Cohen Macaulay Hybrid Graphs

TL;DR

This paper introduces hybrid graphs, a new family of graphs, and proves their Cohen-Macaulay property, also characterizing Cohen-Macaulay chordal graphs as hybrid graphs, expanding understanding of graph properties in algebraic combinatorics.

Contribution

The paper defines hybrid graphs, proves they are Cohen-Macaulay, and characterizes Cohen-Macaulay chordal graphs as hybrid graphs, establishing new links between graph theory and algebraic properties.

Findings

Every hybrid graph associated with a given graph is Cohen-Macaulay.

There are infinitely many hybrid graphs for each graph.

Every Cohen-Macaulay chordal graph is a hybrid graph.

Abstract

We introduce a new family of graphs, namely, hybrid graphs. There are infinitely many hybrid graphs associated to a single graph. We show that every hybrid graph associated to a given graph is Cohen Macaulay. Furthermore, we show that every CohenMacaulay chordal graph is a hybrid graph.

Tables

Table 1. Table 1 . A

Faces of $\Delta$	Corresponding facets of ${\Delta }^{\prime }$
$\varnothing$	$\{5,7,8,9\},\{5,7,8,10\},\{5,7,8,11\},\{6,7,8,9\},\{6,7,8,10\},\{6,7,8,11\}$
$\left\{1\right\}$	$\left\{1\right\}\cup \{7,8,9\},\left\{1\right\}\cup \{7,8,10\},\left\{1\right\}\cup \{7,8,11\}$
$\left\{2\right\}$	$\left\{2\right\}\cup \{5,8,9\},\left\{2\right\}\cup \{5,8,10\},\left\{2\right\}\cup \{5,8,11\},\left\{2\right\}\cup \{6,8,9\},\left\{2\right\}\cup \{6,8,10\},\left\{2\right\}\cup \{6,8,11\}$
$\left\{3\right\}$	$\left\{3\right\}\cup \{5,7,9\},\left\{3\right\}\cup \{5,7,10\},\left\{3\right\}\cup \{5,7,11\},\left\{3\right\}\cup \{6,7,9\},\left\{3\right\}\cup \{6,7,10\},\left\{3\right\}\cup \{6,7,11\}$
$\left\{4\right\}$	$\left\{4\right\}\cup \{5,7,8\},\left\{4\right\}\cup \{6,7,8\}$
$\{1,4\}$	$\{1,2\}\cup \{7,8\}$

Table 2. Table 2 . B

Faces of $\Delta$	Corresponding facets of ${\Delta }^{\prime }$
$\varnothing$	$\{5,6,7\},\{5,6,8\},\{5,6,9\}$
$\left\{1\right\}$	$\left\{1\right\}\cup \{6,7\},\left\{1\right\}\cup \{6,8\},\left\{1\right\}\cup \{6,9\}$
$\left\{2\right\}$	$\left\{2\right\}\cup \{5,7\},\left\{2\right\}\cup \{5,8\},\left\{2\right\}\cup \{5,9\}$
$\left\{3\right\}$	$\left\{3\right\}\cup \{6,7\},\left\{3\right\}\cup \{6,8\},\left\{3\right\}\cup \{6,9\}$
$\left\{4\right\}$	$\left\{4\right\}\cup \{5,6\}$
$\{1,4\}$	$\{1,4\}\cup \left\{6\right\}$

Table 3. Table 3 . C

Faces of $\Delta$	Corresponding facets of ${\Delta }^{\prime }$
$\varnothing$	$\{5,6\},\{5,7\}$
$\left\{1\right\}$	$\left\{1\right\}\cup \left\{6\right\},\left\{1\right\}\cup \left\{7\right\}$
$\left\{2\right\}$	$\left\{2\right\}\cup \left\{6\right\},\left\{2\right\}\cup \left\{7\right\}$
$\left\{3\right\}$	$\left\{3\right\}\cup \left\{6\right\},\left\{3\right\}\cup \left\{7\right\}$
$\left\{4\right\}$	$\left\{4\right\}\cup \left\{5\right\}$
$\{1,4\}$	$\{1,4\}\cup \varnothing$

