \leq m)\mathrm{HS}_{k-2}(I)+ \cdots + (y_1\cdots y_m)\mathrm{HS}_{k-m+1}(I). \end{align*} $$

Furthermore, the ideal $\mathrm {HS}_k(\mathfrak {m} I)$ has homological linear quotients for every k.

Proof. Let $I=\mathrm {HS}_0(I)$ have linear quotients with respect to the ordering $u_1,u_2,\ldots ,u_{\ell }$ of its generators. Then $\mathfrak {m} I$ has simply linear quotients with respect to the order:

$$ \begin{align*} u_1y_1,u_2y_1,\ldots,u_\ell y_1, u_1y_2,u_2y_2,\ldots,u_{\ell}y_2, \ldots, u_1y_m,u_2y_m,\ldots,u_{\ell}y_m. \end{align*} $$

With this ordering of generators,

$$ \begin{align*} \mathrm{set}(u_iy_j)=\mathrm{set}(u_i)\cup\{y_1,\ldots,y_{j-1}\}, \end{align*} $$

where $\mathrm {set}(u_i) = \{x_j \mid x_j \in (u_1,\ldots ,u_{i-1}) : (u_i)\}$ . Using Remark 2.1 to construct $\mathrm {HS}_k(\mathfrak {m} I)$ gives the conclusions in Table 1.

Table 1 Conclusions for $\mathrm {HS}_k(I)$ in Lemma 2.4.

The sum of the ideals in the left column of Table 1 gives

$$ \begin{align*} \mathrm{HS}_k(\mathfrak{m} I)&= (y_1,\ldots,y_m)\mathrm{HS}_{k}(I)+(y_iy_j\mid 1\leq i<j\leq m)\mathrm{HS}_{k-1}(I) \\ &\quad + (y_iy_jy_k\mid 1\leq i<j<k \leq m)\mathrm{HS}_{k-2}(I)+ \cdots \\ &\quad + (y_1\cdots y_m)\mathrm{HS}_{k-m+1}(I). \end{align*} $$

Next we show that $\mathrm {HS}_k(\mathfrak {m} I)$ has linear quotients for every k. Notice that each $\mathrm {HS}_\ell (I)$ has linear quotients by assumption. For each $\ell $ , we fix an admissible ordering on the minimal set of monomial generators of $\mathrm {HS}_\ell (I)$ , and set $u>_\ell v$ for each u and v in $G(\mathrm {HS}_\ell (I))$ if u comes before v in the fixed admissible ordering. Next we show that $\mathrm {HS}_k(\mathfrak {m} I)$ has linear quotients with the following ordering of monomial generators of $\mathrm {HS}_k(\mathfrak {m} I)$ : the monomial $y_{i_1}\cdots y_{i_t}u$ with $u\in \mathrm {HS}_{k-t+1}(I)$ comes before $y_{j_1}\cdots y_{j_s}v$ with $v\in \mathrm {HS}_{k-s+1}(I)$ if either $y_{i_1}\cdots y_{i_t}>_{\mathrm { glex}}y_{j_1}\cdots y_{j_s}$ or if $y_{i_1}\cdots y_{i_t}=y_{j_1}\cdots y_{j_s}$ and $u>_{k-t+1}v$ . Here $>_{\mathrm {glex}}$ denotes the graded lexicographic order on $K[{\boldsymbol {y}}]$ induced by $y_1>\cdots >y_m$ . To see why this is an admissible ordering for $\mathrm {HS}_k(\mathfrak {m} I)$ , consider the colon

$$ \begin{align*} w=y_{i_1}\cdots y_{i_t}u:y_{j_1}\cdots y_{j_s}v \end{align*} $$

of elements of the minimal set of monomial generators of $\mathrm {HS}_k(\mathfrak {m} I)$ in which $y_{i_1}\cdots y_{i_t}u$ comes before $y_{j_1}\cdots y_{j_s}v$ in the ordering just described. Suppose that . We show that there exists $y_{\ell _1}\cdots y_{\ell _s}\tilde {v}$ in the set of generators which appears before $y_{j_1}\cdots y_{j_s}v$ and

$$ \begin{align*} y_{\ell_1}\cdots y_{\ell_s}\tilde{v}:y_{j_1}\cdots y_{j_s}v \end{align*} $$

is a degree one monomial which divides w. We consider two cases.

