https://doi.org/10.26493/1855-3974.1435.c71 (Also available at http://amc-journal.eu)
Vertex transitive graphs G with χ D (G) > χ(G) and small automorphism group ∗
Niranjan Balachandran
Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, India
Sajith Padinhatteeri
†Department of ECE, Indian Institute of Science, Bangalore, India
Pablo Spiga
Dipartimento Di Matematica E Applicazioni, University of Milano-Bicocca, Milano, Italy Received 24 June 2017, accepted 1 October 2019, published online 29 October 2019
Abstract
For a graphGand a positive integerk, a vertex labellingf: V(G)→ {1,2, . . . , k}is said to bek-distinguishing if no non-trivial automorphism ofGpreserves the setsf−1(i) for each i ∈ {1, . . . , k}. The distinguishing chromatic number of a graph G, denoted χD(G), is defined as the minimum k such that there is ak-distinguishing labelling of V(G)which is also a proper coloring of the vertices ofG. In this paper, we prove the following theorem: Givenk ∈ N, there exists an infinite sequence of vertex-transitive graphsGi= (Vi, Ei)such that
1. χD(Gi)> χ(Gi)> k,
2. |Aut(Gi)|<2k|Vi|, whereAut(Gi)denotes the full automorphism group ofGi. In particular, this answers a question posed by the first and second authors of this paper.
Keywords: Distinguishing chromatic number, vertex transitive graphs, Cayley graphs.
Math. Subj. Class.: 05C15, 05D40, 20B25, 05E18
∗The first and second authors would like to thank Ted Dobson for useful discussions.
†Supported by grant PDF/2017/002518, Science and Engineering Research Board, India.
E-mail addresses:[email protected] (Niranjan Balachandran), [email protected] (Sajith Padinhatteeri), [email protected] (Pablo Spiga)
cbThis work is licensed under https://creativecommons.org/licenses/by/4.0/
1 Introduction
LetGbe a graph. An automorphism ofGis a permutationϕof the vertex setV(G)ofG such that, for anyx, y∈V(G),ϕ(x), ϕ(y)are adjacent if and onlyx, yare adjacent. The automorphism group of a graphG, denoted byAut(G), is the group of all automorphisms of G. A graph G is said to be vertex transitive if, for any u, v ∈ V(G), there exists ϕ∈Aut(G)such thatϕ(u) =v.
Given a positive integerr, anr-coloring ofGis a mapf:V(G)→ {1,2, . . . , r}and the setsf−1(i), for i ∈ {1,2. . . , r}, are the color classes off. An automorphismϕ ∈ Aut(G)is said to fix a color classCoff ifϕ(C) = C, whereϕ(C) = {ϕ(v) :v ∈C}.
A coloring ofG, with the property that no non-trivial automorphism ofGfixes every color class, is called a distinguishing coloring ofG.
Collins and Trenk in [5] introduced the notion of the distinguishing chromatic number of a graphG, which is defined as the minimum number of colors needed to color the ver- tices ofGso that the coloring is both proper and distinguishing. Thus, the distinguishing chromatic number ofGis the least integerr such that the vertex set can be partitioned into setsV1, V2, . . . , Vr such that eachVi is independent inG, and for every non-trivial ϕ∈ Aut(G)there exists some color classViwithϕ(Vi) 6=Vi. The distinguishing chro- matic number of a graphG, denoted byχD(G), has been the topic of considerable interest recently (see, for instance, [1,2,3,4]).
One of the many questions of interest regarding the distinguishing chromatic number concerns the contrast between χD(G) and the cardinality ofAut(G). For instance, the Kneser graphsK(n, r)have very large automorphism groups and yet, χD(K(n, r)) = χ(K(n, r))forn ≥ 2r+ 1, and r ≥ 3(see [2]). The converse question is compelling:
Are there infinitely many graphs Gn with ‘small’ automorphism groups and satisfying χD(Gn)> χ(Gn)?
The question as posed above is not actually interesting for two reasons. First, for all evenn,χD(Cn)> χ(Cn) = 2and|Aut(Cn)|= 2n, whereCnis the cycle of lengthn.