Equations

$$\dim (\Delta )=\max \{\dim F:F\in \Delta \}.$$

$$I(G)=(x_i{x}_j:\{i,j\}\in E(G))$$

$$G_{A_1,\dots ,A_r}^{B_1,\dots ,B_r}=G\cup \left(\underset{i=1}{\overset{r}{\cup }}\{F\subset A_i\cup B_i:|F|=2\}\right).$$

$$S=K[\{x_i:i=1,\dots ,n\}\cup \{y_{i,j}:i=1,\dots ,r,j=1,\dots ,s_i\}]$$

$$x_1>\cdots >x_n>y_{1,1}>\cdots >y_{1,s_1}>\cdots >y_{r,1}>\cdots >y_{r,s_r}$$

$$\{j:A_j\cap G=\varnothing \}\subset \{j:A_j\cap G_1=\varnothing \}$$

$$F^{\prime }=\{\dots ,y_{l,k_l^{\prime }},\dots ,y_{m,k_m^{\prime }},\dots \}$$

$$F^{\prime \prime }=\{\dots ,y_{l,k_l},\dots ,y_{m,k_m},\dots \}$$

$$F^{\prime \prime \prime }=\{\dots ,y_{l,k_l^{\prime }},\dots ,y_{m,k_m},\dots \}$$

$$[n]=F_1\cup \dots \cup F_m$$

Full text

introduction

Let $G$ be the simple graph on the vertex set $[n]$ and the edge set $E(G)$. We identify the vertex $i$ with the variable $x_{i}$ and consider the polynomial ring $S=K[x_{1},\ldots,x_{n}]$ over an arbitrary field $K$. Let $I(G)=(x_{i}x_{j}:\{i,j\}\in E(G))\subset S$ denotes the edge ideal of $G$. We say that the graph $G$ is Cohen–Macaulay if $S/I(G)$ is Cohen–Macaulay. Using the Stanley–Reisner correspondence, one can associate to $G$ the simplicial complex $\Delta_{G}$, whose faces are the independent subsets of $[n]$ and whose Stanley–Reisner ideal coincides with the edge ideal of $G$, i.e. $I_{\Delta_{G}}=I(G)$.

According to [8], it is unlikely to have a general classification of Cohen–Macaulay graphs. Therefore, it is natural to study this question for some special classes of graphs. For instance, Villarreal in [11] gave classification of all Cohen–Macaulay trees. Herzog, Hibi and Zheng [8] classified all Cohen–Macaulay chordal graphs later Herzog and Hibi [6] did for all Cohen–Macaulay bipartite graphs.

The potential aim of this paper is to generate new graphs $G^{\prime}$ from a given graph $G$ with the property that $G^{\prime}$ is always Cohen–Macaulay. The classification and construction of Cohen–Macaulay graphs is one of the central problems and enjoys rich literature, for instance [2], [3], [4], [5], [10] and [13]. Here, we generalizes the notion of ”whiskering”, introduced by Villareal[11]. By a whisker to a vertex $x$, we mean to add a new vertex $y\not\in[n]$ and an edge $\{x,y\}$. Villareal proved that if we add whiskers to every vertex of any graph $G$, the resulting graph $G^{\prime}$ is always Cohen–Macaulay. Cook and Nagel [2], extends this work and defined clique-whiskering. They proved that $G^{\prime}$ obtained by clique-whiskering is always Cohen–Macaulay. Recently, J. Biermann and A. V. Tuyl [1] defined $s-$coloring $\chi$ on a simplicial complex $\Delta$ to construct a Cohen–Macaulay simplicial complex $\Delta_{\chi}$, generalizing the previous constructions for simplicial complexes. More recent, A. M. Liu and T. Wu [9] generalized this notion of clique-whiskering for obtaining families of sequentially Cohen–Macaulay graphs.