Case 1. Assume that $y_{i_1}\cdots y_{i_t}>_{\mathrm {glex}}y_{j_1}\cdots y_{j_s}$ . By Remark 2.1, the element v in the minimal set of monomial generators of $\mathrm {HS}_{k-s+1}(I)$ is a product of an element $\hat {v}$ in the minimal set of monomial generators of I and $k-s+1$ pairwise distinct elements of $\mathrm {set}(\hat {v})$ . If $t>s$ , then $k-s+1>k-t+1\geq 0$ . Thus, $k-s+1\neq 0$ . In particular, there exists $x_p$ in the subset of $\mathrm {set}(\hat {v})$ that divides $v/\hat {v}$ . Since $y_{i_r}$ divides $y_{i_1}\cdots y_{i_t}: y_{j_1}\cdots y_{j_s}$ , it follows that

$$ \begin{align*} y_{i_r}\bigg(y_{j_1}\cdots y_{j_s}\frac{v}{x_p}\bigg) \end{align*} $$

has the desired properties, that is, it comes before $y_{j_1}\cdots y_{j_s}v$ and its colon with respect to $y_{j_1}\cdots y_{j_s}v$ is $y_{i_r}$ .

Otherwise, $t=s$ . Suppose that $y_{i_1}\cdots y_{i_s}:y_{j_1}\cdots y_{j_s}=y_{\ell _1}\cdots y_{\ell _p}$ with $\ell _1<\cdots <\ell _p$ . Then

$$ \begin{align*} y_{\ell_1}\frac{y_{j_1}\cdots y_{j_s}}{y_{j_s}}v, \end{align*} $$

where $j_s=\max (y_{j_1}\ldots y_{j_s})$ has the desired properties.

Case 2. Now assume that $y_{i_1}\cdots y_{i_t}=y_{j_1}\cdots y_{j_s}$ and $u>_{k-s+1}v$ . Since $\mathrm {HS}_{k-s+1}$ has linear quotients with respect to the ordering given by $>_{k-s+1}$ , there exists $\tilde {v}$ in the minimal set of monomial generators of $\mathrm {HS}_{k-s+1}(I)$ such that $\tilde {v}>_{k-s+1}v$ and $\tilde {v}: v=x_p$ for some p with $x_p\mid u:v$ . Hence, $y_{j_1}\cdots y_{j_s}\tilde {v}$ is the desired element since it comes before $y_{j_1}\cdots y_{j_s}v$ in the ordering of the generators of $\mathrm {HS}_k(\mathfrak {m} I)$ described before and in addition $y_{j_1}\cdots y_{j_s}\tilde {v}:y_{j_1}\cdots y_{j_s}v=x_p$ .

Let I, J and L be monomial ideals in S such that the minimal set of monomial generators $G(I)$ of I is the disjoint union of $G(J)$ and $G(L)$ . Then $I=J+L$ is called a Betti splitting if

$$ \begin{align*} \beta_{k,{\boldsymbol{a}}}(I)=\beta_{k,{\boldsymbol{a}}}(J)+\beta_{k,{\boldsymbol{a}}}(L)+\beta_{k-1,{\boldsymbol{a}}}(J\cap L) \end{align*} $$

for all k and all multidegrees ${\boldsymbol {a}}$ . In particular, as noted in [4, 7], if $I=J+L$ is a Betti splitting, then for each k,

$$ \begin{align*} \mathrm{HS}_k(I)=\mathrm{HS}_k(J)+\mathrm{HS}_k(L)+\mathrm{HS}_{k-1}(J\cap L). \end{align*} $$

Theorem 2.5 [8, Corollary 2.4].

