Mirror bipartite graphs
Abstract
The concept of mirror bipartite graph appears naturally when studying certain types of products of graphs as for instance the Kronecker product. Motivated by this fact, we study mirror bipartite graphs from the point of view of their degree sequences and of their degree sets. We characterize the sequences of degrees of mirror bipartite graphs. We also show that from a given set of positive integers, we can construct a bipartite graph of order , which is mirror. Furthermore, very little is known for the degree sequences of graphs with loops attached, when the number of loops attached is limited. We show in this paper that mirror bipartite graphs constitute a powerful tool to study the degree sequences of these graphs when the number of loops attached at each vertex is at most .
2010 Mathematics subject classification: 05C07 and 05C40.
keywords:
mirror bipartite graph, graph, mirror bigraphic sequence, bigraphic sequence, bigraphic set1 Introduction
For a bipartite graph , we write to indicate that and are the stable sets of . Define a mirror bipartite graph of order to be a bipartite graph for which there exists a bijective function such that, for every pair , if and only if . We say that and are mirror vertices. An equivalent statement to this is that can be drawn in the Cartesian plane in such a way that, the vertices of are the points , the vertices of are the points , the edges are straight line segments joining adjacent vertices and the resulting configuration is symmetric with respect to the line .
Let be a bipartite graph. The bipartite complement of , denoted by , is the bipartite graph with stable sets and and edges defined as follows: if and only if .
Lemma 1.1
Let be a mirror bipartite graph. Then is also a mirror bipartite graph.
Proof.
Let be a mirror bipartite graph with stable sets and . Then, there exists a bijective function such that if and only if , for every . Thus, if and only if , for every .
One of the motivations for introducing mirror bipartite graphs is that they appear when studying certain types of products as for instance the Kronecker product of graphs. Let and be two graphs. The Kronecker product WhiRus12 () (usually known as direct product) is the graph with vertex set and if and only if and . It is not difficult to check that for an arbitrary graph , when and , the graph is a mirror bipartite graph, with stable sets and and, for each , the pair are mirror vertices. Furthermore, each mirror bipartite graph admits a decomposition of this form. This result appears in the following lemma, which is a particular case of a result found in LopMun13a ().
Another motivation to study mirror bipartite graphs is due to the fact that with the help of these graphs we will be able to provide an interesting characterization for the degree sequences of graphs. We call graphs these graphs without multiple edges and with at most one loop attached to each vertex.
Lemma 1.2
Let be a mirror bipartite graph. Then, there exists a graph such that .
Proof.
Let with and let be a bijective function such that if and only if , for every . We consider a graph with vertex set and edge set defined by if and only if . Let . Then, the function , defined by and is an isomorphism between and .
In this note, we characterize the sequences of degrees of mirror bipartite graphs (Theorem 2.1). We also show that from a given set of positive integers we can construct a bipartite graph of order , which is mirror (Theorem 3.2) and we completely characterize the degree sequences of graphs (Theorem 2.3).
2 Mirror bigraphic sequences
For graphs we define the degree of a vertex to be the number of edges incident with the vertex. That is to say, a loop adds exactly one unit to the degree of the vertex. Let be a sequence of nonnegative integers. We say that is (loop) graphic if there is a ()graph with its vertices having degrees equal to the elements of . In this case, we say that realizes . Similarly, let and be two sequences of nonnegative integers. The pair is bigraphic if there is a bipartite graph with the vertices of having degrees equal to the elements of and the vertices of having degrees equal to the elements of . In this case, we say that realizes the pair . Usually, the elements of and are ordered from biggest to smallest.
The sequence is mirror bigraphic if the pair is bigraphic and there exists a mirror bipartite graph that realizes . Our first goal is to prove the following theorem.
Theorem 2.1
The sequence is mirror bigraphic if and only if the pair is bigraphic.
For the proof of Theorem 2.1, we will use the following bipartite version of HavelHakimi’s theorem Hakimi (); Havel (), see for instance W ().
Theorem 2.2
Suppose and are sequences of nonnegative integers. The pair is bigraphic if and only if is bigraphic, where is obtained from by deleting the largest element from and subtracting from each of the ’s largest elements of .
Next, we are ready to prove Theorem 2.1.
Proof.
By definition, if is mirror bigraphic then the pair is bigraphic. Thus, let us see the converse. We proceed by induction on , where is the length of the sequence . For , the only possible pairs are and . These possibilites produce the following two graphs: and , respectively. Therefore, the statement holds for .
Assume now that is bigraphic, where is a sequence of length at most . Then, by induction hypothesis is also mirror bigraphic. We want to show that if is a sequence of length , namely and is bigraphic then, is mirror bigraphic. Since is bigraphic, it follows that we can apply HavelHakimi’s theorem twice obtaining a bigraphic pair of identical sequences , where is obtained from by eliminating and subtracting to each element of . The rest of the elements remain the same. By induction hypothesis, is mirror bigraphic. Let be a mirror bipartite graph that realizes . Let be the pairs of mirror vertices in , where , for every . Then, adding a new vertex to each stable set of , namely and and joining the vertices and by an edge, and all vertices of the form to and the vertices of the form to , for every , we are done.
Theorem 2.3
Let be a sequence of nonnegative integers. Then the following statements are equivalent.

is loop graphic.

is bigraphic.