In this paper, we give a construction of hybrid graphs (see 2). The well–known constructions mentioned in [2] and [11] appear to be particular cases of our construction. Moreover, by using the construction discussed in [11], one may obtain one Cohen-Macualy Graph corresponding to a given simple graph $G$. Similarly, by using the construction, corresponding to each clique partition of vertex set, one may obtain one Cohen–Macaulay graph. But in our case, corresponding to any simple graph $G$ and a given clique partition of the vertex set of $G$, one may obtain infinitely many Cohen–Macaulay graphs. We call these new graphs as the hybrid graphs associated to $G$, here is our main result.

Theorem 0.1** (Theorem 2.5).**

Let $G$ be a graph on the vertex set $[n]$ and $A_{1},\ldots,A_{r}$ be a partition of $[n]$ into disjoint subsets such that $A_{i}$ is a clique in $G$. Let $B_{i}=\{y_{i,1},\ldots,y_{i,{s_{i}}}\}$ and $G^{\prime}=G_{{A_{1}},\ldots,{A_{r}}}^{{B_{1}},\ldots,{B_{r}}}$ be a hybrid graph of $G$ with respect to $B_{1},\ldots,B_{r}$. Let $\Delta^{\prime}$ be the independence complex of $G^{\prime}$ and $S=\mathbb{K}[\{x_{i}\colon\;i=1,\ldots,n\}\cup\{y_{i,j}\colon\;i=1,\ldots,r,\;j=1,\ldots,s_{i}\}]$ be the polynomial ring. Then,

• (i)

$\Delta^{\prime}$* is pure shellable of dimension $r-1$.*

• (ii)

The ring $\frac{S}{I\left(G^{\prime}\right)}$ is Cohen-Macaulay of dimension $r$.

As a quick consequence of the above theorem, we obtain following well-known results.

Corollary 0.2** (Villareal).**

Suppose $G$ is a graph and let $G^{\prime}$ be the graph obtained by adding a whisker at every vertex $v\in G$. Then the ideal $I(G^{\prime})$ is Cohen–Macaulay.

Corollary 0.3** (Cook and Nagel).**

Let $\pi=\{W_{1},\ldots,W_{t}\}$ be a clique vertex partition of $G$. Then $I(G^{\pi})$ is Cohen–Macaulay.

Finally, we give a complete characterization all Cohen–Macaulay chordal graphs by adding one more equivalent condition to a well known characterization of Cohen–Macaulay chordal graphs due to Herzog, Hibi and Zehng [8],

Corollary 0.4**.**

Let $K$ be a field and $G$ be a chordal graph on the vertex set $[n]$. Let $F_{1},\ldots,F_{m}$ be the facets of $\Delta(G)$ which admit a free vertex. Then the following are equivalent:

1. (1)

$G$* is Cohen–Macaulay;* 2. (2)

$G$* is Cohen–Macaulay over $K$;* 3. (3)

$G$* is unmixed;* 4. (4)

$[n]$* is the disjoint union of $F_{1},\ldots,F_{m}$.* 5. (5)

$G$* is a hybrid graph.*

1. Preliminaries

In this section, we recall some necessary definitions and results.

1.1. Simplicial complexes

A simplicial complex $\Delta$ on the vertex set $V=\{v_{1},\ldots,v_{n}\}$ is a collection of subsets of $V$ such that $\{v_{i}\}\in\Delta$ for all $i$ and, $F\in\Delta$ implies that all subsets of $F$ are also in $\Delta$. The elements of $\Delta$ are called faces and the maximal faces under inclusion are called facets of $\Delta$. We denote by $\mathcal{F}(\Delta)$ the set of facets of $\Delta$. We say that a simplicial complex is pure if all its facets have the same cardinality. The dimension of a face $F$ is $\dim F=|F|-1$, where $|F|$ denotes the cardinality of $F$. A simplicial complex is called pure if all its facets have the same dimension. The dimension of $\Delta$, $\dim(\Delta)$, is defined as:

[TABLE]

Given a simplicial complex $\Delta$ on the vertex set $\{v_{1},\ldots,v_{n}\}$. For $F\subseteq\{v_{1},\ldots,v_{n}\}$ let $\textbf{x}_{F}=\prod_{v_{i}\in F}{x_{i}}$, and let $\textbf{x}_{\varnothing}=1$. The non-face ideal or the Stanley-Reisner ideal of $\Delta$, denoted by $I_{\Delta}$, is an ideal of $S$ generated by square-free monomials $\textbf{x}_{F}$, where $F\not\in\Delta$.

