Algebraic study on Cameron–Walker graphs
Abstract.
Let be a finite simple graph on and the edge ideal of , where is the polynomial ring over a field . Let denote the maximum size of matchings of and that of induced matchings of . It is known that , where is the Castelnuovo–Mumford regularity of . Cameron and Walker succeeded in classifying the finite connected simple graphs with . We say that a finite connected simple graph is a Cameron–Walker graph if and if is neither a star nor a star triangle. In the present paper, we study Cameron–Walker graphs from a viewpoint of commutative algebra. First, we prove that a Cameron–Walker graph is unmixed if and only if is Cohen–Macaulay and classify all Cohen–Macaulay Cameron–Walker graphs. Second, we prove that there is no Gorenstein Cameron–Walker graph. Finally, we prove that every Cameron–Walker graph is sequentially Cohen–Macaulay.
Key words and phrases:
finite graph, edge ideal, Cameron–Walker graph, unmixed graph, Cohen–Macaulay graph, Gorenstein graph, sequentially Cohen–Macaulay graph2010 Mathematics Subject Classification:
05E40, 13H10Introduction
Recently, edge ideals of finite simple graphs have been studied by many authors from viewpoints of computational commutative algebra and combinatorics; see [8, 14, 17], and their references.
Let be a vertex set and a finite simple graph on with its edge set. (A simple graph is a graph with no loop and no multiple edge.) Let denote the polynomial ring in variables over a field . The edge ideal ([10, p. 156]) of is the monomial ideal of generated by those monomials with , viz.,
One of the research topics on is the computation of the Castelnuovo–Mumford regularity ([10, p. 48]) of in terms of the invariants of .
Recall that a subset of is a matching of if, for and belonging to with , one has . The matching number of is the maximum size of matchings of . A matching of is called an induced matching of if, for and belonging to with , there is no with and . Let denote the maximum size of induced matchings of .
For example, if , a complete graph on , then and . If (), a complete bipartite graph with vertex partition , then and . If is the Petersen graph, then and .
It is known ([11, Lemma 2.2] and [9, Theorem 6.7]) that
One has for, e.g., chordal graphs, unmixed bipartite graphs and sequentially Cohen–Macaulay bipartite graphs ([9, 12, 15]; see also [7, 13, 19, 20]).
Cameron and Walker ([3, Theorem 1]) gave a classification of the finite connected simple graphs with , although there is a mistake; see Remark 0.1 below. By modifying their result slightly, we see that a finite connected simple graph satisfies if and only if is one of the following graphs:

a star;

a star triangle;

a finite graph consisting of a connected bipartite graph with vertex partition such that there is at least one leaf edge attached to each vertex and that there may be possibly some pendant triangles attached to each vertex .
Here a star triangle is a graph joining some triangles at one common vertex, e.g., the graph on with the edges , , , , , , , , is a star triangle. A leaf is a vertex of degree and a leaf edge is an edge meeting a leaf. Also a pendant triangle is a triangle whose two vertices have degree and the rest vertex has degree more than . We say that a finite connected simple graph is a Cameron–Walker graph if and if is neither a star nor a star triangle. For example,
Figure 1 (Cameron–Walker graph)
is a Cameron–Walker graph.
Remark 0.1.
The original result of Cameron–Walker [3, Theorem 1] claimed “a triangle” instead of “a star triangle” in the above classification.
The reason why we claimed differently is that the “only if” part of [3, Theorem 1] is a little wrong; concretely, in the second paragraph in the proof of Theorem 1 (Only if) [3, p. 54]. Their argument asserted that when we delete all pendant triangles of , we get a connected bipartite graph . However is possibly an isolated vertex; this case was forgotten, and in such case, should be a star triangle. Indeed a star triangle also satisfies .
In the present paper, we study Cameron–Walker graphs from a viewpoint of commutative algebra. One of the main problems is which Cameron–Walker graphs are Cohen–Macaulay.
Let be a finite simple graph on . A vertex cover of is a subset of such that for all . A vertex cover is called minimal if no proper subset of is a vertex cover of . A finite simple graph is called unmixed if all minimal vertex covers of have the same cardinality. A finite simple graph is Cohen–Macaulay if is Cohen–Macaulay. Every Cohen–Macaulay graph is unmixed ([10, Lemma 9.1.10]). A finite simple graph is called vertex decomposable, shellable or sequentially Cohen–Macaulay if the simplicial complex is vertex decomposable, shellable or sequentially Cohen–Macaulay, respectively ([10, p. 144]), where is the complementary graph of ([10, p. 153]) and is the clique complex ([10, p. 155]) of . Every vertex decomposable graph is shellable ([2, Theorem 11.3]) and every shellable graph is sequentially Cohen–Macaulay ([10, Corollary 8.2.19]). Note that is unmixed if and only if is pure.
In Theorem 1.3, we prove that, for a Cameron–Walker graph , the following five conditions are equivalent:

