Colouring problems for symmetric configurations with block size 3

The study of symmetric configurations v 3 with block size 3 has a long and rich history. In this paper we consider two colouring problems which arise naturally in the study of these structures. The first of these is weak colouring, in which no block is monochromatic; the second is strong colouring, in which every block is multichromatic. The former has been studied before in relation to blocking sets. Results are proved on the possible sizes of blocking sets and we begin the investigation of strong colourings. We also show that the known 2 1 3 and 2 2 3 configurations without a blocking set are unique and make a complete enumeration of all nonisomorphic 2 0 3 configurations. We discuss the concept of connectivity in relation to symmetric configurations and complete the determination of the spectrum of 2‐connected symmetric configurations without a blocking set. A number of open problems are presented.


| INTRODUCTION
In this paper we will be concerned with symmetric configurations with block size 3 and, more particularly, two colouring problems which arise naturally from their study. First we recall the definitions. A configuration v b ( , ) r k is a finite incidence structure with v points and b blocks, with the property that there exist positive integers k and r such that (i) each block contains exactly k points; (ii) each point is contained in exactly r blocks; and (iii) any pair of distinct points is contained in at most one block.
A configuration is said to be decomposable or disconnected if it is the union of two configurations on distinct point sets. We are primarily interested in indecomposable (connected) configurations, and so unless otherwise noted, this is assumed throughout the paper.
If v b = (and hence necessarily r k = ), the configuration is called symmetric and is usually denoted by v k . We are interested in the case where k = 3. Such configurations include a number of well-known mathematical structures. The unique 7 3 configuration is the Fano plane, the unique 8 3 configuration is the affine plane AG (2,3) with any point and all the blocks containing it deleted, the Pappus configuration is one of three 9 3 configurations and the Desargues configuration is one of ten 10 3 configurations. Symmetric configurations have a long and rich history. It was Kantor in 1881 [17] who first enumerated the 9 3 and 10 3 configurations and in 1887, Martinetti [19] showed that there are exactly 31 configurations 11 3 .
It is natural to associate two graphs with a symmetric configuration v 3 . The first is the Levi graph or point-block incidence graph, obtained by considering the v points and v blocks of a configuration as vertices, and including an edge from a point to every block containing it. It follows that the Levi graph is a cubic (3-regular) bipartite graph of girth at least six. The second graph is the associated graph, obtained by considering only the points as vertices and joining two points by an edge if and only if they appear together in some block. Thus the associated graph is regular of valency 6 and order v.
We note that symmetric configurations with block size 3 have also been studied in the context of 3-regular, 3-uniform hypergraphs. In this scenario the points of the configuration are identified with the vertices in the hypergraph and the blocks with the hyperedges; the condition that no pair of distinct vertices should be in more than one hyperedge is usually referred to as a linearity condition in hypergraph terminology.
By a colouring of a symmetric configuration v 3 , we mean a mapping from the set of points to a set of colours. In such a mapping, if no block is monochromatic we have a weak colouring and if every block is multichromatic or rainbow we have a strong colouring. The minimum number of colours required to obtain a weak (resp., strong) colouring will be called the weak (resp., strong) chromatic number and denoted by χ w (resp., χ s ). It is immediate from the definition that the strong chromatic number χ s of a configuration is equal to the chromatic number of its associated graph.
Weak colourings have been studied before in relation to so-called blocking sets and in Section 2.1 we begin the study of the sizes of these. In Section 2.2 we bring together various results concerning symmetric configurations without a blocking set which appears throughout the literature, some of which do not seem to be readily available. Section 2.3 is concerned with the connectivity of configurations and we complete the spectrum of 2-connected symmetric configurations without a blocking set. Our results on enumeration appear in Section 2.4.
In particular we extend known results by enumerating all symmetric configurations 20 3 together with their properties, and prove that the known 21 3 and 22 3 configurations without a blocking set are the unique configurations of those orders with that property. Section 3 is concerned with strong colourings. To the best of our knowledge, both this topic and the sizes of blocking sets in Section 2.1 appear to have been neglected and the results are new. Finally in Section 4 we bring together some of the open problems raised by the work in this paper.

| WEAK COLOURINGS
We begin with the following result which is a special case of Theorem 8 of [9]. A blocking set in a symmetric configuration is a subset of the set of points which has the property that every block contains both a point of the blocking set and a point of its complement. From this definition it is immediate that the complement of a blocking set is also a blocking set, and that the existence of a blocking set in a configuration is equivalent to χ = 2 w . Empirical evidence indicates that almost all symmetric configurations v 3 contain a blocking set. Indeed, Table 2 shows that of the 122,239,000,083 connected configurations with ≤ v 20, only 6 fail to have a blocking set.