Definition 1.1** (Shellable simplicial complexes).**

A simplicial complex $\Delta$ is called shellable if there is a total order of the facets of $\Delta$, say $F_{1},\ldots,F_{t}$, such that $\langle F_{1},\ldots,F_{i-1}\rangle\cap\langle F_{i}\rangle$ is generated by a non-empty set of maximal proper faces of $F_{i}$ for $2\leq i\leq t$. Any such order is called a shelling order of $\Delta$.

To say $F_{1},F_{2},\ldots,F_{t}$ is a shelling order of $\Delta$; it is equivalent to saying that for all $F_{i}$ and all $F_{j}<F_{i}$, there exists $x\in F_{i}\setminus F_{j}$ and $F_{k}<F_{i}$ such that $F_{i}\setminus F_{k}=\{x\}$.

The class of shellable simplicial complexes is important due to the following result.

Theorem 1.2**.**

[Theorem 8.2.6, [7]] A pure shellable simplicial complex is Cohen–Macaulay over any arbitrary field.

Graphs

Throughout in this paper, $G$ will denote a simple graph on $[n]$ vertices which means $G$ has no loops or multiple edges. Given a subset $W$ of $[n]$, we define the induced subgraph of $G$ on $W$ to be the subgraph $G_{W}$ on $W$ consisting of those edges $\{i,j\}\in E(G)$ with $\{i,j\}\subset W$. A walk of length $m$ in $G$ is a sequence of vertices $\{i_{0},\ldots,i_{m}\}$ such that $\{i_{j-1},i_{j}\}$ are edges in $G$ . A cycle of length $m$ is a closed walk $\{i_{0},\ldots,i_{m}\}$ in which $n\geq 3$ and the vertices $i_{1},\ldots,i_{m}$ are distinct. A graph $G$ on $[n]$ is connected if, for any two vertices $i$ and $j$, there is a walk between $i$ and $j$. A connected graph without cycles is said to be a tree. The complete graph $K_{m}$ has every pair of its $m$ vertices adjacent.

A chord of a cycle $C$ is an edge $\{i,j\}$ of $G$ such that $i$ and $j$ are vertices of $C$ with $\{i,j\}\not\in E(C)$. A graph is said to be chordal graph if each of its cycles of length $>3$ has a chord, obviously every tree is a chordal graph. A subset $C$ of $V(G)$ is called a clique of $G$ if induced subgraph $G_{C}$ is complete. The clique complex $\Delta(G)$ of a finite graph $G$ on $V(G)$ is a simplicial complex whose faces are the cliques of $G$. For a finite simple graph $G$ on $n$ vertices, one may associate a square-free monomial ideal $I(G)$ in $S=K[x_{1},\ldots,x_{n}]$, namely, edge ideal of $G$ defined as,

[TABLE]

We say that $G$ is Cohen–Macaulay over the field $K$, if the associated quotient ring $S/I(G)$ is Cohen–Macaulay. A subset $C\in V(G)$ is called a vertex cover of $G$ if $C\cap E\neq\emptyset$ for all $E\in E(G)$. A vertex cover $C$ is minimal if no proper subset of $C$ is a vertex cover of $G$ and if all minimal vertex covers of $G$ have same cardinality, then we say that $G$ is unmixed. All Cohen–Macaulay graphs are unmixed but not vice versa. The following result characterizes all Cohen–Macaulay chordal graphs.