Let I, J and L be monomial ideals in S such that $G(I)$ is the disjoint union of $G(J)$ and $G(L)$ . If both J and L have linear resolutions, then $I=J+L$ is a Betti splitting.

Theorem 2.6. Let G be a graph, and suppose that the graph H is obtained from G by adding a cluster. If the edge ideal $I(G^c)$ has homological linear quotients, then $I(H^c)$ also has homological linear quotients.

Proof. Let $V(G)=\{x_1,x_2\ldots , x_n\}$ and H be obtained by adding a t-cluster to G via $x_n$ . Suppose that $y_1,\ldots ,y_t=x_n$ are vertices of the new clique that is added to G. Then

$$ \begin{align*} I(H^c)=I(G^c)+ (x_iy_j\mid 1\leq i \leq n-1 \, \mathrm{and}\, 1\leq j \leq t-1 ). \end{align*} $$

Set $I=I(H^c)$ , $J=I(G^c)$ and $L=(x_iy_j\mid 1\leq i \leq n-1 \, \mathrm {and}\, 1\leq j \leq t-1 )$ . The ideal L is matroidal. So L has a linear resolution. The ideal J also has a linear resolution by the assumption. Hence, by Theorem 2.5, $I=J+L$ is a Betti splitting. In particular,

$$ \begin{align*} \mathrm{HS}_k(I)=\mathrm{HS}_k(J)+\mathrm{HS}_k(L)+\mathrm{HS}_{k-1}(J\cap L) \end{align*} $$

for each k. Observe that

$$ \begin{align*} J\cap L= (x_ix_jy_\ell \mid\{x_i,x_j\}\in E(G^c)\ \mathrm{and}\ 1\leq \ell\leq t-1) =(y_1,\ldots,y_{t-1})J. \end{align*} $$

Thus, by Lemma 2.4, $\mathrm {HS}_{k-1}(J\cap L)$ has linear quotients and

$$ \begin{align*} \mathrm{HS}_{k-1}(J\cap L)&= (y_1,\ldots,y_{t-1})\mathrm{HS}_{k-1}(J)+(y_iy_j\mid 1\leq i<j\leq t-1)\mathrm{HS}_{k-2}(J)\\ &\quad + (y_iy_jy_k\mid 1\leq i<j<k \leq t-1)\mathrm{HS}_{k-3}(J)+ \cdots\\ &\quad + (y_1\cdots y_{t-1})\mathrm{HS}_{k-t+1}(J). \end{align*} $$

Writing the homological shift ideals of $(x_i \mid 1\leq i \leq n-1 )$ as Koszul complexes and applying Lemma 2.4 yields

$$ \begin{align*} \mathrm{HS}_k(L)=\bigg( x_{i_1}\cdots x_{i_p}y_{j_1}\cdots y_{j_q}\,\bigg|\! \begin{array}{l} 1\leq i_1<\cdots<i_p< n\\ 1\leq j_1<\cdots<j_q< t \end{array} \mathrm{and}\ p+q=k+2\bigg). \end{align*} $$

By our discussion, the ideals $\mathrm {HS}_k(L)$ , $\mathrm {HS}_{k-1}(J\cap L)$ and $\mathrm {HS}_k(J)$ have linear quotients. Suppose that they have linear quotients with respect to the following ordering of their minimal set of monomial generators:

• • $\mathrm {HS}_k(L)=(u_1,\ldots ,u_p)$ ;

• • $\mathrm {HS}_{k-1}(J\cap L)=(v_1,\ldots ,v_q)$ ;

• • $\mathrm {HS}_k(J)=(w_1,\ldots ,w_r)$ .