For any ≥ v 8, a configuration with a blocking set is very easy to construct. For v even, the set of blocks generated by the block {0, 1, 3} under the mapping ↦ i i v + 1(mod ) has a blocking set consisting of all the odd numbers (and hence, another consisting of all the even numbers). For v odd and ≥ v 11, construct the symmetric configuration v ( − 1) 3  The above extension operation can be summarised and generalised as follows.
• Choose two nonintersecting blocks a a a { , , } 1 2 3 and b b b { , , } 1 2 3 such that the points a 1 and b 1 are not in a common block. This construction goes back to Martinetti [19]; see also [8].

| Sizes of blocking sets
Perhaps surprisingly, the cardinalities of blocking sets of those configurations v 3 for which χ = 2 w do not seem to have been studied. Let Q be such a blocking set and let We have the following result.
Theorem 2.2. Let v 3 be a symmetric configuration with a blocking set Q of cardinality q Then v 3 also has a blocking set Q of cardinality q + 1.
Proof. Let A, of cardinality α, be the set of blocks of the configuration which Q intersects in one point and B, of cardinality β, be the set of blocks which Q intersects in two points. Then α β v + = and α β q Bearing in mind that if Q is a blocking set for a configuration v 3 then so is ⧹ V Q, it follows that the range of cardinalities of blocking sets of a configuration is continuous, and that configurations can be categorised by the minimum cardinality of a blocking set; a blocking set of this minimum cardinality will be called a minimal blocking set. Configurations which have a blocking set of cardinality q for all ⌈ ∕ ⌉ ≤ ≤ ⌊ ∕ ⌋ q v q v : 3 2 3 are relatively easy to construct.
Then there exists a configuration v 3 having a blocking set of Proof. In view of Theorem 2.2, it is sufficient to construct a symmetric configuration with a blocking set of cardinality We note that the condition ≥ v 9 in the above theorem is necessary; the unique 7 3 configuration has no blocking set at all, and the unique 8 3 configuration has a minimal blocking set of cardinality 4. Since Theorem 2.3 shows that a configuration v 3 with a minimal blocking set as small as possible exists for all ≥ v 9, it is natural to ask what the range of possible sizes of minimal blocking sets might be. At the minimum end of the range, we are able to prove the following results.  Proof. We deal first with part (a). First observe that from  five points. Otherwise, then by replacing the point a 0 by the point 0 to return to the set , the configuration 8 3 would have a blocking set of size 3. Now consider the set . In the above special case, Q must contain at least s − 1 elements of the set ⧹ V a { } 0 and in all other cases, at least s elements. So Q has at least s + 4 elements; to show that a minimal blocking set has exactly s + 4 elements we may take a blocking set ≤ ≤ Q b i s = {1, 4, 5, 6, : 0 − 1} i . We next deal with configurations s (3 + 9) 3 , ≥ s 3. The procedure is precisely the same as the above case, except that we use the configuration 9 3 as given in the appendix, that is, 012, 034, 056, 135, 147, 248, 267, 368, 578 which also has a minimal blocking set of size 4. In this case we take a blocking set

Finally for configurations s
(3 + 10) 3 we use one of the two configurations 10 3 as given in the appendix with a minimal blocking set of size 5, namely, 012, 034, 056, 135, 178, 247, 268, 379, 469, 589. Again the procedure is as in the above two cases and we can take a blocking set The most interesting of these is possibly the first one which is point-transitive; one of only three such configurations 19 3 , again see [12] and Table 2. Its Levi graph is arcregular and has automorphism group of order 114. It is the unique symmetric graph of order 38 and is graph F038A in the Foster census [13]. The configuration is cyclic and is isomorphic to the configuration generated by the block {0, 1, 8} under the mapping ↦ i i + 1 (mod 19). An example of a symmetric configuration on 20 points having a minimal blocking set of size 9 is as follows.  At the maximum end of the range, the situation appears to be much more difficult. The first of these is one of the two flag-transitive configurations on 16 points; see [6] and Table 2. Indeed its Levi graph is the Dyck graph, which is well known and is the unique arctransitive cubic graph on 32 vertices. The Dyck graph is graph F032A in the Foster census [13].
Although this graph has a number of known constructions, it seems that none of these can be generalised to produce further examples of configurations without small blocking sets. Unfortunately therefore, we cannot provide a construction for an infinite class of symmetric configurations v 3 having a minimal blocking set of maximum cardinality, and this remains a significant open problem.
However, we are able to construct symmetric configurations whose minimal blocking sets have a size as far away as we please from both the minimum or maximum possible cardinalities, as the following theorem and corollary show.