Theorem 1.3**.**

[8, Theorem 2.1]** Let $K$ be a field and $G$ be a chordal graph on the vertex set $[n]$. Let $F_{1},\ldots,F_{m}$ be the facets of $\Delta(G)$ which admit a free vertex. Then the following are equivalent:

1. (1)

$G$* is Cohen–Macaulay;* 2. (2)

$G$* is Cohen–Macaulay over $K$;* 3. (3)

$G$* is unmixed;* 4. (4)

$[n]$* is the disjoint union of $F_{1},\ldots,F_{m}$.*

2. Main result

The following definition is essential to understand the underlined construction.

Definition 2.1**.**

Let $G$ be a graph on the vertex set $[n]$ and $A_{1},\ldots,A_{r}$ be a partition of $[n]$ into disjoint subsets such that $A_{i}$ is a clique in $G$($A_{i}$ can be empty set). For each $i=1,\ldots,r$, let $B_{i}=\{y_{i,1},\ldots,y_{i,{s_{i}}}\}$ be a non-empty set. Define the graph $G_{{A_{1}},\ldots,{A_{r}}}^{{B_{1}},\ldots,{B_{r}}}$ as follows:

[TABLE]

We call the graph $G_{{A_{1}},\ldots,{A_{r}}}^{{B_{1}},\ldots,{B_{r}}}$, the hybrid graph of $G$ with respect to $B_{1},\ldots,B_{r}$.

Remark 2.2**.**

Corresponding to each partition of the vertex set, we have infinitely many choices to choose $B_{i}$’s thus there are infinitely many hybrid graphs corresponding to any given graph.

Let $G$ be a graph and $G^{\prime}=G_{{A_{1}},\ldots,{A_{r}}}^{{B_{1}},\ldots,{B_{r}}}$ be the hybrid graph of $G$ with respect to $B_{1},\ldots,B_{r}$. Let $\Delta$ and $\Delta^{\prime}$ be the independence complexes of $G$ and $G^{\prime}$ respectively. Let

[TABLE]

be the polynomial ring. Let us define the ordering on the variables:

[TABLE]

As the facets of $\Delta^{\prime}$ are maximal independent sets in $G^{\prime}$, it is easy to see that the induced subgraph of $G^{\prime}$ on $A_{i}\cup B_{i}$ is a complete graph. Thus in an independent set, we can select at most one element from $A_{i}\cup B_{i}$ for all $i$. In other words, if $T$ be a facet of $\Delta^{\prime}$, then $|T\cap(A_{i}\cup B_{i})|=1$ for all $i$. Let $F=T\cap[n]$, then $F$ will be an independent set in $G$ and hence a face of $\Delta$. Let us consider $B=\cup_{i=1}^{r}B_{i}$ and $F^{\prime}=T\cap B$, then $T=F\cup F^{\prime}$ and $F\cap F^{\prime}=\emptyset$. It is easy to note that $F^{\prime}=\bigcup_{j:A_{j}\cap F=\emptyset}\{y_{j,k_{j}}\}$ for some $1\leq k_{j}\leq s_{j}$. Let us record this simple observation in the following proposition.

Proposition 2.3**.**

The independence complex $\Delta^{\prime}$ of the hybrid graph $G^{\prime}$ is pure and every facet of $\Delta^{\prime}$ is of the form $F\cup F^{\prime}$, where $F$ is a face of $\Delta$ and $F^{\prime}=\bigcup_{j:A_{j}\cap F=\emptyset}\{y_{j,k_{j}}\}$ for some $1\leq k_{j}\leq s_{j}$.

Note that there are $\prod_{j:A_{j}\cap F=\emptyset}|s_{j}|$ choices for $F^{\prime}$, thus corresponding to each face $F$ of $\Delta$, there is a block of facets of $\Delta^{\prime}$. Here, we explain this fact through the following example.