Second, if one stipulates thatGalso has arbitrarily large chromatic number, then here is a construction for such a graph. Start with a rigid graphGwith a leaf vertexxand having large chromatic number (one can obtain this by minor modifications to a random graph, for instance); then, blow up the leaf vertexxto a new disjoint setXwhose neighbor in the new graphGeis the same as the neighbor ofxinG. In fact one can arrange forχD(G)e −χ(G)e to be as large as one desires. Furthermore, since|Aut(G)|e =|X|!, this provides examples of graphs for which the automorphism groups are relatively ‘small’ in terms of the order of the graph.
In the example above, the fact that χD(G) is larger thanχ(G)is accounted for by a ‘local’ reason, and that is what makes the problem stated above not very interesting.
However, if one further stipulates that the graph is vertex-transitive, then the same question is highly non-trivial. In [1], the first and second authors constructed families of vertex- transitive graphs withχD(G)> χ(G)> kand|Aut(G)|=O(|V(G)|3/2), for any given k. In this paper, we improve upon that result:
Theorem 1.1. Givenk ∈ N, there exists an infinite family of graphs Gn = (Vn, En) satisfying:
1. χD(Gn)> χ(Gn)> k,
2. Gnis vertex transitive and|Aut(Gn)|<2k|Vn|.
Our family of graphs consists of Cayley graphs. To recall the definition, letAbe a group and letS be an inverse-closed subset ofA, i.e.,S =S−1, whereS−1 :={s−1:s∈S}.
The Cayley graphCay(A, S)is the graph with vertex setAand the verticesuandvare adjacent inCay(A, S)if and only ifuv−1∈S.
We start with a brief description of the graphs of our construction. Forq, an odd prime, letFnq denote then-dimensional vector space overFq. Our graphs shall be Cayley graphs Cay(Fnq, S) for some suitable inverse-closed set S ⊂ Fnq which is obtained by taking a union of a certain collection of lines inFnq and then deleting the zero element of Fnq. More precisely, letH0 := {(x1, x2, . . . , xn−1,0) : xi ∈ Fq,1 ≤ i ≤ n−1} and let 0denote the element (0, . . . ,0) ∈ Fnq. For each line (1-dimensional subspace of Fnq)
` ⊂ Fnq satisfying ` ∩ H0 = {0}, pick` independently with probability 1/2 to form the random setS. Our connection sete S for the Cayley graphCay(Fnq, S)is defined by S :={v ∈ Fnq : v ∈ `for some` ∈ S} \ {0}. Our main theorem states that with highe probability,Gn,S := Cay(Fnq, S)satisfies the conditions of Theorem1.1.
To show that these graphs have ‘small’ automorphism groups, we prove a stronger version of Theorem 4.3 of [6] in this particular context, which is also a result of independent interest.
Theorem 1.2. Let qbe a prime power, let n be a positive integer withn ≥ 2 and let Gbe the additive group of the n-dimensional vector space Fnq over the finite fieldFq of cardinalityq, and letF∗q := Fq \ {0}be the multiplicative group of the fieldFq with its natural group action onGby scalar multiplication, and writeK :=Fnq o F∗q. IfSis an inverse-closed subset ofGwithK≤Aut(Cay(G, S)), then either
(i) Aut(Cay(G, S)) =K, or
(ii) there existsϕ∈Aut(Cay(G, S))\KwithϕnormalizingG.
Remark 1.3. Theorem1.2is valid even though the connection setSis not inverse-closed.
Since we deal with Cayley graphs the phrase inverse-closed subset is used in the statement of the theorem.
The rest of the paper is organized as follows. We start with some preliminaries in Section2and then include the proofs of Theorems 1.1and1.2in the next section. We conclude with some remarks and some open questions.
2 Preliminaries
We begin with a few definitions from finite geometry. For more details, one may see [13, 14]. By PG(n, q) we mean the Desarguesian projective space obtained from the affine spaceAG(n+ 1, q).
Definition 2.1. A cone with vertexA⊂PG(k, q)and baseB⊂PG(n−k−1, q), where PG(k, q)∩PG(n−k−1, q) =∅, is the set of points lying on the lines connecting points ofAandB.