The proof of Theorem 2.5 is much simplified by transforming the problem into an equivalent problem concerning the existence of binary words. A binary word b of length n is a We shall be concerned with circular binary words, where the digit b 0 is considered to follow b n−1 ; informally, the word "wraps round" with period n.
+ −1 where the subscripts are taken mod n; in other words, the subword starts at position i and wraps round if necessary. The weight w b ( ) of a word b is simply the number of 1s in b. A sequence of k consecutive 1s in a circular binary word with 0s at either end is called a run of length k; similarly for a sequence of 0s surrounded by 1s.
To make the connection with blocking sets, we let C v be a cyclic configuration as in the statement of the theorem, and identify the point Since each block of C v has the form m m m { , + 1, + 3}, it is immediate that a subset S is a blocking set for C v if and only if the corresponding circular binary word S b( ) does not contain any of the subwords 0000, 0010, 1101 or 1111. The problem of finding a minimal blocking set is therefore equivalent to finding the minimum weight of a circular binary word satisfying this forbidden subword criterion. We begin with two simple lemmas.
= ( ) is a circular binary word corresponding to a blocking set S of the configuration C v . Then any subword of b of length 5 has weight 2 or 3.
Proof. Clearly any subword of length 5 and weight 0 contains the forbidden subword 0000. It is easy to see that the only possible length 5 subword of weight 1 is 01000. The digit immediately to the left of this subword must be 1, otherwise we get the forbidden subword 0010. Then the next digit to the left again must be 0, to avoid the forbidden subword 1101. Continuing in this way, we see that the sequence of digits reading leftwards from 01000 must be 1, 0, 1, 0, 1, 0, …. But b is a circular word containing the subword 000, so this is impossible. Thus no subword of length 5 can have weight 1.
Since the roles of the binary digits 0 and 1 in this problem are symmetric (corresponding to the fact that if S is a blocking set then so is its complement), it follows that no length 5 subword can have weight 4 or 5 either. □ = ( ) is a circular binary word corresponding to a blocking set S of the configuration C v . Then b has one of the following forms: Proof. The proof of Lemma 2.6 shows that whenever the subword 010 appears in b, then b must be of type (a). A similar argument holds for the subword 101. Thus any run length of 1 forces type (a), and this is only possible if v is even. Run lengths of 4 or greater are ruled out by the forbidden subwords 0000 and 1111, so the only remaining possibility is type (b). □ We are now ready to complete the proof of the theorem.
Proof of Theorem 2.5.
= ( ) is a circular binary word corresponding to a blocking set S of the configuration C v . A simple counting argument in conjunction with , it remains to find the minimum value of ε in all cases. We proceed by considering all the congruence classes mod 5.
If ≡ v 2 (mod 5), then we know ≥ ε 1 but an examination of all the possibilities shows that it is not possible to add a single 1 and a single 0 to a word of the form b = 11000 11000 11000… without creating a run of length 1 or 4. Thus ≥ ε 2, and b = 111000 111000 11000…11000 satisfies the conditions of Lemma 2.7 and so ε = 2.
If ≡ v 4 (mod 5), then we know ≥ ε 2 and b = 11000 1100 11000…11000 satisfies the conditions of Lemma 2.7 and so ε = 2. □ Corollary 2.8. Let ≥ k 1. Then there exist: (a) a configuration v 3 with a minimal blocking set of size exactly Proof. We use Theorem 2.5. For (a), take v k = 15 and for (b), take v k = 10 . □

| Configurations without blocking sets
We now turn our attention to the case where χ = 3 w , that is, to symmetric configurations with block size 3 which have no blocking set. This is an old problem which goes back some 30 years. It has appeared three times as a problem at the British Combinatorial Conference. The [4], by which time the five largest values had been resolved positively due to the work of Kornerup [18]. Finally in the Proceedings of the 16th Conference [5], Problem 333, Gropp asked whether there exists a symmetric configuration 16 3 without a blocking set, having reported that the case of such a configuration 15 3 had been resolved negatively.
A det( ) = per( ). (In our context, the biadjacency matrix of the Levi graph of a configuration v 3 is simply the v v × incidence matrix of the configuration.) Thomassen [23] pointed out that a symmetric k-configuration is blocking set free if and only if its Levi graph is det-extremal. In [14] the following theorem was proved from which it is an immediate corollary that there are no symmetric configurations v 3 without a blocking set for v = 20, 23, 24, 26. The four values 20, 23, 24, 26 indeed seem to be the most problematic. If ≥ v 27, it is easy to give a short self-contained account to prove that there exists a symmetric configuration v 3 with no blocking set and we do this below beginning with two constructions from [9] which we present as theorems. Proof. Denote the points and blocks of the ′′ are the points and blocks of a configuration v v ( + ′ − 1) 3 which it is easily verified has no blocking set. Proof. Denote the points and blocks of the configuration v ( ) i 3 by V i and i  , respectively, are the points and blocks of a configuration Again it is easy to verify that this has no blocking set. □ We note that the construction of Theorem 2.11 was reported independently by Abbott and Hare [1], referencing an earlier paper of Abbott and Liu [2].