Example 2.4**.**

Consider the graph $G$ with vertex set $V(G)=\{1,2,3,4\}$ and edge set $E(G)=\{\{1,2\},\{1,3\},\{2,3\},\{2,4\},\{3,4\}\}$. Consider the vertex partition as $V(G)=\{1\}\cup\{2\}\cup\{3\}\cup\{4\}$ and take $B_{1}=\{5,6\},B_{2}=\{7\},B_{3}=\{8\},B_{4}=\{9,10,11\}$ then $G^{\prime}=G_{{A_{1}},\ldots,{A_{4}}}^{{B_{1}},\ldots,{B_{4}}}$ will be,

1$$3$$2$$4$$\bullet$$\bullet$$\bullet$$\bullet$$G$$1$$3$$2$$4$$5$$6$$7$$8$$9$$10$$11$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$G^{\prime}

The facets of $\Delta^{\prime}$ are shown in Table A,

If we consider another partition of vertex set as $V(G)=\{1,3\}\cup\{2\}\cup\{4\}$ and $B_{1}=\{5\},B_{2}=\{6\},B_{3}=\{7,8,9\}$, then $G^{\prime}=G_{{A_{1}},\ldots,{A_{3}}}^{{B_{1}},\ldots,{B_{3}}}$ will be,

1$$3$$2$$4$$5$$6$$7$$8$$9$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$G^{\prime}

In this case, Table B describes the facets of $\Delta^{\prime}$,

Lastly, assume the partition of vertex set as, $V(G)=\{1,2,3\}\cup\{4\}$ and $B_{1}=\{5\},B_{2}=\{6,7\}$, then $G^{\prime}=G_{{A_{1}},\ldots,{A_{2}}}^{{B_{1}},\ldots,{B_{2}}}$ will be,

1$$3$$2$$4$$5$$6$$7$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$G^{\prime}

See the table C for complete list of facets of $\Delta^{\prime}$ in this case.

Now, we present the main result of this paper.

Theorem 2.5**.**

Let $G$ be a graph on the vertex set $[n]$ and $A_{1},\ldots,A_{r}$ be a partition of $[n]$ into disjoint subsets such that $A_{i}$ is a clique in $G$. Let $B_{i}=\{y_{i,1},\ldots,y_{i,{s_{i}}}\}$ and $G^{\prime}=G_{{A_{1}},\ldots,{A_{r}}}^{{B_{1}},\ldots,{B_{r}}}$ be a hybrid graph of $G$ with respect to $B_{1},\ldots,B_{r}$. Let $\Delta^{\prime}$ be the independence complex of $G^{\prime}$ and $S=K[\{x_{i}\colon\;i=1,\ldots,n\}\cup\{y_{i,j}\colon\;i=1,\ldots,r,\;j=1,\ldots,s_{i}\}]$ be the polynomial ring. Then,

• (i)

$\Delta^{\prime}$* is pure shellable of dimension $r-1$.*

• (ii)

The ring $S/{I\left(G^{\prime}\right)}$ is Cohen-Macaulay of dimension $r$.

Proof.

Lemma 2.3 guarantees that the independence complex $\Delta^{\prime}$ of $G^{\prime}$ is pure and has dimension $r-1$, thus it is sufficient to show that $\Delta^{\prime}$ is shellable. From above discussion, we know that corresponding to every face of $\Delta$, there is a block of facets of $\Delta^{\prime}$. Now we define an order the facets of $\Delta^{\prime}$ to show that $\Delta^{\prime}$ is shellable.

We order the faces of $\Delta^{\prime}$ in terms of increasing dimensions. If two faces have same dimension, we order them by (2). Thus, associated to each face $F$ of $\Delta^{\prime}$, we consider the block associated to $F$, ordered as in (2).

Let us assume $S$ and $T$ be two distinct facets of $\Delta^{\prime}$ with $S<T$, here arises two cases:

Case:1. When $S$ and $T$ belongs to different blocks We can write $S=F\cup F^{\prime}$ and $T=G\cup G^{\prime}$ where $F,G$ are different faces of $\Delta$ and $F^{\prime}=\bigcup_{j:A_{j}\cap F=\emptyset}\{y_{j,k_{j}}\}$ for some $1\leq k_{j}\leq s_{j}$, $G^{\prime}=\bigcup_{j:A_{j}\cap G=\emptyset}\{y_{j,p_{j}}\}$ for some $1\leq p_{j}\leq s_{j}$.

