ON THE STRUCTURE OF SELF-COMPLEMENTARY GRAPHS ROBERT MOLINA DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE ALMA COLLEGE ABSTRACT

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1 ON THE STRUCTURE OF SELF-COMPLEMENTARY GRAPHS ROBERT MOLINA DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE ALMA COLLEGE ABSTRACT A graph G is self complementary if it is isomorphic to its complement G. In this paper we define bipartite self-complementary graphs, and show how they can be used to understand the structure of self-complementary graphs. For G a selfcomplementary graph of odd order, we describe a decomposition of G into edge disjoint subgraphs, one of which is a bipartite self-complementary graph of order G. A method of constructing self-complementary graphs of odd order based on this decomposition is presented. The bipartite self-complementary graphs of order up to 2 are also presented. SELF-COMPLEMENTARY GRAPHS All graphs in this paper are simple, finite and undirected. Let G = G(V, E) be a graph with vertex set V and edge set E. If X V, then the induced subgraph X is the maximal subgraph of G with vertex set X. If X and Y are disjoint subsets of V then the bipartite induced subgraph X, Y, is the maximal subgraph of G with vertex set X Y where every edge joins a vertex in X to a vertex in Y. A graph G is self-complementary (s.c.) if it is isomorphic to its complement G. If G is a self-complementary graph with vertex set {, 2,,, n}, and φ is an isomorphism from G to G, then φ can be viewed as an element of the symmetric group S n, and is referred to as a complementing permutation of G. The permutationsa in this paper will be expressed as the product of disjoint cycles. Fig. shows a s.c. graph and its complement with associated complementing permutation. 2 2 Fig. ()(2) Let G be a s.c. graph of order p with complementing permutation φ. Since G has p(p )/ edges (exactly half as many as K p ), p 0 or (mod). It is well known that the nontrivial cycles of φ have length divisible by, and that φ has a unique -cycle if and only if p is odd. Proofs of this can be found in [] or []. We wish to characterize those vertices in a s.c. graph of odd order fixed by some complementing permutation. Two vertices x and y in a graph G are similar if there is an automorphism of G that maps x to y. Theorem. Let G be a s.c. graph of odd order, and let φ and π be two complementing permutations of G. If φ(x) = x and π(y) = y for vertices x and y in G, then x and y are similar. Proof: We first note that φ 2 is an automorphism of G with a unique cycle of length, namely the cycle containing x, and all other cycles of even length. Consider the partition of the vertex set of G into similarity classes. Clearly all vertices in a cycle of φ are in the same class. Therefore, the class that x lies in has odd order and all other classes have even order. A similar argument which uses the cycle structure of π shows that y also lies in a similarity class of odd order. Since there is only one class of odd order, x and y are in the same similarity class. By the proof of theorem we see that a s.c. graph has a unique similarity class of odd order, and a vertex is in this class if and only if it is fixed by some complementing permutation.

2 2. BIPARTITE SELF-COMPLEMENTARY GRAPHS Let B = B(X Y, E) be a bipartite graph with vertex set X Y and edge set E, where each edge in E joins a vertex in X to a vertex in Y. The bipartite complement of B with respect to the partition X Y is the bipartite graph B(X Y, E ), where E contains all edges not in E that join a vertex in X to a vertex in Y. If B is disconnected then it is possible that the partition X Y is not unique, and hence B might have two nonisomorphic bipartite complements. The graph 2K m,n, m n, is such a graph. If B is connected then the partition X Y is unique and there is just one bipartite complement. When it is clear from the context which sets X and Y are being considered, we simply write B for the bipartite complement of B. If there is an isomorphism φ from B to B where φ(x) = Y and φ(y ) = X, then we say that B is a bipartite self-complementary graph (b.s.c. graph), and we refer to φ as a bipartite complementing permutation, or simply a complementing permutation if the meaning is clear from the context. Fig. 2 shows a b.s.c. graph and its bipartite complement with associated complementing permutation. Here the elements of X and Y are colored black and white respectively. (2)() Fig. 2 Theorem 2. Let B be a bipartite self-complementary graph with bipartite complementing permutation φ. Then the number of vertices in B is divisible by and all cycles of φ have length divisible by. Proof: Let X = m. Since φ(x) = Y, Y = m and thus B has 2m vertices and m 2 /2 edges (half as many edges as K m,m ). Thus m is even and divides B. Now if C is the set of vertices from some cycle of φ, then C is a b.s.c. graph with complementing permutation φ restricted to C. By an argument similar to the one used to show that divides B, we see that divides C.. DECOMPOSITION OF SELF-COMPLEMENTARY GRAPHS Let G be a s.c. graph of odd order with complementing permutation φ, and let x be the vertex of G fixed by φ. Let X be the set of vertices adjacent to x and let Y be the set of vertices other than x that are not adjacent to x. Consider the following decomposition of G into edge disjoint subgraphs: G = x, X X X, Y Y We refer to this decomposition as the bipartite decomposition of G at x. Since φ is a map from G to G that fixes x, we see that φ( X ) = Y and φ( X, Y ) = X, Y. Thus X and Y are complements of each other and X, Y is a b.s.c. graph. If < X > A and < X, Y > B, then we say that G is of type (A, B). By theorem, the graphs A and B are unique up to isomorphism. The bipartite decomposition gives us a useful way to think about the structure of b.s.c. graphs of odd order. This is illustrated in Fig.. x A B A Fig. 2

