Counting Two-graphs Related to Trees

In an earlier paper, I showed that the classes of pentagon-free two-graphs and of pentagon-and-hexagon-free two-graphs could be represented in terms of trees. This paper gives formulae for the numbers of labelled objects in each of these classes, as well as the numbers of labelled reduced two-graphs in each class. The proofs use various enumeration results for trees. At least some of these results are well-known. To make the paper self-contained, I have included proofs. MOS classification: Primary 05 C 30; secondary 05 C 05, 05 A 18. 1. Trees and two-graphs A two-graph is a pair (X, V ), where X is a set of points, and V a set of 3-subsets of X, having the property that any 4-subset of X contains an even number of members of V . Given a graph G on the vertex set X, the set of 3-sets carrying an odd number of edges of G forms a two-graph on X. Every two-graph can be represented in this way; and graphs G1 and G2 represent the same two-graph if and only if they are related by switching with respect to a set Y of vertices. (This operation consists of interchanging adjacency and non-adjacency between Y and its complement X \ Y , while leaving edges within or outside Y unchanged.) When I speak of the pentagon and hexagon two-graphs below, I mean the two-graphs obtained from the pentagon and hexagon graphs by this procedure. All this material can be found in Seidel [9]. Let (X,V ) be a two-graph. Define a relation ≡ on X by setting x ≡ y if either x = y or no member of V contains both x and y. This is an equivalence relation. The two-graph is called reduced if the relation just defined is equality. In any two-graph, the three points of any triple in V belong to different classes; and replacing a point by an equivalent point does not affect membership in V (that is, ≡ is a congruence). Thus we have a ‘canonical projection’ onto a reduced two-graph. The original two-graph is uniquely determined by this reduced image and the sizes of the equivalence classes. In [2], I gave two constructions leading from trees to two-graphs. Construction 1. Let T be a tree with edge set X. Let V be the set of 3-subsets of X not contained in paths in T . Then (X,V ) is a two-graph. Proposition 1.1. A two-graph arises from a tree by Construction 1 if and only if it contains neither the pentagon nor the hexagon as an induced substructure. Trees T1 and T2 yield isomorphic two-graphs if and only if they are themselves isomorphic. Construction 2. Let T be a series-reduced tree (one with no divalent vertices). Let X be the set of leaves of T . Now T , being bipartite and connected, has exactly two vertex 2-colourings; select one, and call the colours black and white. Let V consist of all 3-subsets of X such that the paths joining these vertices meet at a black vertex. Then (X,V ) is a two-graph. (If we use the other colouring, we obtain the complementary two-graph.)