To implement the constructions we begin with three basic systems. From [6], in the range ≤ ≤ v 7 1 8 there exist only two symmetric configurations v 3 with no blocking set: the unique 7 3 configuration (Fano plane) and a 13 3 configuration obtained from two copies of it using Theorem 2.10.
The blocks of the latter system can be represented by the following triples: . 012 034 056 135 146 236 278 49c 5ab 79b 7ac 89a 8bc A symmetric configuration 22 3 with no blocking set was given by Dorwart and Grünbaum [11]; it is illustrated in Figure 1 and as is evident, is obtained by merging three Fano planes. Its blocks are as follows: As reported in [16], Kornerup [18] constructed blocking set free configurations v 3 for the values v = 29, 30, 32. These are contained in a thesis of the University of Aarhus which we have been unable to see, and the configurations found do not seem to be published elsewhere. Thus to give a complete account in one place, we have also constructed configurations 29 3 , 30 3 and 32 3 without a blocking set. We show their Levi graphs in Figure 2, and include the blocks below.
A blocking set free configuration 29 3 : Again we have used the "merging" technique and claim no originality for these. They may very well be the same systems discovered by Kornerup.

| Connectivity of configurations
Here we introduce the idea of the connectivity of a symmetric configuration and derive some results. First recall that in a cubic graph, the vertex connectivity is equal to the edge connectivity. Further if a connected cubic graph is also bipartite, then the connectivity cannot be 1 and so is equal to either 2 or 3. Define the connectivity of a symmetric configuration to be the connectivity of its Levi graph. Funk et al. [14] present the following operation. Let G 1 and G 2 be cubic bipartite graphs which are disjoint, and let ∈ y V G ( ) 1 with neighbour set x x x { , , } 1 2 3 and ∈ x V G ( ) 2 with neighbour set y y y { , , } 1 2 3 . Then the graph is said to be a vertex-sum of G 1 and G 2 . They then quote the following theorem which they attribute to McCuaig [20].  This naturally raises the question of the spectrum of 2-connected symmetric configurations without a blocking set. From our account above it is clear that the systems v 3 with ≡ v 1 (mod 6) arising from Theorem 2.13 are 3-connected. There are no 2-connected systems for ∈ v {7, 13, 19} since all have been enumerated and arise from Theorem 2.13; see Table 2 and The Levi graph of a 25 3 configuration with no blocking set ERSKINE ET AL.
| 409 the discussion in Section 2.4. So to complete the spectrum, what is needed is a 2-connected configuration 25 3 without a blocking set. Such a configuration does exist and its Levi graph is shown in Figure 3. The blocks are listed below.

| Enumeration of configurations
Finally in this section we present some enumeration results. As stated above, for ≤ ≤ v 7 1 8 there exist just two symmetric configurations with no blocking set; unique 7 3 and 13 3 systems. Gropp [16] reported that there exist at least four configurations 19 3 without a blocking set.
Recently the present authors [12] have enumerated all configurations 19 3 and we confirm that there are exactly four without a blocking set. These have a nice description as follows. Because 19 = 13 + 7 − 1, it must be true that at least some of the four configurations 19 3 can be obtained by using Theorem 2.10 with the unique 13 3 and 7 3 configurations without blocking sets. We may use the construction of Theorem 2.10 with v v = 7, ′ = 13, taking all possible choices for the distinguished point and block in the two constituent configurations. To this set of configurations we may add those obtained by taking v v = 13, ′ = 7 in the same way. Finally, this set of configurations can be reduced to isomorphism class representatives using the GAP package DESIGN [15,22]. In this way we were able to determine that the method results in exactly four isomorphism classes of configurations 19 3 with no blocking set. Thus these correspond precisely to the four in the enumeration; this description of the four 19 3 configurations was known to Gropp and the construction is described in [10,16]. The blocks of these four 19 3 systems are as follows: Although there is no symmetric configuration 20 3 without a blocking set, increases in computer power allowed us to extend the enumeration of symmetric configurations to the case where v = 20 and this information is summarised in Table 2. Our enumeration, in common with our previous results [12], was carried out using the program confibaum as used in [6].
We are grateful to G. Brinkmann for this program and for assistance in our previous enumeration.