Definition 2.2. LetV be an(n+ 1)-dimensional vector space over a finite fieldF. A subsetSofPG(V)is called anFq-linear set if there exists a subsetU ofV that forms an Fq-vector space, for someFq ⊂F, such thatS=B(U), where
B(U) :={huiF:u∈U\ {0}}
and wherehuiFdenotes the projective point ofPG(V), corresponding to the vectoruof U ⊂V.
Further details aboutFq-linear sets can be found in [14], for instance.
The projective spacePG(n, q)can be partitioned into an affine spaceAG(n, q)and a hyperplane at infinity, denoted byH∞.
Definition 2.3. Following [13], we say that a set of pointsU ⊂AG(n, q)determines the directiond∈H∞,if there is an affine line throughdmeetingU in at least two points.
We now state the main theorem of [13] which will be relevant in our setting.
Theorem 2.4. Let U ⊂ AG(n,Fq),n ≥ 3, |U| = qk. Suppose thatU determines at most q+32 qk−1+qk−2+· · ·+q2+qdirections and suppose thatU is anFp-linear set of points, whereq = ph,p > 3prime. Ifn−1 ≥(n−k)h,thenU is a cone with an (n−1−h(n−k))-dimensional vertex atH∞and with base aFq-linear point setU(n−k)h of sizeq(n−k)(h−1), contained in some affine(n−k)h-dimensional subspace ofAG(n, q).
We end this section by recalling another result that appears in [6] as Theorem 4.2.
Theorem 2.5. LetGbe a permutation group onΩwith a proper self-normalizing abelian regular subgroup. Then|Ω|is not a prime power.
3 Proofs of the Theorems
In this section we prove Theorems1.1and1.2starting with the proof of Theorem1.2. We believe that this result is only the tip of an iceberg: its current statement has been tailored to the context of our setting, and uses some ideas that appear in [6, Section 3] and [9].
Proof of Theorem1.2. We suppose that(i)does not hold, that is,Kis a proper subgroup ofAut(Cay(G, S)); we show that(ii)holds. WriteΓ := Cay(G, S).
LetBbe a subgroup ofAut(Γ)withK < Band withKmaximal inB. Suppose that KCB. AsGis characteristic inK, we getGCB. In particular, every elementϕinB\K satisfies(ii).
Suppose then that Kis not normal in B. SinceK is maximal inB andGCK, we haveNB(G) = K. Suppose that there existsb ∈ B\K such thatL := hG, Gbi(the smallest subgroup ofBcontainingGandGb) satisfiesL∩K =G. We claim that we are now in the position to apply Theorem2.5(and implicitly some ideas from [9]). Indeed, as NL(G) =NB(G)∩L=K∩L=G,Lis a transitive permutation group on the vertices of Γwith a proper regular self-normalizing abelian subgroupG. (Observe thatGis a proper subgroup ofLbecauseb /∈ NB(G) = K.) By Theorem2.5,|G|is not a prime power, which is a contradiction because|G|=qn. This proves that, for everyb∈B\K, we have hG, Gbi ∩K > G.
Fixb∈B\K. Now,GandGbare abelian and henceG∩Gbis centralized byhG, Gbi.
From the preceding paragraph, there existsk∈ hG, Gbi∩Kwithk /∈G. Observe now that K =Fnq o F∗q is a Frobenius group with kernelG=Fnq and complementF∗q. Therefore, kacts by conjugation fixed-point-freely onG\ {0}. AskcentralizesG∩Gb, we deduce
|G∩Gb|= 1.
LetC:=T
x∈BKxbe the core ofKinB. AsG∩Gb= 1for allb∈B\K,K∩Kb has no non-identity q-elements. ThereforeC∩G = 1. AsCCB andC ≤ K,C is
a normal subgroup of the Frobenius groupKintersecting its kernel on the identity. This yieldsC= 1.
