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 [Reference Bayati1, 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 [Reference Herzog, Moradi, Rahimbeigi and Zhu13, Corollary 3.6], Veronese-type ideals [Reference Herzog, Moradi, Rahimbeigi and Zhu13, Theorem 3.3], polymatroidal ideals generated in degree two [Reference Ficarra and Herzog7, Theorem 4.5] and for the first homological shift ideal of any polymatroidal ideal [Reference Ficarra6, Theorem 2.2]. In [Reference Bayati, Jahani and Taghipour3, Proposition 3.1], analogues of these results for the property of being equigenerated squarefree Borel are presented and in [Reference Bayati2], a quasi-additive property of homological shift ideals is studied.
Having linear quotients is another property that has received considerable attention. Following [Reference Ficarra and Herzog7], 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 [Reference Bayati, Jahani and Taghipour3, Theorem 2.4] and [Reference Bayati, Jahani and Taghipour3, Theorems 2.4 and 3.3] that principal Borel ideals as well as squarefree Borel ideals have homological linear quotients (see also [Reference Herzog, Moradi, Rahimbeigi and Zhu14]). It is shown in [Reference Herzog, Moradi, Rahimbeigi and Zhu13, Theorem 2.2] that even
$\mathbf {c}$
-bounded principal Borel ideals have homological linear quotients. It is also proved in [Reference Ficarra and Herzog7, 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 [Reference Fröberg, Balcerzyk, Józefiak, Krempa, Simson and Vogel10] and [Reference Herzog and Hibi12, Theorem 10.2.6],
$I(G)$
has linear quotients if and only if
$G^c$
is a chordal graph. It is shown in [Reference Ficarra and Herzog7, 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 [Reference Herzog and Takayama15, 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 [Reference Dirac5, Reference Fulkerson and Gross11] as those graphs whose vertices admit a perfect elimination ordering.
Remark 2.2. While it is known by [Reference Fröberg, Balcerzyk, Józefiak, Krempa, Simson and Vogel10] and [Reference Herzog and Hibi12, 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 [Reference Herzog, Moradi, Rahimbeigi and Zhu13, 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 [Reference Herzog, Moradi, Rahimbeigi and Zhu13, 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 [Reference Crupi and Ficarra4, Reference Ficarra and Herzog7], 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 [Reference Francisco, Há and Van Tuyl8, 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:
$$ \begin{align} u_1,\ldots,u_p ,v_{j_1},\ldots,v_{j_s} ,w_1,\ldots,w_r. \end{align} $$
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 [Reference Ficarra and Herzog7, 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 [Reference Ficarra and Herzog7] 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
$$ \begin{align} \mathrm{HS}_k(I)= \bigg(u \,\Big|\!\! \begin{array}{l} u~\text{is a squarefree monomial of degree}\ k+2 \\ \text{which has a source with respect to}~G^c \end{array}\!\!\bigg). \end{align} $$
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
$$ \begin{align} i_1<\ell_1. \end{align} $$
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,
$$ \begin{align} \{x_j\in \mathrm{N}_G(x_{i_1}) \mid j>i_1\}=\{ x_{t_1},x_{t_2}\} \end{align} $$
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 [Reference Frank, Harary and Plantholt9] if
$\lambda (G-e)=\lambda (G)-1$
for each edge e.
Theorem 3.3 [Reference Regonati and Salvi16, 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.
