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International Journal of Combinatorics
Volume 2010 (2010), Article ID 767361, 17 pages
Classification of Triangle-Free Configurations
1Department of Mathematics, Faculty of Science, Kuwait University, P.O. Box 5969, Safat-13060, Kuwait
2Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA
Received 20 May 2010; Accepted 29 July 2010
Academic Editor: Laszlo A. Szekely
Copyright © 2010 Abdullah Al-Azemi and Anton Betten. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The 157, 211 triangle-free symmetric configurations are classified and some of their properties are examined. We conclude that each such configuration has a blocking set. Further properties like transitivity on lines, self-duality, and self-polarity are discussed.
A finite incidence structure is a pair , where and are finite sets. In particular, is a set of points and is a set of blocks (or lines) such that for .
The number of blocks containing a point is called the degree, denoted by . The number of points that are contained in a block is called the size of , denoted by . A pair with is called a flag. In this case, we say that lies on , or that and are incident.
A (combinatorial) configuration (see ) of type is an incidence structure with for , for , any two distinct points being incident with at most one line.
The last axiom implies that two points determine at most one line and that two lines intersect in at most one point. We say that two points are collinear if they lie on a line (and two lines are concurrent if they intersect). Note that these structures are defined purely combinatorially (and hence sometimes called combinatorial configurations). It is a different question whether or not a given configuration can be embedded in projective space (such that the blocks arise from lines in that space). This question leads to the notion of geometric configurations. In this paper, we are concerned with combinatorial configurations only. We do not discuss the problem of whether or not they can be embedded in projective space. Clearly, every geometric configuration is also combinatorial. Therefore, the results in this paper may be seen as a starting point to classify the corresponding geometric configurations. For the sake of simplicity, henceforth we will simply talk about configurations. In all cases, this will mean combinatorial configurations.
Configurations are closely related to graphs. Let be a configuration. The Levi graph (or incidence graph), denoted , associated with , is the cubic bipartite graph with vertex set with and adjacent if and only if ; see [3–7]. Alternatively, it is the cubic bipartite graph with black vertices representing the points, with white vertices representing the lines, and with an edge joining two vertices if and only if the corresponding point and line are incident. According to Coxeter , configurations can be characterized in the following way. Recall that the girth in a graph is the length of the shortest cycle.
Proposition 1.1. An incidence structure is a configuration if and only if its Levi graph is cubic and has girth at least .
An isomorphism between two incidence structures and is a bijection which takes to (where for ). If such an isomorphism exists, the incidence structures are isomorphic. It is well known that isomorphism of incidence structures is an equivalence relation. The equivalence classes are known as isomorphism types. The problem of classifying a class of incidence structures is determining the classes of isomorphic objects.
Furthermore, for an incidence structure the isomorphisms from to are known as automorphisms. They form a group, the automorphism group of
If is any subgroup of the automorphism group of a configuration , then may be seen as acting on the set of points, the set of blocks, and the set of flags; see  for more details. An orbit of on points, blocks, flags (resp.) is known as a point-orbit, block-orbit, flag-orbit (resp.). It is well known that a flag transitive automorphism group is also transitive on points and blocks (but not conversely).
A blocking set in a configuration is a subset of points such that each block contains at least one element from and one element not from . Not every configuration has a blocking set. An example of a blocking set free configuration is the unique configuration (or Fano plane) shown in Figure 1. The figure also shows the Levi graph of this configuration. This graph is also known as the Heawood graph.
To each configuration , we may associate another configuration known as the dual configuration. The dual configuration is , with the roles of points and blocks reversed, but with the same incidence. That is, a “point” is on a “block” in the dual configuration if in Clearly, is a configuration. Also, and have the same Levi graph, except that the color classes are reversed. Applying duality twice in a row, we obtain a configuration that is isomorphic to the original configuration .
If is isomorphic to its dual , we say that is self-dual and a corresponding isomorphism is called duality. Moreover, a duality of order is called a polarity. A self-polar configuration is a configuration that admits a polarity. The Fano plane of Figure 1 is both self-dual and self-polar. To see this, notice that the bijective mapping for preserves the Levi graph of the Fano plane.
In this paper, we consider configurations that contain no triangles. A triangle in a configuration is a triple of points that are pairwise collinear but not with the same line. A configuration is triangle-free if and only if the girth of the incidence graph is at least (configurations in general have girth at least ). In terms of points and lines, we have that a point that is not on a line is collinear to at most one point of . This is a weakening of the axiom of a generalized quadrangle: in a generalized quadrangle, a point not on a line is collinear to exactly one point of .
So far, triangle-free configurations have been classified for . In the current work, a classification for is carried out. Table 1 shows the known results of the nonisomorphic triangle-free configurations for . The entries for were determined previously in [8, 9]. The entries in the last row of Table 1 are new and were constructed in our search.
2. Search and Results
Necessary existence conditions for a configuration are obtained by counting in two ways the incidences of points with lines and the incidences of pairs of points containing a given point with lines, respectively. For , these conditions are also sufficient; see Gropp .
In this paper, we use computer search to classify the triangle-free configurations. We find exactly nonisomorphic configurations of this type. Moreover, we also verified the results in  for triangle-free configurations with at most points.
Gropp  has stated that there is a blocking set free configuration. Our search shows that there is no blocking set free configuration which is also triangle-free. Thus, Gropp's configuration must contain triangles.
A configuration with and can be represented by a incidence matrix , say, with rows and columns. The entry is one if and only if Clearly, there are three ones in each row and column. Also, the dot product of any two distinct rows is at most . Those properties are equivalent to those that were mentioned earlier in Section 1, namely, (C1), (C2), and (C3). Moreover, two matrices are isomorphic if one can be obtained from the other by permuting the rows and the columns. In the incidence matrices that are displayed below, we write “x” if and “empty square” if
Let us now describe the algorithm that we use to classify the triangle free configurations. The algorithm is an instance of the method of orderly generation .