We claim that $\mathrm {HS}_k(I)$ has linear quotients with respect to the ordering of generators:

(2.1) $$ \begin{align} u_1,\ldots,u_p ,v_{j_1},\ldots,v_{j_s} ,w_1,\ldots,w_r. \end{align} $$

Here $1\leq j_1<\cdots <j_s\leq q$ and the elements $v_{j_1},\ldots ,v_{j_s}$ are those elements of $G(\mathrm {HS}_{k-1}(J\cap L))=\{v_1,\ldots ,v_q\}$ which do not appear among $u_1,\ldots ,u_p$ , that is, those elements of $G(\mathrm {HS}_{k-1}(J\cap L))$ divided by $x_n$ . Let v be a squarefree monomial in $K[{\boldsymbol {x}},{\boldsymbol {y}}]$ . Denote by the number of $y_j$ which divide v for $j=1,\ldots ,t$ .

First consider $u:v_{j_i}$ for some $i=1,\ldots ,s$ and $u\in \{u_1,\ldots ,u_p,v_1,\ldots ,v_{j_{i-1}}\}$ . Let z be a variable dividing $u:v_{j_i}$ . Then

$$ \begin{align*} \tilde{u}=\frac{v_{j_i}}{x_n} z \end{align*} $$

is a monomial appearing among $u_1,\ldots ,u_p$ in (2.1) and $\tilde {u}:v_{j_i}=z$ .

Next consider $u:w_j$ for some $j=1,\ldots ,r$ and $u\in \{u_1,\ldots ,u_p,v_1,\ldots ,v_{j_{s}}\}$ (see (2.1)). Since , one deduces that $y_{j_\ell }$ divides $u:w_j$ for some $\ell $ . So $u=(w_j/\max w_j)y_{j_\ell }$ is simply an element of $\{u_1,\ldots ,u_p,v_1,\ldots ,v_{j_{s}}\}$ with $u:w_j=y_{j_\ell }$ , as desired.

Corollary 2.7. Let G be a block graph. Then the edge ideal $I(G^c)$ has homological linear quotients.

Corollary 2.8 [7, Corollary 3.3].

Let G be a tree. Then the edge ideal $I(G^c)$ has homological linear quotients.

3 $\lambda $ -minimal graphs

Let $e=\{x_i,x_j\}$ be an edge of a graph G. By adding a pinnacle on e, we mean adding a new vertex y, and edges $\{x_i,y\}$ and $\{x_j,y\}$ to G. We call the subgraph induced on these two new edges a pinnacle and the vertex y its tip (see Figure 2).

Herzog and Ficarra, using an inductive argument by adding whiskers, showed in [7] that if G is a tree, then the edge ideal $I(G^c)$ has homological linear quotients. Here, generalising their result, we determine a labelling on the vertices of trees with some pinnacles to find an admissible ordering of generators for every $\mathrm {HS}_k(I(G^c))$ .

Theorem 3.1. Let G be either a tree or obtained by adding some pinnacles to a tree. Then the edge ideal $I(G^c)$ has homological linear quotients.

Proof. We may assume that $\{x_1,x_2,\ldots ,x_n\}$ is the vertex set of G such that for some t, the induced subgraph H on $\{x_t ,x_{t+1},\ldots ,x_n\}$ is a tree and G is obtained by adding some pinnacles to H with tips $\{x_1,x_2,\ldots ,x_{t-1}\}$ . By a suitable relabelling of vertices, we may also assume that if $t \leq i,j\leq n$ and , then $i<j$ .

One can see that the labelling described above gives a perfect elimination ordering on the vertices of G. In fact, if $x_i$ is the tip of a pinnacle on an edge $\{ x_{j_1},x_{j_2}\}\in E(H)$ , then

$$ \begin{align*}\{x_j\in \mathrm{N}_G(x_i) \mid j>i\}=\{ x_{j_1},x_{j_2}\}\end{align*} $$

