Bounds for the boxicity of Mycielski graphs

Bounds for the boxicity of Mycielski graphs

111Interdisciplinary Faculty of Science and Engineering, Shimane University, Shimane 690-8504, Japan.    E-mail address: kamibeppu@riko.shimane-u.ac.jp This work was supported by Grant-in-Aid for Young Scientists (B), No.25800091.  Akira Kamibeppu
Abstract

A box in Euclidean -space is the Cartesian product , where is a closed interval on the real line. The boxicity of a graph , denoted by , is the minimum nonnegative integer such that can be isomorphic to the intersection graph of a family of boxes in Euclidean -space.
Mycielski [10] introduced an interesting graph operation that extends a graph to a new graph , called the Mycielski graph of . In this paper, we observe behavior of the boxicity of Mycielski graphs. The inequality holds for a graph , and hence we are interested in whether the boxicity of the Mycielski graph of is more than that of or not. Here we give bounds for the boxicity of Mycielski graphs: for a graph with universal vertices, the inequalities hold, where is the edge clique cover number of the complement . Further observations determine the boxicity of the Mycielski graph , if has no universal vertices or odd universal vertices and satisfies .
We also present relations between the Mycielski graph and its analogous ones and in the context of boxicity, which will encourage us to calculate the boxicity of or .

Keywords: boxicity; chromatic number; cointerval graph; edge clique cover number; Mycielski graph
2010 Mathematics Subject Classification: 05C62, 05C76

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1 Introduction

The notion of boxicity of graphs was introduced by Roberts [12]. It has applications in some research fields, like niche overlap in ecology (see [13, 14]) and fleet maintenance in operations research (see [11]). Roberts [12] proved that the maximum boxicity of graphs with vertices is (also see [6]), where denotes the largest integer at most . Cozzens [5] proved that the task of computing boxicity of graphs is NP-hard. Some researchers have attempted to calculate or bound boxicity of graphs with special structure. Roberts [12] showed that the boxicity of a complete -partite graph is the number of which is at least 2. Scheinerman [15] proved that the boxicity of outer planar graphs is at most 2. Thomassen [16] proved that the boxicity of planar graphs is at most 3. Cozzens and Roberts [6] investigated the boxicity of split graphs. As Chandran et al. [1] say, not much is known about boxicity of most of the well-known graph classes. They proved that the boxicity of a graph is at most , where is the treewidth of , and presented upper bounds for chordal graphs, circular arc graphs, AT-free graphs, co-comparability graphs, and permutation graphs. Recently, Chandran et al. [2] found the following relation between boxicity and chromatic number. 

Theorem 1.1 ([2], Theorem 6.1).

Let be a graph with vertices. If for , the inequality holds, where is the chromatic number of .

Theorem 1.1 implies that, if the boxicity of a graph with vertices is very close to the maximum boxicity , the chromatic number of the graph must be very large. The converse does not hold in general; there is a graph whose boxicity is small, even if the chromatic number of the graph is large, like a complete graph. Also there are bipartite graphs with arbitrary large boxicity (see section 5.1 in [2] and also see [3]). However, a graph operation increasing chromatic number may admit increasing boxicity. For example, the join of two graphs, taking the disjoint union of two graphs and adding all edges between them is desired one. Behavior of boxicity has been studied in the context of various graph operations (see [4, 17] for example). This paper is another attempt in this direction that studies behavior of boxicity in the context of Mycielski’s graph operation.

One of the purpose of this paper is to consider whether behavior of boxicity is similar to that of chromatic number under Mycielski’s graph operation. Mycielski [10] invented an interesting graph operation that extends a graph to a new graph , called the Mycielski graph of or the Mycielskian of . It is well-known that the chromatic number of the Mycielski graph of is more than that of , actually, holds. We can construct (triangle-free) graphs with arbitrary large chromatic number by using the graph operation. Here we present the definition of the graph . Let be a copy of the vertex set of a graph , where . For each vertex , the symbol denotes the vertex in corresponding to . The vertex set of is defined to be , the disjoint union of the set of a single new vertex and copies and , and the edge set of is defined to be the union , where

and denotes the edge set of (see Fig. 1 for example). Note that the inequality holds for a graph since contains the subgraph induced by , isomorphic to .

Fig. 1: The Mycielski graph of a cycle and its complement .

So, first we are interested in whether the boxicity of the Mycielski graph is more than that of , the same as behavior of chromatic number under the graph operation, as mentioned at the beginning of this paragraph. Many researchers have studied Mycielski graphs and have compared a graph with under various graph invariants (see [7, 9] for example).