is unmixed;

is Cohen–Macaulay;

is unmixed and shellable;

is unmixed and vertex decomposable;

consists of a connected bipartite graph with vertex partition such that there is exactly one leaf edge attached to each vertex and that there is exactly one pendant triangle attached to each vertex .
When is Cohen–Macaulay, we call the Cohen–Macaulay type of the Cohen–Macaulay type of . A finite simple graph is called Gorenstein if is Gorenstein. Note that is Gorenstein if and only if is Cohen–Macaulay with Cohen–Macaulay type 1. We also consider the problem which Cameron–Walker graphs are Gorenstein. The answer is that there is no Gorenstein Cameron–Walker graph. We draw this conclusion by computing the Cohen–Macaulay types of all Cohen–Macaulay Cameron–Walker graphs (Theorem 2.1).
In addition, we also consider the problem which Cameron–Walker graphs are sequentially Cohen–Macaulay. In Theorem 3.1, we prove that every Cameron–Walker graph is sequentially Cohen–Macaulay. Actually, it is vertex decomposable and thus, shellable. We also give a shelling for a Cameron–Walker graph whose supporting connected bipartite graph is a complete bipartite graph (Proposition 3.6).
Acknowledgments: The authors are grateful to Seyed Amin Seyed Fakhari, Russ Woodroofe and Siamak Yassemi for giving them helpful comments on the first version of this paper. The authors would also like to appreciate the anonymous referee for suggesting the crucial ideas of the proof of Theorem 2.1.
The second author is partially supported by JSPS Research Fellowship for Young Scientists. The third author is partially supported by JSPS GrantinAid for Young Scientists (B) 24740008.
1. Cohen–Macaulay Cameron–Walker graphs
In this section, we classify all Cohen–Macaulay Cameron–Walker graphs. It will turn out that such graphs are of the form that a connected bipartite graph each of whose vertex has exactly one leaf edge () or exactly one pendant triangle (). We will first prove that more generally, for a finite simple graph , the graph obtained by attaching complete graphs to each vertex of is unmixed and vertex decomposable, in particular, Cohen–Macaulay. This is a generalization of the result by Villarreal [16, Proposition 2.2] and Dochtermann and Engström [6, Theorem 4.4].
We recall the definition of a vertex decomposable simplicial complex. For a simplicial complex and its vertex , let and . A simplicial complex is called vertex decomposable if is a simplex, or if there exists a vertex of such that

and are vertex decomposable and

no face of is a facet of .
We also recall the definition of a shellable simplicial complex. A simplicial complex is called shellable if all of its facets can be listed
in such a way that
is a pure simplicial complex of dimension for every . Here . When this is the case, we call the order a shelling of .
For simplicial complexes, the following implications are known:

vertex decomposable shellable sequentially Cohen–Macaulay;

pure and vertex decomposable pure and shellable Cohen–Macaulay.
Let be a graph on the vertex set . For , we denote by , the induced subgraph of on . For a vertex , we use notation instead of . For , let (or ) denote the neighbourhood of in and let .
Let be a finite simple graph on a vertex set . Villarreal [16, Proposition 2.2] proved that the graph obtained from by adding a whisker to each vertex is Cohen–Macaulay. Dochtermann and Engström [6, Theorem 4.4] proved that such a graph is unmixed and vertex decomposable. Adding a whisker to each vertex is the same as saying that attaching the complete graph to each vertex. We generalize the above results as follows.
Theorem 1.1.
Let be a finite simple graph on a vertex set . Let be integers. Then the graph obtained from by attaching the complete graph to for is unmixed and vertex decomposable. In particular, is shellable and Cohen–Macaulay.
Before proving Theorem 1.1, we recall the result by Cook and Nagel [4, Theorem 3.3], which is another generalization of [16, Proposition 2.2] and [6, Theorem 4.4]. A clique vertexpartition of is a set of disjoint (possibly empty) cliques of such that their disjoint union forms . For a clique vertexpartition , let denotes the graph on the vertex set with the edge set .
Lemma 1.2 (Cook and Nagel [4]).
Let be a clique vertexpartition of . Then is unmixed and vertex decomposable.
Now we prove Theorem 1.1.
Proof of Theorem 1.1.
For , let be the vertex set of the attached complete graph , where . We consider the graph obtained from by attaching the complete graph to for whose vertex set is , that is, . Then has a clique vertexpartition , where , and . By Lemma 1.2, we conclude that is unmixed and vertex decomposable, as desired. ∎
Now we classify all Cohen–Macaulay Cameron–Walker graphs. Actually we have the following theorem for Cameron–Walker graphs.
Theorem 1.3.
For a Cameron–Walker graph , the following five conditions are equivalent:

is unmixed.