The enumeration confirms the fact that there is no symmetric configuration 20 3 without a blocking set. For completeness we describe here the properties enumerated in Table 2, following the notation of [6]. For a configuration  , an automorphism is a permutation of the points and blocks of  which preserves incidence. The dual of  is the configuration obtained by reversing the roles of the points and blocks of  . If  is isomorphic to its dual, we say it is self-dual, and an isomorphism between  and its dual is an anti-automorphism. An antiautomorphism of  of order two is called a polarity, and a configuration admitting such an isomorphism is self-polar. Note that for consistency with previously published results, the counts in Table 2 include disconnected configurations.
The next case to consider is v = 21. A 21 3 configuration without a blocking set can be constructed from three 7 3 configurations by Theorem 2.11. Because the automorphism group of the Fano plane is flag-transitive, all systems constructed by this method are isomorphic. We show that this is the unique system of this order without a blocking set. From Theorem 2.13, any such system is 2-connected.
The first observation to make is that a cubic bipartite graph with edge connectivity 2 and edge cutset ab cd { , } must take the form illustrated in Figure 4. In the diagram, the circles represent the components C C , 1 2 following the edge cut and the black/white colouring of the vertices represents the bipartition of the graph.
Suppose now that the graph in Figure 4 is the Levi graph of a symmetric configuration 21 3 . Say the components C C , 1 2 following the edge cut have respective orders n 1 and n 2 , with n n + = 42 1 2 . Then C 1 has n − 2 1 vertices of valency 3, and two vertices (a and d) of valency 2. In other words, it is a subcubic bipartite graph with The problem of constructing all cubic bipartite graphs with edge connectivity 2 can therefore be reduced to finding all possible components C C , 1 2 . Note that a component is not necessarily an edge-deleted Levi graph of some configuration; this will be the case for C 1 , for example, if and only if the distance between the distinguished vertices a and d is at least 5. But these vertices may be at distance 3 or even 1. However the component can contain no 4-cycles. By using the genbg utility provided in the nauty package [21], we may use a computer to construct all possible components. This computer search shows that the smallest possible one of these has order 14 and is unique; it is an edge-deleted Heawood graph. At order 16 there are three possible components: one with a d , at distance 5 which is an edge-deleted Levi graph of the 8 3 configuration; one with a d , at distance 3 and one with a d , adjacent. In principle then, all cubic bipartite graphs with edge connectivity 2, girth at least 6 and order 42 can be constructed by finding all possible components C C , 1 2 such that n n + = 42 1 2 and joining their distinguished vertices as in Figure 4. The join can be done in two (possibly) nonisomorphic ways and is subject to the constraint that at least one of C C , 1 2 must have its distinguished vertices nonadjacent (to avoid creating a 4-cycle). ERSKINE ET AL.

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We therefore proceed as follows. For n = 14, 16, …, 28 we generate using genbg all subcubic bipartite graphs of order n and girth at least 6 with ∕ n 3 2 − 1 edges. Then the idea is that we connect up a graph of order n with a graph of order n 42 − as above, subject to the constraint noted. The resulting cubic graph will have girth at least 6; this process therefore generates the entire population of 2-edge-connected cubic bipartite graphs of order 42. Any blocking set free configuration at v = 21 must have a Levi graph within this population.
Although there are a large number of possible components, it turns out that with modern computers the generation of the components and hence the enumeration of all possible Levi graphs of configurations 21 3 could be completed. Exactly one of the resulting Levi graphs arose from a configuration which failed to have a blocking set. It is illustrated in Figure 1 and its blocks are as follows. We therefore have the following result.
Theorem 2.15. There is a unique symmetric configuration 21 3 having no blocking set; it is the configuration obtained by using three Fano planes in the construction of Theorem 2.11.
As noted above, the symmetric configuration 22 3 with no blocking set illustrated in Figure 1 was found by Dorwart and Grünbaum [11]. In fact we can show that this also is the unique such configuration. We use the same procedure as for the 21 3 configuration, but the search can be considerably shortened by the following simple lemma.  Thus an enumeration of blocking set free configurations on 22 points can be achieved by an exhaustive enumeration of such graphs.
We use the same basic search methodology as in the 21 3 case, but this time we extend the generation of the components C C , 1 2 up to order 30. To guarantee that the resulting graph of order 44 will be non-Hamiltonian, we require that at least one of C C , 1 2 must fail to have a Hamiltonian path between its distinguished vertices. (To check the existence of a Hamiltonian path, we create an augmented graph in which the two distinguished vertices of valency 2 are joined to a new vertex; then the augmented graph is Hamiltonian if and only if there is a Hamiltonian path between the distinguished vertices in the original graph. This technique allows us to use the well-tested cubhamg utility in the nauty package, rather than writing new software for the Hamiltonicity test.) Restricting our search to pairs C C , 1 2 such that at least one component fails to have a Hamiltonian path between the distinguished vertices gives a very substantial reduction in the number of component pairs to be considered. We were thus able to complete the enumeration of the 2-connected non-Hamiltonian cubic bipartite graphs of order 44, and found that only one of these is the Levi graph of a blocking set free configuration 22 3 . Thus we have the following result.