In section 3, we improve the trivial lower bound for the boxicity of the Mycielskian of a graph in terms of the number of universal vertices of . This implies that the boxicity of the Mycielski graph is more than that of if the graph has universal vertices. Also note that there is a graph without universal vertices such that the boxicity of the Mycielski graph is more than that of . While such examples of graphs appear, there is also a graph such that . As a conclusion, behavior of boxicity is not similar to that of chromatic number under Mycielski’s graph operation in general. We reach the next purpose: Classify as many graphs as possible into or .

In section 4, we discuss upper bounds for the boxicity of Mycielski graphs. Chandran et al. [2] proved that the inequality holds for a graph , where is the minimum cardinality of a vertex cover of . It is easy to see that for a graph , and hence we have . Here we present another upper bound for the boxicity of the Mycielskian of a graph in terms of the edge clique cover number of the complement . We also consider graphs that satisfy the equality . The family of graphs satisfying contains complete multi-partite graphs, for example. Other examples of such graphs appear at the end of section 4. As a result, our observations determine the boxicity of their Mycielski graphs if original graphs have no universal vertices or odd universal vertices.

In section 5, we consider relations between the Mycielski graph and its analogous one , called the generalized Mycielski graph of , in the context of boxicity, where . We present upper bounds for the boxicity of the generalized Mycielski graph in terms of that of for a bipartite graph or in terms of that of for a graph . These results will become our motivation a bit to calculate the boxicity of or .

2 Preliminary

In this paper, all graphs are finite, simple and undirected. We use for the vertex set of a graph . We use for the edge set of a graph . An edge of a graph with endpoints and is denoted by . A vertex of is said to be universal if is adjacent to all vertices in . A graph is said to be trivial if is empty. For a subset of , let be the subgraph induced by . For a subset of , let be the subgraph on with as its edge set. A subset of that induces a complete subgraph of is called a clique of . For a graph , its complement is denoted by . The intersection graph of a nonempty family of sets is the graph whose vertex set is and is adjacent to if and only if for , . The intersection graph of a family of closed intervals on the real line is called an interval graph. A graph can be represented as the intersection graph of a family if there is a bijection between and such that two vertices of are adjacent if and only if the corresponding sets in have nonempty intersection. A box in Euclidean -space is the Cartesian product , where is a closed interval on the real line. The boxicity of a graph , denoted by , is the minimum nonnegative integer such that can be represented as (isomorphic to) the intersection graph of a family of boxes in Euclidean -space. The boxicity of a complete graph is defined to be 0. If is an interval graph, . If is an induced subgraph of , holds by the definition.

A graph is a cointerval graph if its complement is an interval graph. Lekkerkerker and Boland [8] presented the forbidden subgraph characterization of interval or cointerval graphs. Cointerval graphs do not contain the complement of a cycle of length at least 4 as an induced subgraph, for example. It is easy to see that the union of a cointerval graph and isolated vertices is also a cointerval graph. A cointerval edge covering of a graph is a family of cointerval spanning subgraphs of such that each edge of is in some graph of . For a set , the cardinality of is denoted by . The following theorem is useful to calculate of the boxicity of graphs. 

Theorem 2.1 ([6], Theorem 3).

Let be a graph. Then, if and only if there is a cointerval edge covering of with .

3 A lower bound for the boxicity of Mycielski graphs

For a complete graph , it is easy to see that since is not complete by the definition. We determine the boxicity of next section (see Lemma 4.1). First we consider if the boxicity of the Mycielski graph of a graph is more than that of in general.

Question 1.

For a graph , does the inequality hold?

The following example shows that there exists a graph such that the equality holds. Here denotes a cycle with vertices. 

Example 3.1.

The boxicity of the Mycielski graph of a cycle is equal to 2. To check this, we give a cointerval edge covering of the complement (see Fig. 1).

Let and be the graphs appeared in Fig. 2. Both graphs are cointerval spanning subgraphs of .

Fig. 2: Cointerval spanning subgraphs and of and an interval representation of .

Note that the disjoint union of a cointerval graph and isolated vertices is also cointerval since these isolated vertices become pairwise adjacent universal vertices in the complement. Hence, we may prove that and are cointerval, instead of and , respectively. A family of intervals on the real line with intersection graph isomorphic to can be found as in the bottom of Fig. 2. Similar arguments work for . Also see that and cover all edges of . The family is a desired cointerval edge covering of , and hence, by Theorem 2.1. Also note that .

Question 2.

Is there a graph such that the inequality holds?

The distance between two vertices and in a graph is defined by length of the shortest path from to in and is denoted by . If there exist no paths from to in , define . Let and be subgraphs of . The distance between two subgraphs and in , denoted by , is defined to be the minimum distance . The following lemma is a generalization of Corollary 3.6 in [6]

Lemma 3.2.