is a clique. Otherwise, if $x_i$ is a vertex of the tree H with $i<n$ , the set

$$ \begin{align*} \{x_j\in \mathrm{N}_G(x_i) \mid j>i\} \end{align*} $$

has exactly one element. In contrast, assume that distinct elements $x_{j_1}$ and $x_{j_2}$ belong to $\{x_j\in \mathrm {N}_G(x_i) \mid j>i\}$ . Then by labelling the vertices as described above, both $d(x_{j_1},x_n)$ and $d(x_{j_2},x_n)$ are less than or equal to $d(x_i,x_n)$ . Hence, there exist a path $P_1$ from $x_{j_1}$ to $x_n$ and a path $P_2$ from $x_{j_2}$ to $x_n$ neither of which contains $x_i$ . This yields the existence of two paths from $x_i$ to $x_n$ , one via the adjacent vertex $x_{j_1}$ and $P_1$ , and the other via the adjacent vertex $x_{j_2}$ and $P_2$ , a contradiction to the fact that H is a tree. Thus, the labelling of $V(G)$ gives a perfect elimination ordering and, by Theorem 2.3, for each k, the kth homological shift ideal of $I=I(G^c)$ is

(3.1) $$ \begin{align} \mathrm{HS}_k(I)= \bigg(u \,\Big|\!\! \begin{array}{l} u~\text{is a squarefree monomial of degree}\ k+2 \\ \text{which has a source with respect to}~G^c \end{array}\!\!\bigg). \end{align} $$

Fix k in the set $\{0,\ldots ,\mathrm {proj\, dim\,} I\}$ . We will show that $\mathrm {HS}_k(I)$ has linear quotients with respect to the lexicographic ordering of generators with $x_1>\cdots >x_n$ . For this purpose, suppose that u and v are two monomials in the minimal set of monomial generators of $\mathrm {HS}_k(I)$ , $u>_{\mathrm {lex}} v$ , and

$$ \begin{align*} u:v= x_{i_1}\cdots x_{i_p} \end{align*} $$

with $p>1$ and ${i_1}<\cdots <{i_p}$ . Since $\mathrm {HS}_k(I)$ is generated in a single degree, the monomial $v:u$ is also of degree p, say

$$ \begin{align*} v:u=x_{\ell_1}\cdots x_{\ell_p} \quad \mathrm{with}\ \ell_1<\cdots< \ell_p. \end{align*} $$

Notice that $u>_{\mathrm {lex}} v$ implies that

(3.2) $$ \begin{align} i_1<\ell_1. \end{align} $$

We show that there exists a monomial w in the minimal set of monomial generators of $\mathrm {HS}_k(I)$ , such that $w>_{\mathrm {lex}} v$ , and

$$ \begin{align*} w:v= x_{i_s} \end{align*} $$

for some $s= 1,\ldots ,p$ .

Figure 2 A tree on the vertex set $\{x_1,\ldots,x_4\}$ with three pinnacles.

First, suppose that $i_1\geq t$ , so that $x_{i_1}$ is a vertex of the tree H. As discussed above, the vertex $x_{i_1}$ is adjacent to at most one vertex $x_j$ of the tree H with $j>i_1$ . From (3.2), $x_{i_1}$ is adjacent to at most one of $x_{\ell _1}$ or $x_{\ell _2}$ ; say

$$ \begin{align*} \{x_j\in \mathrm{N}_G(x_{i_1}) \mid j>i_1\}\cap \{\ell_1,\ell_2\} \quad \mathrm{is\ either\ \emptyset\ or\ \{\ell_1\}}. \end{align*} $$

Then the variable $x_{i_1}$ becomes a source of $w=(v/x_{\ell _1})x_{i_1}$ with respect to $G^c$ ; here, ${i_1<\ell _2}$ guarantees that $i_1\neq \max w$ . Furthermore, by (3.2), $w>_{\mathrm {lex}}v$ . Thus, w is a monomial with the desired properties.