Theorem 2.17. There is a unique symmetric configuration 22 3 having no blocking set; it is the configuration of Dorwart and Grünbaum [11].
Next, a 25 3 configuration without a blocking set can be constructed by Theorem 2.10 using either two 13 3 s or a 7 3 with one of the 19 3 s. Again in [16], Gropp reports that there are at least 19 such configurations. With the assistance of computers in a similar way to the construction of the 19 3 configurations, in fact we find 23 isomorphism classes of configurations 25 3 arising from Theorem 2.10 in this way. The blocks of these are given in the appendix.
All of these systems have connectivity 3 and we now know that there is at least one further system which is 2-connected; thus 25 is the smallest order for which there exist both 3-connected and 2-connected blocking set free systems. Using Theorem 2.13 we can now make an enumeration of 3-connected symmetric configurations v 3 without a blocking set for ∈ v {7, 13, 19, 25, 31, 37, 43}. We do this by repeated application of the v v + ′ − 1 construction in all possible ways, and reducing the resulting configurations to a set of isomorphism class representatives. The results are shown in Table 3.

| STRONG COLOURINGS
In this section we turn our attention to the strong chromatic number χ s of a symmetric configuration, and also investigate its relationship to the weak chromatic number χ w . Our first observation is that the strong chromatic number of a configuration is equal to the chromatic number of its associated graph. Since the associated graph is regular of valency 6 and contains triangles, it follows from Brooks' Theorem that ∈ χ {3, 4, 5, 6, 7} s , and χ = 7 s if and only if the associated graph is a complete graph; that is to say, the configuration is the Fano plane.
The first case to consider is χ = 3 s . An immediate observation is that each block of the configuration must contain exactly one point from each of the three colour classes, and so ≡ v 0 (mod 3). By colouring two classes in the strong colouring (say) red and the third blue, we see that χ = 3 s implies χ = 2 w . ERSKINE ET AL.

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A nice description of strongly 3-chromatic configurations is as follows. From the associated graph of the configuration, form the subgraph induced by the points from any two of the three colour classes. It is easy to see that this induced subgraph is a cubic bipartite graph (not necessarily connected), and a given strong 3-colouring of a configuration gives rise to three cubic bipartite graphs in this way by deleting each of the colour classes. Given any cubic bipartite graph Γ, it is natural to ask whether Γ can arise in this way. Our next result answers this in the affirmative. Then there exists a strongly 3-chromatic symmetric configuration  on m 3 points, and a strong 3-colouring of  , such that the induced subgraph of the associated graph of  formed by deleting the points of one colour class is isomorphic to Γ.
Proof. Our aim is to construct a new 6-regular graph Γ′ on m 3 vertices to be the associated graph of our configuration  . We begin by creating three sets of vertices V V V , , 1 2 3 , each of order m. Between the vertices of V 1 and V 2 we add edges such that the induced subgraph on ∪ V V 1 2 is isomorphic to Γ. We now note that by [7], the edges of Γ can be decomposed into a collection of m copies of the graph K 3 2 , that is, a collection of m sets of three disjoint edges. Each of the m sets of three edges contains exactly six vertices; we construct Γ′ by joining each of the m vertices in V 3 to all the vertices in exactly one of these sets.
Since Γ′ is a 6-regular tripartite graph, any decomposition of its edge set into triangles will yield a strongly 3-chromatic configuration on m 3 points, where the colour classes are the sets V V V , , 1 2 3 . A suitable triangle decomposition is given by using each edge between vertices in V 1 and V 2 together with the two edges joining its endpoints to a vertex in V 3 . By construction, the configuration  represented by this decomposition has the required properties, taking the colour class assigned to V 3 as the one to be deleted. □ In general, the three cubic bipartite graphs formed by deleting a colour class from a strongly 3-chromatic configuration in this way will not be isomorphic. Another natural question is whether we can construct strongly 3-chromatic configurations in such a way that, with a suitable colouring, the resulting colour class deleted graphs are actually isomorphic. It turns out that we can do this for any ≥ v 9 which is a multiple of 3.