is Cohen–Macaulay.

is unmixed and shellable.

is unmixed and vertex decomposable.

consists of a connected bipartite graph with vertex partition such that there is exactly one leaf edge attached to each vertex and that there is exactly one pendant triangle attached to each vertex .
Proof.
follows from Theorem 1.1. As mentioned above, and follow. is well known. We prove .
Since is a Cameron–Walker graph, consists of a connected bipartite graph with vertex partition such that there is at least one leaf edge attached to each vertex and that there may be possibly some pendant triangles attached to each vertex . Assume that there are leaves in total and total pendant triangles. Also assume that out of each vertex (), there is at least one triangle and there is no triangle out of each vertex .
The following three subsets of the vertex set of are minimal vertex covers of :

the subset of the vertex set of the connected bipartite graph and two vertices of degree of each pendant triangle;

the subset of the vertex set of the connected bipartite graph, all leaves and exactly one vertex of degree of each pendant triangle;

the subset of the vertex set of the connected bipartite graph and exactly one vertex of degree of each pendant triangle.
The cardinalities are (i) ; (ii) ; (iii) . Since is unmixed, these are equal. By (i) and (iii), we have ; it follows that there is just one triangle out of each vertex . By (i) and (ii), we have , i.e., . Since there is at least one leaf out of each vertex , we have . On the other hand . Therefore we have . Then we have the desired assertion. ∎
2. Gorenstein Cameron–Walker graphs
In this section, we consider the problem which Cohen–Macaulay Cameron–Walker graph is Gorenstein. In order to attack the problem, we compute the Cohen–Macaulay type of a Cohen–Macaulay Cameron–Walker graph.
Let be a graph on the vertex set . We say that a subset is an independent set if no two vertices in are adjacent in .
The following theorem is the main result in this section.
Theorem 2.1.
Let be a Cohen–Macaulay Cameron–Walker graph with pendant triangles. Then the Cohen–Macaulay type of is equal to . In particular, there is no Gorenstein Cameron–Walker graph.
Proof.
Let be a Cohen–Macaulay Cameron–Walker graph. Then is a graph described in Theorem 1.3 (5); let be the vertex partition of the supporting connected bipartite graph of , the leaf of attached to and the two vertices of degree 2 which form the pendant triangle with . Then is an ideal of . (We identify each vertex of with the variable of .) Since the cardinality of the minimal vertex cover is equal to , it follows from the proof of Theorem 1.3 that . Thus, we obtain that .
Consider the sequence
of elements of . Let us consider the polynomial ring . Set
Also let be the induced subgraph of on . Then is an ideal of . Modulo the sequence x, one has , and thus . Since and the sequence x consists of elements, x is a linear system of parameter. Hence x is a regular sequence of because is Cohen–Macaulay.
The Cohen–Macaulay type of coincides with ([10, Proposition A.6.1]). Therefore, we compute . Set . Since , where , a set of elements , where is a monomial in , such that and forms a basis for the vector space . By Lemma 2.2 below, one can compute by counting the maximal independent sets of .
A maximal independent set of is uniquely determined by the intersection . In fact, for a subset of , there exists a unique maximal independent set of : . Hence it follows that there are exactly maximal independent sets of , as desired. ∎
Lemma 2.2.
With the same notation as in the proof of Theorem 2.1, there is a onetoone correspondence between each monomial in such that and and each maximal independent set of .
Proof.
Take a monomial in such that and . Then are distinct elements of because . Moreover, since , it follows that is an independent set of . Now we prove that is maximal. Take with . Since , we have . This means that there is such that is an edge of . Hence, is a maximal independent set.
On the other hand, take a maximal independent set of . Then the corresponding squarefree monomial does not belong to . Also take . If , then . Otherwise, because is a maximal independent set of . Therefore . ∎
3. Sequentially Cohen–Macaulayness of Cameron–Walker graphs
In this section, we prove that every Cameron–Walker graph is sequentially Cohen–Macaulay. Actually, we prove that it is vertex decomposable and thus, shellable. We also provide a shelling for a Cameron–Walker graph whose supporting connected bipartite graph is a complete bipartite graph.
The following theorem is the main result in this section.
Theorem 3.1.
Every Cameron–Walker graph is vertex decomposable, in particular, shellable and sequentially Cohen–Macaulay.
To prove Theorem 3.1, we use the following.
Lemma 3.2 (Woodroofe [18, Theorem 1]).
Let be a graph with no chordless cycles of length other than or . Then is vertex decomposable.
The following is a rephrase of the definition of the vertex decomposability of graphs.
Lemma 3.3 (cf. [18, Lemma 4]).
A finite simple graph is vertex decomposable if and only if is totally disconnected (i.e., has no edge), or if there is a vertex of such that