We carry out a row-by-row (or point-by-point) backtrack search over all incidence matrices of triangle free configurations. We start with the all-zero matrix and augment it one row at a time, subject to the properties (C1), (C2), and (C3). Augmenting means deciding on the positions of the three ones in one particular row. The rows are augmented in order. Moreover, we use an algorithm from [12, 13] for testing the girth condition (of the partially filled incidence matrix). In this way, we ensure that all of the considered structures are triangle-free configurations. Once a row has been completed, one of two actions is taken. If the number of completed rows is between 17 and 21, no further action is taken. In the other cases, we perform a test whether or not the lexicographically least form of the incidence matrix agrees with the matrix that was created. If yes, we keep the row that was just added and proceed with the search. If no, the row is rejected and we backtrack. This test is our way to solve the isomorphism problem. Namely, each isomorphism class is represented by its lexicographically least representative. Since we perform this test after 22 rows, the resulting objects are pairwise nonisomorphic. The reason for not testing after rows 18, 19, and 20 is the following. Computing the lexicographically least incidence matrix is expensive. Therefore, the isomorphism test described above slows down the search quite a bit. The benefits of using the isomorphism test are worth the effort in the early rows (up to row 17). Namely, the isomorphism test keeps the number of possibilities down and thereby reduces the size of the search space. On the other hand, for rows 18–20, the number of possibilities of partial incidence matrices increases dramatically. We observe, however, that many of these matrices do not complete and hence do not contribute to the classification. Therefore, switching off the isomorphism test for rows 18–20 means that we do not spend time on classifying partial incidence matrices that do not complete anyway. This simple trick saves us a lot of time. In fact, we can classify the triangle free configurations in about hours CPU time (on a single CPU machine).
We remark that we use our own algorithm to compute the lexicographically least representative of the isomorphism class of a matrix. The complexity of this algorithm is exponential in the size of the input. No fast algorithm to solve this problem is known.
Alternately, the idea of canonical augmentation due to McKay  can be used. In this method, the lexicographically least representative is replaced by a canonical representative (that in almost all cases is different from the lexicographically least representative). We also tried this method, using nauty  to compute the canonical representative. We found that orderly generation using lexicographically least representatives worked better for us. This may not be seen as a critique of “canonical augmentation”. We simply did not try very hard, so a comparison is unfair. We remark that while canonical representatives of an isomorphism class may be computed faster than lexicographically least representatives, the procedure is still exponential. Again, no fast (i.e., polynomial) algorithm to solve this problem is known.
A. Some Selected Configurations
In what follows, we will present some examples and discuss some properties of the configurations that were found in our search. We start by listing the distribution of automorphism group orders Here, means that there are configurations with an automorphism group of order .
In the following, we will look at some of the configurations with nontrivial automorphism groups.
At first, we determine the type of the automorphism group. We use , and , and to denote the cyclic group of order , the Dihedral group of order and the elementary abelian group of order , respectively. For groups and , let be a split extension of by (with normal subgroup ). For groups of order other than a prime, we find the following types in Table 2.
It is often convenient to identify the blocks of a configuration with the triple of points that are incident with it. In what follows, we write for points and we write for blocks. Also, we give the corresponding incidence matrix with row indices and column indices corresponding to the set of points and blocks, respectively. The orbits of the automorphism group acting on points and blocks (resp.) form a partition (of and of , resp.). In the figures, we group points and blocks according to this partition. The boundaries of the classes of the partition are indicated by boldface lines.
A.1. Triangle-Free Configuration with Automorphism Group of Order 4
As mentioned in Table 1, there are exactly seven configurations that are self-dual but are not self-polar. Six configurations have a group of order while one has a group of order . The latter configuration has the following blocks:
Its partitioned incidence matrix is presented in Figure 2. The automorphism group is generated by and where
It has ten point-orbits of three different sizes:
A.2. Triangle-Free Configuration with Automorphism Group of Order 11
The unique triangle-free configuration with automorphism group of order has the following blocks:
Its partitioned incidence matrix is presented in Figure 3. Its Levi graph is shown in Figure 4. This configuration is self-polar by the correspondence Its automorphism group is generated by It has two point-orbits of size , shown in Figure 3.
A.3. Triangle-Free Configuration with Automorphism Group of Order 16
There are six triangle-free configurations with an automorphism group of order . Four of those structures are self-dual and self-polar. Here we present only one of those four configurations. It has the following blocks: Its partitioned incidence matrix is shown in Figure 5. The Levi graph is shown in Figure 6. The configuration is self-polar under the correspondence The automorphism group is generated by the following three elements:
A.4. Triangle-Free Configuration with Automorphism Group of Order 22
The unique triangle-free configuration with automorphism group of order has the following blocks:
Figure 7 shows its partitioned incidence matrix. It is self-polar via the correspondence
The automorphism group is generated by the following two elements: It has two point-orbits of equal sizes: Its Levi graph is shown in Figure 8.
A.5. Triangle-Free Configuration with Automorphism Group of Order 24
Altogether, there are three configurations with an automorphism group of order 24. In this section, we present those three configurations, say , , and . The blocks for configuration are
The blocks for configuration are
Finally, the blocks for configuration are
The automorphism group of configuration is generated by and for is generated by and for is generated by
Configuration is self-polar under the correspondence
The authors would like to thank the two referees for the helpful suggestions and comments. The first author was partially supported by the Research Administration at Kuwait University Grant number ZS 02/09.
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