LetΩbe the set of right cosets ofKinB. From the paragraph above,Bacts faithfully onΩ. Moreover, asK is maximal inB, the action ofB onΩis primitive. ThereforeB is a finite primitive group with a solvable point stabilizerK. In [11], Li and Zhang have explicitly determined such primitive groups: these are classified in [11, Theorem 1.1] and [11, Tables I–VII]. Now, using the terminology in [11], a careful (but not very difficult) case-by-case analysis on the tables in [11] shows thatBis a primitive group of affine type, that is,B contains an elementary abelian normalr-subgroupV, for some primer. For this analysis it is important to keep in mind that the stabilizerKis a Frobenius group with kernel the elementary abelian groupG∼=Fnq andn≥2.
Let|V|=rt. Now, the action ofBonΩis permutation equivalent to the natural action ofB = V oK onV, withV acting via its regular representation and withK acting by conjugation. Observe thatq 6= r, becauseK acts faithfully and irreducibly as a linear group onV and henceKcontains no non-identity normalr-subgroups. Observe further that|B|=|V||K|=rt·qn·(q−1).
We are finally ready to reach a contradiction and to do so, we go back studying the action ofBon the vertices ofΓ. Observe thatBis solvable becauseV is solvable and so is B/V ∼=K. We writeB0for the stabilizer inBof the vertex0ofΓ. AsGacts regularly on the vertices ofΓ, we obtainB=B0GandB0∩G= 1. In particular,|B0|=rt·(q−1).
Observe thatB0is a HallΠ-subgroup of the solvable groupB, whereΠis the set of all the prime divisors ofq−1together with the primer. AsV is aΠ-subgroup, from the theory of Hall subgroups (see for instance [7], Theorem 3.3),V has a conjugate contained inB0. SinceV CB, we haveV ≤B0. This is clearly a contradiction becauseV is normal inB, butB0is core-free inB, being the stabilizer of a point in a transitive permutation group.
For the next lemma, recall that
H0:={(x1, x2, . . . , xn−1,0) :xi∈Fq,1≤i≤n−1}.
In what follows, Gn,S will denote the Cayley graph Cay(Fnq, S)andS = Se\ {0} for some setSe =S
`∈L`, whereLis a collection of lines inFnq with each` ∈ Lsatisfying
`∩ H0={0}.
Lemma 3.1. IfL 6=∅, thenχ(Gn,S) =q.
Proof. Observe that each line that belongs to the setSgives rise to a clique of sizeqin the graphGn,S. Thereforeχ(Gn,S) ≥q. On the other hand, for a fixedv ∈S, the partition (Cλ)λ∈Fq, whereCλ :={w+λv : w ∈ H0}, of the vertex setFnq is a proper coloring of the graphGn,S. Indeed, for any distinctx= w1+λv,y =w2+λvinCλ, we have x−y=w1−w2∈/Sbecausew1−w2∈ H0andS∩ H0=∅. Therefore the setsCλare independent inGn,S for eachλ∈Fq.
Lemma 3.2. Assume thatqis prime. LetSebe the random set corresponding to a union of lines`inFnq with`∩ H0 ={0}and where each`∈Fnq is chosen independently with probability12; and letS=Se\ {0}. Then
P(χD(Gn,S)> q)≥1−exp
−qn−3 4
.
Proof. First, note thatE(|S|) = qn−12 , so takingδ = 1q andµ = E(|S|)in the Chernoff bound (see (2.6) on page26of [10]) we obtain
P
|S|<qn−1−qn−2 2
≤exp
−qn−3 4
.
In particular, with probability at least1−exp(−qn−3/4), we have|S|> qn−1−q2 n−2. We may thus assume|S|> qn−1−q2 n−2 in what follows.
We claim that every color class in a properq-coloring ofGn,Sis an affine hyperplane of Fnq. To see why, letC1, . . . , Cq be independent sets inGn,Switnessing a properq-coloring ofGn,S. Fixv ∈ Sand consider the line`v := {λv : λ ∈Fq}along with its translates
`v +w := {λv+w : λ ∈ Fq}, forw ∈ H0. Each set`v +wis a clique of sizeqin Gn,S, and these cliques partition the vertex set ofGn,S, so in particular eachCicontains at most one vertex from each of these translates`v+w. Consequently,|Ci| ≤qn−1for all i∈ {1, . . . , q}. By size considerations, it follows that|Ci|=qn−1for eachi∈ {1, . . . , q}.