and are vertex decomposable and

no independent set in is a maximal independent set in .
Now we prove Theorem 3.1.
Proof of Theorem 3.1.
Let be a Cameron–Walker graph on the vertex set whose supporting connected bipartite graph has a vertex partition such that there is at least one leaf edge attached to each vertex and that there may be possibly some pendant triangles attached to each vertex . We prove the assertion by induction on .
When , the supporting bipartite graph is a star. Thus contains no cycle except for pendant triangles. Hence, by Lemma 3.2, is vertex decomposable.
Assume that . We take the vertex and consider the graphs and . Since a union of an independent set of and a leaf adjacent to is an independent set of , the condition (ii)’ in Lemma 3.3 is satisfied with . Therefore, in order to prove that is vertex decomposable, it is sufficient to prove that and are vertex decomposable by Lemma 3.3. Also, in order to prove that and are vertex decomposable, it is sufficient to prove that all of their connected components are vertex decomposable (see [18, Lemma 20]). Each of connected components of and is one of the following four graphs:

an isolated vertex;

an edge;

a triangle;

a Cameron–Walker graph whose supporting connected bipartite graph has a vertex partition with and .
Clearly, the first three graphs (i), (ii) and (iii) are vertex decomposable. Moreover, by the inductive hypothesis, the graph (iv) is also vertex decomposable, as desired. ∎
Remark 3.4.
Next we consider the Cameron–Walker graphs whose supporting connected bipartite graphs are complete bipartite graphs. We provide a shelling for these graphs though we have already known that these are shellable by Theorem 3.1.
Let be a Cameron–Walker graph on the vertex set whose supporting connected bipartite graph is with vertex partition such that out of each vertex there is at least one leaf and out of each vertex there may be a pendant triangle. Assume that there are leaves in total and out of each vertex there are leaves so that . Furthermore assume that there are total pendant triangles and that out of each vertex there are triangles so that . Let be the number of vertices in with at least one adjacent pendant triangle. Then by relabelling we may assume for . We will use notation by referring to vertices of as in Figure 2.
Figure 2 (the vertices of )
Since the vertices form a complete bipartite graph , it is easy to check that the facets of the clique complex are of one of the following two forms; note that a clique of is equivalent to an independent set in :

where and for each ;

where and for each .
Note that each defines a family of facets of the first form, which we will denote . Similarly, each defines a family of the second form, . For a fixed note that all facets contain and so is determined completely by the in . Thus we have
where Similarly, we have for
We first order facets within (resp. ). Consequently, we have that the simplicial complex with facets (resp. ) is shellable. To do this, we determine a total order on which will induce the order for our shelling. Let be the number of ’s in and denote the entry of . We say that if and only if either of the following two conditions hold

, or

, , and where is the first entry and differ when reading from the left.
For example
Lemma 3.5.
For each fixed ,
is a shelling for the simplicial complex with facets , where for all .
A similar order on gives a shelling for the simplicial complex with facets for each fixed .
Proof.
We first note that all have the same dimension, denoted by . To show the assertion, we need merely verify that
is a pure simplicial complex of dimension .
Suppose that such that . Let be a subset of such that if and only if . Note that . Clearly for some otherwise we would not have . Let be the index such that and agrees with in every other entry. Then and
A similar argument shows the assertion for . ∎
Now we define an order on the all facets of .
We define an order on the families of facets, with (respectively with ) by imposing a total order on the subsets (respectively ). For any subsets , we say that if and only if either or and the first nonzero entry of the vector
reading from left to right is a . Here we take to be the standard basis vectors. For example
We define a similar order for all subsets .
Proposition 3.6.
Let be a Cameron–Walker graph whose supporting connected bipartite graph is a complete bipartite graph. Then
is a shelling of , where the order of the indexing sets is given by the order defined above and the order within each family or is given by the shelling order in Lemma 3.5.
Proof.
Let with and let and such that . We then proceed as in the proof of Lemma 3.5 to show that there exists with such that and . Since , one has , i.e., . Choose and set . Thus we have . Take a vector as an extension of . Then Furthermore, our choice of guarantees that A similar argument holds for with , , and . In this case we consider where for some and proceed as above.
Finally, let , , , and such that . First we observe that for all . Similarly for all . If , i.e., for some , then we consider where the subscript for denotes the same sign vector as that of . (Here we note .) Then . If , then let such that . It is then easy to check that where and furthermore . ∎
We finish the paper by noting that we can describe the projective dimension for a Cameron–Walker graph on in terms of the graph. Let be a minimum cardinality of independent sets with . Since a Cameron–Walker graph is sequentially Cohen–Macaulay by Theorem 3.1, we have the following corollary by the results of Dao and Schweig [5].
Corollary 3.7 ([5, Corollary 5.6 and Remark 5.7]).
Let be a Cameron–Walker graph on the vertex set . Then .
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