T A B L E 3 Numbers of 3-connected blocking set free configurations v 3 v Configurations Self-dual Self-polar Theorem 3.2. Let ≥ s 3 and let v s = 3 . Then there is a cubic bipartite graph Γ of order s 2 , and a strongly 3-chromatic symmetric configuration  on v points, such that deleting any of the three colour classes in a suitable colouring of  we obtain a graph isomorphic to Γ. Then the monochromatic triangles in the above edge-coloring form a triangle decomposition of Γ′. The configuration  represented by this decomposition is strongly 3-chromatic (since Γ′ is tripartite) and deleting any of the three sets in the tripartition leaves a graph isomorphic to Γ. □ Note that the symmetric configuration constructed in the above theorem is resolvable, the sets of monochromatic triangles of the three colours forming the resolution classes. The graph Γ′ is a Cayley graph of the group × s 3   .
We next turn our attention to the case χ = 4 s . It is easy to see that a strongly 4-chromatic configuration is weakly 2-chromatic; if we strongly colour the configuration with colours 1, 2, 3, 4 then we can colour the points in colour classes 1 and 2 blue, and the remainder red. Then no block is monochromatic.
In Table 4 we give computer calculations of the strong chromatic numbers of all connected configurations with ≤ v 15; the numerical evidence is that the case χ = 4 s seems to be the most common. Indeed, our next result shows that we can construct a symmetric configuration with χ = 4 s for all ≥ v 8.
Proof of this theorem is facilitated by the following lemma.  The only values of ≥ v 7 which cannot be written in the form a b 4 + 5 are 7 and 11. If v = 7 then the associated graph of C v is the complete graph K 7 and this has chromatic number 7. If v = 11 then Lemma 3.5 cannot be applied, and computer testing shows that χ C ( ) = 6 s 11 . In fact as Table 4 shows, this is the unique 6-chromatic configuration  Table 4 shows that such a configuration also exists for v = 11 but not v = 7. It remains to determine existence for ≡ v 0 (mod 4), which is more appropriate for us to do later in Theorem 3. (in other words, the configuration contains a blocking set). However, we have been unable to find a proof of this, and so the existence of a symmetric configuration with χ = 5 s and χ = 3 w remains an open question. As a partial result in this direction, we can show that all configurations which are "almost" strongly 4-colourable have weak chromatic number 2.
Theorem 3.6. Suppose that we have a strongly 5-chromatic configuration v 3 in which all but at most two points can be coloured using four colours. Then the weak chromatic number of the configuration is 2.
Proof. First, note that in any 5-colouring each of the v blocks is coloured with one of the ( ) = 10 5 3 possible sets of three colours; and each of these sets must appear at least once if the weak chromatic number is 3. (If a set of three colours does not appear in any block, we can assign blue to these three and red to the other two to get a weak 2-colouring.) Now suppose that we can assign four colours (say 1, 2, 3 and 4) to v − 1 points so that no colour is repeated in a block. Clearly we can assign a fifth colour 5 to the remaining point, and in this 5-colouring at least three sets of three colours must fail to appear in any block, since there are six possible sets containing this colour but only three blocks containing the single vertex with this colour. Thus the configuration has weak chromatic number 2.
If we can only assign four colours to v − 2 points the position is more awkward. Let the two uncoloured points be a and b. Suppose first that a and b do not appear in the same block. We seek an assignment of two of the existing colours 1, 2, 3, 4 to red and the remaining two to blue, such that we can choose red or blue for a and b to obtain a weak 2-colouring. There are exactly three ways to do this initial red/blue assignment: 12/34, 13/24 and 14/23. We shall call an assignment compatible with a if it leaves a possible red/blue choice for a such that no monochromatic block is created. Since a appears in three blocks, it is easy to see that at most one of the three possible assignments is not compatible with a. For example, if the colours of the other points in the blocks containing a are {1, 2}, {3, 4} and {1, 4}, then the assignment 12/34 is incompatible with a but the assignments 13/24 and 14/23 are compatible. Since the same argument holds for b, at least one possible assignment is compatible with a and b and so the configuration has a weak 2-colouring.