As $F\neq G$, we must have some $x_{t}\in G\setminus F$. Let $G_{1}=G\setminus\{x_{t}\}$, then $G_{1}$ will also be a face of $\Delta$. As $G_{1}\subset G$ so

[TABLE]

thus $G^{\prime}\subset G_{1}^{\prime}:=\bigcup_{j:A_{j}\cap G=\emptyset}\{y_{j,p_{j}}\}$ for some $1\leq p_{j}\leq s_{j}$. In fact, if $x_{t}\in A_{p}$ then $G_{1}^{\prime}=G^{\prime}\cup\{y_{p,k_{p}}\}$ for some $1\leq k_{p}\leq s_{p}$. Let us take $T_{1}=G_{1}\cup G_{1}^{\prime}$, then $T_{1}<T$ with $T\setminus T_{1}=\{x_{t}\}$.

Case:2. When $S$ and $T$ belongs to same block In this case, $S=F\cup F^{\prime}$ and $T=F\cup F^{\prime\prime}$. As $S\neq T$, we have $F^{\prime}\neq F^{\prime\prime}$ and $T\setminus S\neq\emptyset$. Let $l$ be the least number, such that $y_{l,k_{l}}\in F^{\prime\prime}\setminus F^{\prime}$, thus $y_{l,k_{l}}\in F^{\prime\prime}$ and $y_{l,k_{l}}\notin F^{\prime}$ which further implies the existence of a $y_{l,k_{l}^{\prime}}\in F^{\prime}$ for some $1\leq k_{l}^{\prime}\neq k_{l}\leq s_{l}$ with $y_{l,k_{l}^{\prime}}<y_{l,k_{l}}$ as $S<T$.

If $y_{l,k_{l}}$ is the only vertex in $F^{\prime\prime}\setminus F^{\prime}$, we are done, otherwise suppose $y_{m,k_{m}}\in F^{\prime\prime}\setminus F^{\prime}$. We have ordered $F^{\prime}$ and $F^{\prime\prime}$ as defined in (2) and as we have assumed $l$ to be least such number, thus the first $l-1$ components in $F^{\prime}$ and $F^{\prime\prime}$ will be the same. Thus $F^{\prime}$ and $F^{\prime\prime}$ will be of the form,

[TABLE]

[TABLE]

where $y_{m,k_{m}^{\prime}}\neq y_{m,k_{m}}$. Let us consider,

[TABLE]

and suppose $T_{1}=F\cup F^{\prime\prime\prime}$, then $T_{1}$ will be a facet of $\Delta^{\prime}$ by Lemma 2.3 with $T_{1}<T$. Note that $y_{m,k_{m}}\notin F^{\prime\prime}\setminus F^{\prime\prime\prime}$ and $y_{l,k_{l}}\in F^{\prime\prime}\setminus F^{\prime\prime\prime}$, thus $y_{m,k_{m}}\notin T\setminus T_{1}$ and $y_{l,k_{l}}\in T\setminus T_{1}$. If $y_{l,k_{l}}$ is the only element in $T\setminus T_{1}$, we are done, otherwise we shall repeat the same process. As we have finite element, this process will terminate in finite steps and we shall have a $T_{i}<T$ such that $T\setminus T_{i}=\{y_{l,k_{l}}\}$, as required.

∎

Here, we give the descriptive definition of a hybrid graph.

Definition 2.6**.**

A graph $G$ is said to be hybrid if there exists some graph $H$ such that $G$ is a hybrid graph associated to $H$.

The following result shows that the whiskering of a graph, given by Villarreal in [11] is a particular case of our construction.

Corollary 2.7** (Villareal).**

Suppose $G$ is a graph and let $G^{\prime}$ be the graph obtained by adding a whisker at every vertex $v\in G$. Then the ideal $I(G^{\prime})$ is Cohen–Macaulay.

Proof.