is mirror bigraphic.
Proof.
By Theorem 2.1, conditions (ii) and (iii) are equivalent. We will prove that (i) implies (iii) and viceversa. Suppose there exists a graph that realizes . Then, is a mirror bipartite graph that realizes . Hence, by definition, is mirror bigraphic. Suppose now that is mirror bigraphic. Then, by definition there exists a mirror bipartite graph that realizes . Thus, by Lemma 1.2, there exists a graph such that . Hence, realizes .
Next, let be a sequence of nonnegative integers for which is bigraphic. Let Bipp be the bipartite graphs (modulo isomorphism) that realize , and Mirr be the mirror bipartite graphs (modulo isomorphism) that realize .
It is clear that, for each , the sequence of length , is a sequence for which Bipp=Mirr. The next lemma introduces another sequence for which all bipartite graphs that realizes the pair are mirror bipartite graphs.
Lemma 2.1
Let , for each . Then, Bipp=Mirr.
Proof.
It is easy to show that the pair is bigraphic for every . Next, we will show that, for every , there exists a unique bipartite graph (modulo isomorphisms) that realizes . Suppose that is a bipartite graph that realizes , where and deg. Let . Then, is adjacent to every vertex of . Since , one of the vertices in should have degree . Without loss of restriction, assume that deg. Then, should be adjacent to every vertex of . Since , one of the vertices in should have degree . Without loss of restriction, assume that deg. Then, the vertex should be adjacent to every vertex of . We proceed in this way until we complete the adjacencies of all vertices in .
The bipartite graph (modulo isomorphisms) that realizes is shown in Figure 1.
It is trivial that regular graphs of even order and graphs of the form are mirror bipartite graphs. The following lemma is also easy to prove.
Lemma 2.2
Every regular bipartite graph is a mirror bipartite graph.
Proof.
Since every regular graph is the disjoint union of cycles, it suffices to prove that every cycle of even order is mirror. Let , , and Then, clearly and , for , is a function from to such that if and only if , for all .
At this point, we are ready to state and prove the following proposition.
Proposition 2.1
Let be a bipartite regular graph with . If is not a mirror bipartite graph then .
Proof.
It is clear that . By previous comments and Lemma 2.2, if then cannot be neither , , nor regular. Thus, is regular. But in this case, and is mirror. If or , then is regular, with . Again, it is not possible for to be either , or regular. However, if is regular, with , then the bipartite complement of is regular, with , and hence, mirror. Therefore, by Lemma 1.1, is also mirror, a contradiction. This implies that .
Example 2.4
Figure 2 shows a regular nonmirror bipartite graph of order . This fact is clear since the vertices and that appear in one of the stable sets are twin vertices (they share the same set of neighbors), whereas in the other stable set there are not twin vertices.
Open problem 2.1
Characterize the sequences of length for which Bipp=Mirr.
3 Bigraphic sets
Let be a set of positive integers. We say that is a graphic set if there exists a sequence of the form , which is graphic. It is an easy observation that every set of positive integers is a graphic set. However, Kapoor et al. introduced in Kapoor () the following result. A short proof of it can be found in TriVij07 ().
Theorem 3.1
Kapoor () Let be a set of positive integers. Then there exists a graph of order with degree set .
Let and be two sets of integers. We say that the pair is bigraphic if there exists a bipartite graph with stable sets and such that the vertices of have degrees and the vertices of have degrees . We say that realizes the pair . Next, we have the following theorem.
Theorem 3.2
Let be a set of integers. Then, is bigraphic and there exists a bipartite graph that realizes the pair, with each stable set of size . Furthermore, such a graph can be chosen to be a mirror bipartite graph.
Proof.
In order to prove the theorem, we will consider two cases.
Case 1. Assume that all elements of form a set of consecutive integers. That is to say, . In this case, we consider the sequence obtained from in such a way that all elements of are the elements of minus , that is . By Lemma 2.1 we obtain that the sequence is mirror bigraphic, and hence there is a mirror bipartite graph that realizes , with each stable set of size . Let such a graph be and let and be the stable sets of . Add a new vertex to and a new vertex to , namely and , and join with all vertices of , including . Similarly, join with all vertices of . In this way, we obtain a new bipartite graph with vertex set, and such that the vertices of and of have the following degree sequence: From we can obtain a new graph with stable sets and , in a similar way, where the vertices of and of , have degree sequence: We proceed in this way until we reach the graph with the required degree set. This concludes case 1.
Case 2. Assume that the elements of do not form a set of consecutive integers. Then, there exists such that . Let be the smallest such . Consider the new set . By Theorem 3.1, there is a graph that has order and degree set . Call such a graph . Assume that the vertices of are and consider the graph . By construction, this new graph is a mirror bipartite graph of order and with degree sequences in each stable set equal to the degree sequence of . Moreover, we can add to a set of edges of the form , where , in order to obtain a graph that realizes the pair .
4 Conclusions
We have introduced the concept of mirror bipartite graphs that naturally appears when studying certain types of products of graphs as for instance the Kronecker product. We have studied mirror bipartite graphs from the point of view of their degree sequences and of their degree sets. The main contributions of this note are Theorem 2.1, Theorem 2.3 and Theorem 3.2. Theorem 2.1 establishes that, for a given sequence of positive integers , the pair is bigraphic if and only if there exists a mirror bipartite graph that realizes . Theorem 2.3 presents a characterization of loop graphic sequences, in terms of bigraphic sequences and of mirror bigraphic sequences. In fact, mirror bigraphic sequences can be thought as the link between bigraphic sequences and loop graphic sequences. In Theorem 3.2, we prove that, for a given set of positive integers , the pair can be realized by a bipartite graph of minimum order, that is, , which in fact can be chosen to be a mirror bipartite graph.
Acknowledgements The research conducted in this document by the first author has been supported by the Spanish Research Council under project MTM201128800C0201 and by the Catalan Research Council under grant 2009SGR1387.
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