Next, suppose that $i_1< t$ , so that $x_{i_1}$ is the tip of a pinnacle. From the labelling given to the vertices of G,

(3.3) $$ \begin{align} \{x_j\in \mathrm{N}_G(x_{i_1}) \mid j>i_1\}=\{ x_{t_1},x_{t_2}\} \end{align} $$

for vertices $x_{t_1}$ and $x_{t_2}$ on an edge of the tree H which is on a pinnacle with the tip $x_{i_1}$ . We consider three cases.

Case 1. If neither of the vertices $x_{t_1}$ and $x_{t_2}$ divides v, then $x_{i_1}$ is a source of the monomial $w=(v/x_{\ell _1})x_{i_1}$ with respect to $G^c$ . By (3.2), $i_1<\ell _2\leq \max w$ . Hence, the squarefree monomial $w=(v/x_{\ell _1})x_{i_1}$ of degree $k+2$ is an element of $\mathrm {HS}_k(I)$ by (3.1). Furthermore, $w:v=x_{i_1}$ and, by (3.2), $w>_{\mathrm {lex}} v$ , as desired.

Case 2. Assume that exactly one of the variables $x_{t_1}$ and $x_{t_2}$ divides v, say $x_{t_1}$ . Then by (3.3), the variable $x_{i_1}$ is a source of the monomial $w=(v/x_{t_1})x_{i_1}$ with respect to $G^c$ . Here, $i_1<\max w$ is a consequence of $i_1<\ell _1<\ell _2\leq \max v$ by (3.2). Now since $w=(v/x_{t_1})x_{i_1}$ has a source with respect to $G^c$ , this squarefree monomial of degree $k+2$ is an element of $\mathrm {HS}_k(I)$ . Moreover, from the labelling of vertices, $i_1<t_1$ because $i_1$ is a tip, while $t_1$ is a vertex of the tree H. So $w>_{\mathrm {lex}} v$ and w is a desired monomial.

Case 3. Finally, assume that $x_{t_1}$ and $x_{t_2}$ both divide v. Suppose that $t_1<t_2$ . Since v belongs to the minimal set of monomial generators of $\mathrm {HS}_k(I)$ , by (3.1), it has a source variable with respect to $G^c$ . Suppose that $x_{\ell }$ is a source of v for some $\ell $ . Since $\{x_{t_1},x_{t_2}\}$ is an edge of H and we have assumed that $t_1<t_2$ , it follows that $x_{t_1}$ is not a source of v with respect to $G^c$ . In particular, $x_{t_1}\neq x_{\ell }$ . We show that if either $t_1<\ell $ or $\ell <t_1$ , the variable $x_{\ell }$ remains a source in $w=(v/x_{t_1})x_{i_1}$ . When $t_1<\ell $ , it is clear that $x_{\ell }$ is still a source of $w=(v/x_{t_1})x_{i_1}$ because the replacement of $x_{i_1}$ by $x_{t_1}$ in w occurs before $x_\ell $ . However, if $\ell <t_1$ , then $\ell $ is not adjacent to $i_1$ because $i_1$ is a tip in G with $N_G(x_{i_1})=\{x_{t_1},x_{t_2}\}$ . So the set

$$ \begin{align*} \{x_j \mid x_j \text{ divides}\ w\ \text{and}\ \ell<j \} \end{align*} $$

is still the empty set. Moreover, $x_{\ell }\neq \max w$ because $x_{t_2}$ divides w. Thus, $x_\ell $ is a source of w as well. Again, note that we have set the tip $x_{i_1}$ lexicographically greater than the vertex $x_{t_1}$ of H. Hence, $w>_{\mathrm {lex}}v$ , as desired.

Proposition 3.2. Let G be either the complete graph $K_3$ or obtained by adding some pinnacles to $K_3$ . Then the edge ideal $I(G^c)$ has homological linear quotients.

