Abstract. Figure 1. No. of nodes No. of SC graphs

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1 CATALOGING SELF-COMPLEMENTARY GRAPHS OF ORDER THIRTEEN Myles F. McNally and Robert R. Molina Department of Mathematics and Computer Science Alma College Abstract A self-complementary graph G of odd order has a unique decomposition into edge disjoint subgraphs, one of which is a bipartite self-complementary graph of order G 1. This decomposition allows the efficient generation of self-complementary graphs. In this paper we report the generation of the 500 self-complementary graphs of order thirteen using techniques based on this decomposition. We describe the bipartite construction method, the algorithmic solution employed, and give details of the generation process. Programmed on a personal computer, total running time was less than 30 minutes. 1. Introduction General progress in the construction and cataloging of self-complementary graphs has been slow. Self-complementary graphs exist only for orders congruent to 0 or 1 modulo. Read [] provided a formula for the enumeration, i.e., counting, of such graphs in 193. Figure 1 gives Read s enumeration. No. of nodes No. of SC graphs Figure 1 In 199 the self-complementary graphs on and 9 vertices were cataloged by Morris [5] and those on vertices independently by Alter [1]. A new construction algorithm and its implementation on a computer allowed Kropar and Read [3] to report in 1979 the cataloging of self-complementary graphs on 1 vertices. In this article we describe the cataloging of the self-complementary graphs on 13 vertices. As with Kropar and Read, our approach used a new construction algorithm and a computer. The method of construction was suggested in [] and is based on the bipartite decomposition of a selfcomplementary graph. The computer was a microcomputer, a Macintosh Quadra 50, and the total computation time was under 30 minutes.. Definitions All graphs in this paper are finite, simple 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 bipartite subgraph of G with parts X and 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 {1,,3,, 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 permutations in this paper will be expressed as the product of disjoint cycles. Let B be a bipartite graph with parts X and Y. The bipartite complement of B with respect to the parts X and Y is the bipartite graph B (X,Y ), with edge set E containing all edges not in B 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. If B is connected then the parts X and Y are determined and there is just one bipartite complement. When it is

2 clear from the context which sets X and Y are being considered, we simply write B for the bipartite complement of B. A bipartite graph B is bipartite self-complementary (b.s.c.) if B and B are isomorphic. If φ is an isomorphism from B to B then we will refer to φ as a complementing permutation. We say that φ is even if φ(x) = X and odd if φ(x) = Y. 3. The Bipartite Construction for Self-Complementary Graphs In this section we introduce the bipartite construction for self-complementary graphs of odd order as described in []. Let B be a b.s.c. graph with parts X and Y and φ an odd complementing permutation of B where φ(x) = Y. Let A be any graph with vertex set X such that φ (A) = A, and let C be the graph with vertex set Y defined by C = φ(a). Take a vertex x that is not in X or Y and let x+x be the star obtained by joining x to all vertices in X. Then G = (x+x) A B C is a s.c. graph of odd order. Moreover, any s.c. graph of odd order can be constructed in this way. Note that in the above construction, C A, and what we have done is to essentially glue A and A on either side of B, and then add the vertex x. This is illustrated in Figure. Such a graph is said to be of type (A, B). x A B A Figure If we wish to construct all nonisomorphic s.c. graphs of a given odd order, we can construct a s.c. graph for each triple (B, φ, A) where B, φ, and A satisfy the conditions described in the bipartite construction. If B has the property that there is no automorphism of B mapping X to Y, then it will make a difference whether we attach x to X or Y, and we must consider both cases in our construction. Many of the graphs we generate in this way will be isomorphic and hence isomorphism checks must be made to reduce the collection to one of nonisomorphic graphs. However, we need only check for isomorphism between an (A, B) type graph and an (A, B ) type graph if A A and B B, and if an isomorphic check is needed, then we need only consider isomorphisms fixing the sets {x}, X and Y. These are key points in the efficiency of our approach. If vertex x is not added in our construction, the resulting graph is a s.c. graph of even order. Regrettably, the bipartite decomposition of the resulting graph may not be unique, and hence isomorphism checks between all pairs of s.c. graphs generated would be required, greatly increasing the complexity of the algorithm. However the s.c. graphs of even order can be obtained by first cataloging the s.c. graphs of order one greater, then deleting from each graph the fixed vertex x and boiling down under isomorphism.. The Overall Approach We can now describe the approach taken in the construction of the s.c. graphs of order 13. Rather than code one monolithic program, we decided to tackle the construction in stages using a number of programs written using the C language. First we generated the b.s.c. graphs on 1 vertices and with each of them a set of odd complementing permutations. There are 13 such graphs. It was important that we reduce the number of complementing permutations associated with each graph as much as we could. From each b.s.c. graph with its complementing permutations we construct a distinct set of s.c. graphs. We then used this output to construct the 500 s.c. graphs of order 13 using the method described in section three. We lastly used these graphs to generate the 70 s.c. graphs of order 1. The situation is schematically described in figure 3. Produce BSCs w/ Perms Order 1 B w/ Comp Perms Generate Odd Order Order 13 Reduce to Even Order Order 1 Figure 3