Let be a graph and , induced subgraphs of the complement . If , the following inequality holds:

Proof.

If either or is trivial, say , then is complete. Hence, . Since is an induced subgraph of , we see that

holds. In what follows, we may assume that and are nontrivial.

The assumption means that for a vertex of and a vertex of . Hence, an edge of and an edge of form , the disjoint union of two edges, as an induced subgraph of . Moreover, we claim the following.
Claim (1): no cointerval spanning subgraphs of contain an edge of and an edge of , and
Claim (2): we need at least cointerval spanning subgraphs of to cover all edges of , where .
Claim (1) follows from the forbidden subgraph characterization of cointerval graphs. Actually, cointerval graphs do not contain as an induced subgraph. Claim (2) follows from Theorem 2.1. A cointerval graph with edges of does not contain edges of . Thus, the inequality holds. ∎

We can derive a positive answer to Question 2 by using Lemma 3.2. The following lemma is useful to make our answer more precise. Here, denotes the smallest integer at least .

Lemma 3.3 ([6], Lemma 3).

Let be a graph. Let and be disjoint subsets of such that the only edges between and in are the edges , where . Then, .

Theorem 3.4.

For a graph with universal vertices, the following inequality holds:

Proof.

Let be a graph and universal vertices of . Let be the subgraph of induced by . Note that holds. We consider the Mycielski graph and its complement . Let , the set of vertices in corresponding to universal vertices of . Let be the subgraph of induced by the union of and . Note that and are disjoint by their definition. It is not difficult to check that the only edges between and in are the edges , where . Actually, the vertex is adjacent to all vertices in in and the vertex is adjacent to all vertices in in since is a universal vertex of . We see that by Lemma 3.3.

We prove that holds. Let be a vertex of and a vertex of . The vertex is in and the vertex is in or . We may represent as , where . Since is a universal vertex of , the vertex is not adjacent to in . This implies that for a vertex of and a vertex of , that is, . Thus, the inequality

holds by Lemma 3.2. ∎

Remark 3.5.

We note the proof of Theorem 3.4 works on the generalized Mycielski graph (see section 5 for definition), that is, holds for a graph with universal vertices. Further observations on appear in section 5.

In the proof of Theorem 3.4, we prove that by using Lemma 3.3. Actually, note that . Any two vertices in are not adjacent in since they are adjacent in . Hence, is independent in . Also note that is a clique in by the definition of Mycielski graphs, that is, in . Also see the argument behind the proof of Theorem 5 in [6].

If we restrict our attention to the graph with only one universal vertex or only two universal vertices in the proof of Theorem 3.4, then Lemma 3.3 is superfluous. Note that since is the trivial graph with two vertices and is the path with four vertices.

Theorem 3.4 implies that for a graph with universal vertices, holds. Also note that Mycielski’s graph operation can be used to construct graphs with arbitrary large boxicity (and chromatic number) the same as the join of graphs.

At the end of this section, we note that there is a graph without universal vertices such that the boxicity of the Mycielski graph is more than that of . We give a simple example here. Also see section 6. 

Example 3.6.

Let be a path with vertices, where . We see that . We can give a representation of by a family of boxes in Euclidean 2-space. See Fig. 3 below, where we write and and for a vertex , denotes a box in Euclidean 2-space corresponding to the vertex . Also note that contains an induced cycle .

Fig. 3: A representation of by a family of boxes in Euclidean 2-space.

4 An upper bound for the boxicity of Mycielski graphs

In this section, we give an upper bound for the boxicity of Mycielski graphs. Moreover we calculate the boxicity of Mycielski graphs of some of complete multi-partite graphs. First we determine the boxicity of Mycielski graphs of complete graphs.

Lemma 4.1.

For a complete graph , the following equalities hold:

Proof.

Let be the subgraph of induced by . We have the inequality by Lemma 3.3.

Let . To see , we give cointerval subgraphs of . Let be the subgraph of induced by . We define to be the subgraph of induced by , where . Moreover, let be the subgraph of induced by . It is easy to see that the family is a cointerval edge covering of , and hence holds.

If is odd, the family is a cointerval edge covering of , because the edge is covered with the graph . Hence we have the equality .

If is even, that is, , we show that . Suppose to the contrary that can be covered with cointerval (spanning) subgraphs of . Let for . The graph contains at most two edges in since is cointerval. Actually, the graph must contain two edges in . Otherwise there is a graph in which contains only one edge in or which contains no edges in . Hence the family of cointerval subgraphs of must cover at least edges in , but this is impossible. On the other hand, there is a cointerval graph in which contains an edge , where the vertex is in . We may assume that the graph contains two edges and in . Hence we see . We note that

by the definition of Mycielski’s construction.