3 There is a similar decomposition for s.c. graphs of even order. Let G be such a graph with complementing permutation φ, and assume that the vertices of G are labeled in such a way that the numbers in any cycle of φ appear in increasing order. Let X be the set of even numbered vertices and Y the set of odd numbered vertices. Then G = X X, Y Y, the graphs X and Y are complements of each other, and X, Y is a b.s.c. graph. In this case, unlike the case G odd, the subgraphs X and X, Y are not determined up to isomorphism.. CONSTRUCTION OF SELF-COMPLEMENTARY GRAPHS Construction algorithms for self complementary graphs are well known (see [], [] or []), and most are a variation on the following procedure. STANDARD CONSTRUCTION. Let φ be a permutation in S n with all nontrivial cycles of length divisible by, and at most one cycle of length. 2. Let K n be the complete graph with vertex set {, 2,, n}, and let E i, i =,2, k, be the edge orbits of K n under φ.. Two color the edges of each edge orbit E i by arbitrarily selecting an edge e in E i to color red, and then coloring the rest of the edges in E i by the rule φ j (e) is red if j is even and blue if j is odd.. Select from each E i either the red edges or the blue edges, and call this edge set C i.. For each i, delete from K n the edges in E i that are not in C i. The resulting graph G is self-complementary. The reason that the algorithm works is that the graph constructed has φ as a complementing permutation. This is because in each edge orbit of G, red edges are mapped onto blue edges, and blue edges are mapped onto red edges. In the graph obtained by deleting the non-c i edges, φ maps edges onto non-edges, and non-edges onto edges. The algorithm generates 2 k s.c. graphs for each permutation φ (k being the number of edge orbits under φ), many of which are isomorphic. We describe a new construction algorithm based on the bipartite decomposition described in the previous section. The idea is to start with a b.s.c. graph B with complementing permutation φ, and use φ to extend B to a s.c. graph with an odd number of vertices. BIPARTITE CONSTRUCTION. Let B = B(X Y, E) be a b.s.c. graph with complementing permutation φ. 2. Let A be a graph with vertex set X such that φ 2 (A) = A.. Let C be the graph with vertex set Y defined by C = φ(a) = φ(a).. Let x be a vertex that is not in X Y, and let G = x, X A B C. Then G is a b.s.c. graph of type (A, B). If we extend φ by defining φ(x) = x, then φ is a complementing permutation of G. To see this, note that G = < x, X > A B C G = < x, Y > C B A and that φ maps subgraphs of the bipartite decomposition of G at x onto the corresponding subgraphs in the bipartite decomposition of G at x. For example φ(c) = φ 2 (A) = φ 2 (A) = A. The bipartite construction technique can be used to enumerate s.c. graphs with an odd number of vertices. For example, the author of this paper used this technique to enumerate the s.c. graphs of order 9. We give a brief account of this exercise. Since the number was known in advance (a formula for the number of nonisomorphic s.c. graphs of a given order can be found

4 in [2]), the task was complete when nonisomorphic graphs were generated. First, the b.s.c. graphs of order were constructed. For each b.s.c. graph B, a complementing permutation φ was found. For each pair B, φ, all graphs A with vertex set X fixed by φ 2 were generated. Finally, for each triple A, B, φ, a self-complementary graph of type (A, B) was formed. We note that when we performed this computation, only one complementing permutation was needed for each b.s.c. graph. However if nonisomorphic s.c. graphs had not been generated, then it would have been necessary to consider other complementing permutations. What seems to work best is to select a complementing permutation φ with as many -cycles as possible, since more -cycles in φ tend to yield more graphs A fixed by φ 2. Although this method seems more complicated than the standard construction, the advantage is that one need only check for isomorphism between an (A, B) type graph and an (A, B ) type graph if A A and B B. If an isomorphic check is needed, then one need only consider isomorphisms fixing x, X and Y. As an example, we will construct 2 s.c. graphs of order 9. We start with the pair B and φ shown in figure 2. The black vertices in B will become the neighbors of the fixed vertex x. Next, we construct all graphs with vertex set X = {,,,} fixed by φ 2. These graphs are shown in Fig.. Fig. We select two of these graphs as choices for A to demonstrate the construction. The union of the graphs A, B and C = φ(a) form a graph of order as is illustrated in Fig.. A A B A B C Fig. These two graphs are s.c. graphs of order. To obtain the associated s.c. graphs of order 9, we need only add a vertex x that is adjacent to the black vertices. Note that the resulting graphs are not isomorphic, but are of the same type (A, B).

5 . B.S.C GRAPHS OF ORDER 2 OR LESS In this section we list the b.s.c. graphs of order 2 or less. These graphs were generated using a modification of the standard construction technique for s.c. graphs described in section. Fig. shows the unique b.s.c. graph of order and the b.s.c. graphs of order, and Fig. shows the b.s.c graphs of order 2. Fig. Fig. We note that there are 00 s.c. graphs on vertices, and it should be possible to enumerate these graphs, starting with the graphs in Fig., and extending these by implementing the bipartite construction technique on a computer. Because of the large number of isomorphism checks that would be required, further refinements to this algorithm may be necessary. Acknowledgment I would like to thank Dr. Myles McNally for writing the computer program which generated the b.s.c. graphs of order 2. References [] R.A. Gibbs, Self-complementary graphs. J. Combin. Theory (B) (9) 0-2 [2] R.C. Read, On the number of self-complementary graphs and digraphs, J. London Math. Soc. (9), 99-0 [] G. Ringel, Selbstkomplementare graphen. Arch. Math. (Basel) (9) - [] H. Sachs, Uber selbstkomplementare graphen. Publ. Math. Debrecen 9 (92) 20-2