If a and b do appear in the same block, then each has two other blocks in which it appears. In this case, not only is there an assignment compatible with both a and b, but also the choice of red/blue for a and b may be made freely. So we can choose red for a and blue for b and again there is a weak 2-colouring. □ Now we come to the case χ = 6 s . Table 4 shows that this is uncommon; of the 269,049 connected configurations v 3 with ≤ ≤ v 8 15, only 18 are strongly 6-chromatic. These are given in the appendix. Nevertheless, we are able to deduce the existence of strongly 6-chromatic configurations for almost all values of v as the next result shows. Proof. The cases v = 11 and 13 follow from Table 4. So let ≥ v 14 and let C 7 be the cyclic configuration on seven points generated by the block {0, 1, 3} under the mapping ↦ i i + 1 (mod 7); this is of course the unique 7 3 configuration and is strongly 7-chromatic. Now choose any connected configuration v ( − 7) 3 and number the points from 7 to v − 1. By relabelling if necessary, we may assume without loss of generality that this configuration contains the block {7, 8, 9}. Now create a new configuration  with the blocks of these two configurations, but replacing the blocks {0, 1, 3} and {7, 8, 9} with {1, 3, 7} and {0, 8, 9}. Suppose  can be strongly coloured with five colours. Then the colours assigned to points 1-6 together with a sixth colour for point 0 would give a strong 6-colouring for the original configuration C 7 , which is impossible. Thus ≥ χ ( ) 6 s  and since ≤ χ ( ) 6 s  by Brooks' Theorem,  is a strongly 6-chromatic connected configuration on v points as required. □ As noted above, all the blocking set free configurations of which we are aware have χ = 6 s . However, only the single example at v = 13 in Table 4  Finally in this section we complete the proof of the existence spectrum for χ = 5 s which we earlier deferred until later. It follows the proof of Theorem 3.7 but is more intricate.  Now let ≥ v 20 and let C 11 be the cyclic configuration on 11 points generated by the block {0, 1, 3} under the mapping ↦ i i + 1 (mod 11); from Theorem 3.4 this is strongly 6-chromatic. Now choose any connected configuration v ( − 11) 3 with χ = 5 s and number the points from 11 to v − 1. From Theorem 3.4 this is possible. By relabelling if necessary, we may assume without loss of generality that this configuration contains the block {11, 12, 13} and that in the strong 5-colouring, these points receive colours red, yellow and blue, respectively. Now create a new configuration  with the blocks of these two configurations but replacing the blocks {0, 1, 3} and {11, 12, 13} with {1, 3, 11} and {0, 12, 13}. Suppose  can be strongly coloured with four colours. Then the colours assigned to points 1-10 together with a fifth colour assigned to point 0 would give a strong 5-colouring of the original configuration C 11 , which is impossible. Thus ≥ χ ( ) 5 s  , and since ≤ χ ( ) 6 s  by Brooks' Theorem,  is either strongly 5-or 6-chromatic. It remains to show that it is the former by exhibiting a colouring.
Colour the blocks of the v ( − 11) 3 configuration without the block {11, 12, 13} with five colours red, yellow, blue, green and white, respecting that colours have already been assigned to points 11, 12 and 13. Colour the remaining points as follows: 4 and 8 red; 2 and 9 yellow; 3 and 7 blue; 0, 1 and 5 green; 6 and 10 white. □

| OPEN QUESTIONS
We gather here some of the interesting open questions arising from this study. The first of these relates to symmetric configurations 25 3 without a blocking set. We now have enumerations of all symmetric configurations v 3 for ≤ ≤ v 7 2 0 and all 3-connected symmetric configurations without a blocking set for ≤ ≤ v 7 4 3. There are unique configurations 21 3 and 22 3 without a blocking set, both necessarily 2-connected, and a 2-connected configuration 25 3 without a blocking set is known. The question remains whether this is unique.
The second problem is to extend the work on the sizes of minimal blocking sets, possibly along the lines of Theorems 2.4 and 2.5. In particular it would be interesting to find constructions of symmetric configurations v 3 whose minimal blocking set has maximum cardinality, that is, if v is even. The admittedly limited evidence from Table 1 suggests that such configurations exist except for v = 7 (where there is no blocking set) and v = 14, though amongst the set of all symmetric configurations they may be relatively rare. However, given the long history of blocking set free symmetric configurations, finding those with only minimal blocking sets of maximum cardinality may also be quite challenging.
The third problem concerns the relationship between the strong and the weak chromatic numbers. We have observed that if χ = 3 s or 4 then χ = 2 w and that there are configurations with χ χ ( , ) s w equal to both (6,2) and (6,3). However all of the known systems with χ = 5 s have χ = 2 w . So we ask does there exist a symmetric configuration v 3 with strong chromatic number 5 and weak chromatic number 3? Equivalently, does every blocking set free configuration have strong chromatic number 6?
Finally, as we observed, a given strong 3-colouring of a strongly 3-chromatic configuration gives rise to three cubic bipartite graphs by deleting each of the colour classes from the associated graph of the configuration. Denote these graphs by Γ 1 , Γ 2 and Γ 3 . In Theorem 3.1 we proved that one of these graphs, say Γ 1 , can be any cubic bipartite graph. Now suppose that Γ 1 , Γ 2 and Γ 3 are all specified. Does there exist a symmetric configuration v 3 whose three cubic bipartite graphs constructed as above are isomorphic to Γ 1 , Γ 2 and Γ 3 ? If not, what are the constraints on these three graphs for this to be possible? The case where Γ 1 , Γ 2 and Γ 3 are isomorphic would be of particular interest.
There are of course other problems on symmetric configurations and we hope that this paper will encourage colleagues to work on these.