3 5. Construction of Bipartite Self-Complementary Graphs The construction for bipartite self-complementary graphs can be described as follows. Let φ be an element of S n whose cycles all have length a multiple of. Assume that numbers in any cycle of φ appear in consecutive order. Let B be the complete bipartite graph with vertex set {1,,, n} where the parts of B are the even and odd numbers in this set. Let E i, 1 i k, denote the edge orbits of B 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. If we delete from B the edge sets C i, 1 i k, the resulting graph is a b.s.c. graph. Thus k b.s.c. graphs, many of which are isomorphic, are associated with each complementing permutation φ. Ringel [] has shown that only permutations containing cycles whose lengths are powers of need be considered. Thus as an example, if we wish to generate the b.s.c. graphs of order 1 we would consider the complementing permutations ( ) ( ) ( 1 3 ) ( 5 7 ) ( ) and for each of these construct the b.s.c graphs associated with them. The last and most time consuming step is to reduce the resulting set of graphs to a set of mutually nonisomorphic graphs. Figure gives the resulting graphs as first announced in []. Graph 1 Graph Graph 3 Graph Graph 5 Graph Graph 7 Graph Graph 9 Graph 10 Graph 11 Graph 1 Graph 13 Figure. Which Complementing Permutations Need to be Considered We can reduce the complexity of the bipartite construction if for a given b.s.c graph, we don t have to consider all of its complementing permutations. To see that this is an important issue, consider the extremal case of the b.s.c. graph K n,n which has (n!) odd complementing permutations.

4 Lemma 1. Let B be a b.s.c. graph with parts X and Y. Let C(B) denote the set of complementing permutations of B mapping X to Y, and let A(B) denote the set of automorphisms of B that map X to X. Let φ be an element of C(G) and f an element of A(G). If in the bipartite construction algorithm we have constructed all s.c. graphs corresponding to the pair (B, φ), then we need not construct s.c. graphs corresponding to the pair (B, fφf 1 ). Proof: If we are considering the pair (B, fφf 1 ), then we will construct a s.c. graph G by attaching graphs A and C = fφf 1 (A) to B, where A satisfies (fφf 1 ) (A) = A. Since (fφf 1 ) (A) = fφ f 1 (A) = A, we have φ (f 1 (A)) = f 1 (A). Thus A = f 1 (A) and C = φ(f 1 (A)) would be a pair of graphs attached to B to form a s.c. graph G when considering the pair (B, φ). But it is easily seen that f maps A, B and C to A, B and C respectively, and thus f is an isomorphism from G to G. Hence by considering the pair (B, fφf 1 ), we gain only s.c. graphs isomorphic to the ones we obtained by considering the pair (B, φ). Hence we can remove from consideration certain conjugates of complementing permutations that have already been considered. By comparing the third and fourth columns of the table in Figure 5 we can see the impact this lemma has on the running time of the algorithm. 7. Constructing the Odd Order Graphs Given an input file of the b.s.c graphs and their associated complementing permutations, this step is fairly easy. Each b.s.c. graph B generates a distinct set of graphs, and no isomorphism checks need to be made between distinct sets. Given a complementing permutation φ for B, we first generate the set of A (and associated C) subgraphs. The As are generated by considering edge orbits induced by φ and taking various combinations of them. We then proceed to glue each (A, C) pair on either side of B, using an adjacency matrix representation. Last, we add the fixed vertex x, and test the resulting graph to see if it is isomorphic to previously constructed graphs. First we test if corresponding A subgraphs are isomorphic, and only if they are do we need to test the entire graphs. If we need to, we consider only isomorphisms fixing the sets {x}, X and Y. This plus the use of iterative vertex classification algorithm (described in [7]) gives very fast isomorphism checks. Figure 5 shows the number of s.c. graphs generated from each of the input b.s.c. graphs, and notes whether the b.s.c. graph was symmetric. Here, symmetric means there is an automorphism of B mapping X to Y, and for such graphs we don t need to attach x to both parts of B. Bipartite Graph Number Is Graph Symmetric Total Comp. Perms. After Conj. Reduction Self-Comp. Graphs Generated Figure 5. Constructing the Even Order Graphs If we remove the fixed vertex x from the odd-order graphs generated by the above method, an even-order s.c. graph results. To obtain the order 1 s.c. graphs we ran though the 500 order 13 graphs, removing the fixed vertex in each. We used our isomorphism checker to reduce that set

5 of graphs to the 70 graphs which are unique up to isomorphism. Since we could not restrict the permutations considered as in the odd-order case, this step took.5 times as long as the previous steps combined. References [1] R. Alter, A Characterization of self-complementary graphs of order. Portugal. Math. 3 (1975) [] R.A. Gibbs, Self-complementary graphs. J. Combin. Theory (B) 1 (197) [3] M. Kropar and R.C. Read, On the construction of the self-complementary graphs on 1 nodes. J. Graph Theory 3 (1979) [] R.Molina, On the structure of self-complementary graphs, Congressus Numerantium 10 (199), [5] P.A. Morris, Self-complementary graphs and digraphs, Math. Comput. 7 (1973), 1-17 [] R.C. Read, On the number of self-complementary graphs and digraphs, J. London Math. Soc. 3 (193), [7] R.C. Read and D.G. Corneil, The graph isomorphism disease, J. Graph Theory 1 (1977), [] G. Ringel, Selbstkomplementäre graphen. Arch. Math. (Basel) 1 (193) [9] H. Sachs, Uber selbstkomplementäre graphen. Publ. Math. Debrecen 9 (19) 70-