Proof. Set $I=I(G^c)$ . Assume that $V(G)=\{x_1,\ldots ,x_n\}$ for some $n\geq 3$ , the subgraph H induced on $\{x_{n-2},x_{n-1},x_n\}$ is a 3-clique, and G is constructed by adding pinnacles with tips $\{x_1,\ldots ,x_{n-3}\}$ . Fixing $0\leq k\leq \mathrm {proj\, dim\,} I(G^c)$ , we are going to show that $\mathrm {HS}_k(I(G^c))$ has linear quotients with respect to the lexicographic ordering of its minimal set of monomial generators induced by $x_1>\cdots >x_n$ . For this purpose, first we see that if w is an element of the minimal set of monomial generators of $\mathrm {HS}_k(I)$ , then at most one of the vertices $x_{n-2}$ , $x_{n-1}$ , $x_{n}$ can divide w. Indeed, the neighbourhood of each vertex of G intersects $\{x_{n-2},x_{n-1},x_n\}$ exactly in two vertices, and if more than one variable among $x_{n-2}$ , $x_{n-1}$ or $x_{n}$ divides w, then w does not have a source with respect to $G^c$ , which is a contradiction. (See Theorem 2.3 where the generators of $\mathrm {HS}_k(I)$ are described.)

Next, let u and v be two monomials in the minimal set of monomial generators of $\mathrm {HS}_k(I)$ , $u>_{\mathrm {lex}} v$ , and $ u:v= x_{i_1}\cdots x_{i_p} $ with $p>1$ and ${i_1}<\cdots <{i_p}$ . Since $\mathrm {HS}_k(I)$ is generated in a single degree, we may write

$$ \begin{align*} v:u=x_{\ell_1}\cdots x_{\ell_p} \quad \mathrm{with}\ \ell_1<\cdots< \ell_p. \end{align*} $$

Now on the one hand, $u>_{\mathrm {lex}} v$ implies that $i_1<\ell _1<\ell _2$ . On the other hand, since at most one of the variables $x_{n-2}$ , $x_{n-1}$ or $x_{n}$ divides v, we deduce that $\ell _1\leq n-3$ . Hence, $i_1<n-3$ . In particular, the vertices $x_{\ell _1}$ and $x_{i_1}$ are the tips of two pinnacles in G. Consequently, if $m=\max v$ , then $x_{i_1}$ is a source of $w=(v/x_m)x_{i_1}$ . To see this, note that $i_1<\ell _1<\ell _2\leq \max v$ and, by removing $x_m$ from v, the monomial w can have only variables corresponding to some tips in its support. So $x_{i_1}$ is not adjacent to any of the $x_j$ in the support of $v/x_m$ . So w is a monomial in $\mathrm {HS}_k(I)$ , as described in Theorem 2.3, with $w>_{\mathrm {lex}} v$ and $w:v=x_{i_1}$ .

The statement of Proposition 3.2 does not hold if we replace $K_3$ by an arbitrary complete graph. For example, consider the graph G in Figure 1 obtained by adding two pinnacles to $K_4$ , and refer to Remark 2.2 where $\mathrm {HS}_2(I(G^c))$ is determined.

Let G be a graph and k be a positive integer. A k-colouring of G is a mapping from $V(G)$ to $[k]$ . If f is a k-colouring of G, then the colour of each edge $\{x_i,x_j\}$ is defined to be $\{f(x_i),f(x_j)\}$ . A k-colouring f of the graph G is called a line-distinguishing colouring if every two distinct edges of G have distinct colours. The minimum number k for which G has a line-distinguishing k-colouring, denoted by $\lambda (G)$ , is called the line-distinguishing chromatic number of G. The graph G is called $\lambda $ -minimal in [9] if $\lambda (G-e)=\lambda (G)-1$ for each edge e.

Theorem 3.3 [16, Theorem 2.4].

Let G be a chordal graph. Then G is $\lambda $ -minimal if and only if G is either constructed by adding at least one pinnacle to each edge of a star or constructed by adding at least one pinnacle to each edge of the complete graph  $K_3$ .

Corollary 3.4. Let G be a $\lambda $ -minimal chordal graph. Then the edge ideal $I(G^c)$ has homological linear quotients.