Suppose $V(G)=\{v_{1},\ldots,v_{n}\}$ and consider the trivial clique partition of $V(G)$ into singleton sets as, $V(G)=\{v_{1}\}\cup\ldots\cup\{v_{n}\}$. If we take $B_{i}=\{y_{i}\}$ for all $1\leq i\leq n$, then $G^{\prime}=G_{{A_{1}},\ldots,{A_{n}}}^{{B_{1}},\ldots,{B_{n}}}$, thus $\Delta^{\prime}$ is pure and $I(G^{\prime})$ is Cohen–Macaulay by Theorem 2.5. ∎

R. Woodroof [13] and D. Cook and U. Nagel [2] generalized the the concept of whiskering, and defined the terms clique-whiskering and fully clique-whiskering respectively. Recall from [2] that a vertex clique-partition $\pi$ of a graph $G$ is a partition $\pi=\{W_{1},\ldots,W_{t}\}$ of $V(G)$ such that each subgraph induced on $W_{i}$ is a nonempty clique, see [13]. The clique-whiskering is particular case of our construction.

Corollary 2.8** (Theorem 3.3, [2]).**

Let $\pi=\{W_{1},\ldots,W_{t}\}$ be a vertex clique-partition of $G$. Then $I(G^{\pi})$ is Cohen–Macaulay.

Proof.

If $\pi=\{W_{1},\ldots,W_{t}\}$ be a clique vertex partition of $G$. Let us take $B_{i}=\{y_{i}\}$, singleton sets for all $i$. Then $G^{\pi}=H_{{W_{1}},\ldots,{W_{t}}}^{{B_{1}},\ldots,{B_{t}}}$ is a particular hybrid graph associated to $G$ and hence Cohen–Macaulay by Theorem 2.5. ∎

Herzog et al. [8] characterize Cohen–Macaulay chordal graphs. One can see easily that every Cohen–Macaulay chordal graph is in fact a hybrid graph associate to some graph. Thus, it characterizes all Cohen–Macaulay chordal graphs. By a free vertex in a simplicial complex, we mean a vertex which belongs to exactly on facet of the simplcial complex.

Corollary 2.9**.**

Let $K$ be a field and $G$ be a chordal graph on the vertex set $[n]$. Let $F_{1},\ldots,F_{m}$ be the facets of $\Delta(G)$ which admit a free vertex. Then the following are equivalent:

1. (1)

$G$* is Cohen–Macaulay;* 2. (2)

$G$* is Cohen–Macaulay over $K$;* 3. (3)

$G$* is unmixed;* 4. (4)

$[n]$* is the disjoint union of $F_{1},\ldots,F_{m}$.* 5. (5)

$G$* is a hybrid graph.*

Proof.

$(4)\Rightarrow(5)$ As,

[TABLE]

where $F_{i}$ are cliques of $\Delta(G)$ containing free vertices. Let $A_{i}$ and $B_{i}$ denote the non-free and free vertices of $F_{i}$ respectively. Let $A=\cup_{i=1}^{r}A_{i}$ and $H:=G_{A}$ be the induced graph. Then $G=H_{{A_{1}},\ldots,{A_{m}}}^{{B_{1}},\ldots,{B_{m}}}$.

$(5)\Rightarrow(1)$ Follows from Theorem 2.5. ∎

In particular, one obtains

Corollary 2.10** (Corollary 6.3.5, [12]).**

If $G$ is a tree then the following are equivalent:

1. (1)

$G$* is Cohen–Macaulay.* 2. (2)

$G$* is unmixed.* 3. (3)

$G$* is a hybrid graph.*

Remark 2.11**.**

In small graphs, diagrammatically it is quite easy to check whether a given graph is hybrid or not. It is pertinent to mention that there exist graphs that are not hybrid but still Cohen–Macaulay. For example, $5-$cycle is not hybrid but Cohen–Macaulay.

Acknowledgement

We would like to thank the Higher Education Commission, Pakistan and Abdus Salam School of Mathematical Sciences, Lahore Pakistan for supporting and facilitating this research. We would also like to thank Ali Akbar Yazdan Pour for valuable remarks during his visit to the Abdus Salam School of Mathematical Sciences and for proposing the name ”hybrid graph”.