Fig. 4: The subgraph of containing edges and .

If , it follows from Lemma 3.3 that since , a contradiction. Hence we may assume that . We reach the four cases on the graph indicated in Fig. 4. These cases imply that , which contradicts our assumption that is cointerval. Thus we have . Hence we obtain the equality if is even. ∎

Remark 4.2.

We proved that the inequality holds at the second paragraph of the proof of Lemma 4.1. We can also derive this inequality by using the minimum cardinality of a vertex cover of , that is, using the inequality . A subset of the vertex set of a graph is a vertex cover of if for each , there is a vertex such that is in . Note that

The edge clique cover number of a graph , denoted by , is the minimum cardinality of a family of cliques that covers all edges of . The following theorem gives us an upper bound for the boxicity of Mycielski graphs. 

Theorem 4.3.

For a graph with universal vertices, the inequality

holds. If is zero or odd, we have the inequality

Proof.

Let be a family of cliques in that covers all edges of . Let be all isolated vertices of and write . Note that . We define to be the subgraph of induced by and let and , where . We can check that is a cointerval graph (see Fig. 5).

Fig. 5: The subgraph and an interval representation of .

Note that the subgraph of induced by is isomorphic to . Hence the edge set of the subgraph of isomorphic to can be covered with at most cointerval subgraphs as in the proof of Lemma 4.1. Let be the subgraph of induced by and the subgraph of induced by for . Moreover, let be the subgraph of induced by . We can check that cointerval subgraphs cover all edges of .

Let be an edge of . If , we see . Hence there is an such that or . If , we have . Hence, if , especially, , we see . If and for any , we see , and hence for . If , we reach the following two cases:

(i) or (ii) .

In the case (i), the edge is in some since the family of cliques covers all edges of , and hence we have

Now we focus on the case (ii). Let be a vertex in and the union of cliques in containing the vertex . If is an isolated vertex in , let be the set . Then we note is never adjacent to vertices in on by the definition of Mycielski graphs. Hence the following two cases occur:

(ii-1) the edge connects a vertex of and a vertex of for some or
(ii-2) the edge connects a vertex and a vertex , where .

Under the case (ii-1), we notice . Under the case (ii-2), we see . These arguments complete the proof of our first statement.

If , the graphs cover all edges of . If is odd, cover all edges of , because the edge is covered with the graph . Our second statement follows from similar arguments as above. ∎

Theorem 3.4 and Theorem 4.3 pretty much narrow the boxicity of Mycielskians of graphs that satisfy the equality . They also determine the boxicity of some Mycielski graphs. 

Corollary 4.4.

For a graph with universal vertices that satisfies the equality , the inequalities

hold. Moreover if be zero or odd, the equality

holds. ∎

We can give examples of graphs that satisfy . Recall that the boxicity of a complete -partite graph is the number of which is at least 2. If has universal vertices, we obtain . Hence we have

Corollary 4.5.

For a complete -partite graph with universal vertices, the inequalities

hold. Especially, if is zero or odd, the equality holds. ∎

We present other examples of graphs that satisfy . The graph whose complement is a chain of cliques is a desired one, where neighboring cliques share exactly one vertex and each clique has at least 4 vertices. Note that the graph contains a complete multi-partite graph as an induced subgraph and the number of its partite sets is equal to that of maximal cliques of the complement .

Moreover if we consider a graph operation that extends a graph to a new graph , called the -suspension of , we can get more examples that we desire. The vertex set of is the union of and the set of new vertices . The edge set of is the union of and the set . Here we assume that is an integer at least 2. We see that and for a graph by Theorem 2.1 and Lemma 3.2. Hence if the graph satisfies , the equality holds. We note that the family of graphs satisfying is not narrow at all.

5 Relation between boxicity of Mycielski graphs and generalized Mycielski graphs

In this section, we consider relations between Mycielski graphs and their analogous ones in the context of boxicity.

Let be a graph and an integer at least 2. Let be a copy of , where . For each vertex , the symbol denotes the vertex in corresponding to . The generalized Mycielski graph of , denoted by , is the graph whose vertex set is , the disjoint union of the set of an additional new vertex and copies of , and whose edge set is , where

Note that the graph is identical to . First, we present a relation between and for a bipartite graph

Theorem 5.1.

For a bipartite graph and , the inequality holds.

Proof.

We partition into two partite sets and . Fix a family of boxes in the optimal dimensional space which represents . Note that , , and for distinct two vertices and of by the definition of . Moreover we note that , , , and are equivalent each other. First we define the family of boxes in -dimensional space to give a box-representation of the graph