Consider a color classC. SupposeCdetermines at leastq+32 qn−2+qn−3+· · ·+q2+ q+ 1directions. Then ifhCidenotes the set of all vertices in the affine lines intersecting at least two points inC, we have|hCi|+|S| > 1 +q+· · ·+qn−1, sohCi ∩S 6= ∅.
However, this contradicts the assumption thatCis an independent set inGn,S. Therefore C determines at most q+32 qn−2+qn−3+· · ·+q2+qdirections. Sinceq is prime, by Corollary 10 in [13], it follows thatCis anFq-linear set. Hence, by Theorem2.4, the color classCis a cone with ann−2(projective) dimensional vertexVatH∞and an affine point u1as base. In particular, the affine plane corresponding to theFq-subspace spanned byV passing through the affine pointu1is contained inC. Since|C|=qn−1, it follows thatC is this affine hyperplane, and this proves the claim.
To complete the proof, observe that for eachλ∈F∗q\{1}, the mapϕλ(x) =λx,x∈Fnq
fixes each color class. Moreover,ϕλfixes the setSandϕλ(u)−ϕλ(v) =ϕλ(u−v), soϕλ
is a non-trivial automorphism which fixes each color class. ThereforeχD(Gn,S)> q.
Lemma 3.3. Ifn≥6andq≥5is prime, thenAut(Gn,S)∼=Fnq o F∗qwith probability at least
1−2−qn
−1 3 .
Proof. SinceGn,S is a Cayley graph on the additive group G = Fnq, by Theorem 1.2, eitherAut(Gn,S) =K∼=Fnqo F∗qor there existsϕ∈Aut(Gn,S)\Kwithϕnormalizing G=Fnq. We show that with probability at least1−2−qn
−1
3 , there is noϕsatisfying the latter condition.
Suppose ϕ ∈ Aut(Gn,S)normalizesFnq. Ifa = ϕ(0)andλa : Fnq → Fnq is the right translation viaa, thenλ−1a ϕis an automorphism ofGn,S normalizingFnq and with (λ−1a ϕ)(0) = (λ−1a )(ϕ(0)) = (λ−1a )(a) =a−a=0. Therefore, without loss of gener- ality, we may assume thatϕ(0) =0. SinceSis the neighbourhood of0inGn,S, we get ϕ(S) =S. Moreover, sinceϕacts as a group automorphism onFnq, we haveϕ∈GLn(q).
Now, for ϕ ∈ GLn(q), letEϕ denote the event ϕ(S) = S. Let L denote the set of all lines` with`∩ H0 = ∅. Also, letOrbϕ(`) = {`, ϕ(`), ϕ2(`), . . . , ϕk(`)} where ϕk+1(`) =`. Then
P(Eϕ)≤
Nϕ
Y
i=1
21−|Orbϕ(`i)|= 2Nϕ−|L|,
whereNϕdenotes the number of distinct orbits ofϕinL. SettingG = GL(n, q)\ {λI : λ∈F∗q}, we have
P
[
ϕ∈G
Eϕ
≤X
ϕ∈G
P(Eϕ)≤2−|L|X
ϕ∈G
2Nϕ. (3.1)
LetFϕ:=|{`∈ L:ϕ(`) =`}|andF := maxϕ∈GFϕ. NowNϕ≤F+|L|−F2 =F+|L|2 . Thus, it suffices to give a suitable upper bound forF. Towards that end, we note that, if Fϕ =F forϕ ∈ G, then every line`fixed byϕcorresponds to an eigenvector ofϕ. If E1,E2, . . . ,Ek denote the eigenspaces ofϕfor some distinct eigenvaluesλ1, . . . , λk, then
Fϕ≤
k
X
i=1
dimEi
1
q
−
dim(Ei∩ H0) 1
q
!
≤qn−2+ 1.
Similarly, we have|L|= n1
q− n−11
q =qn−1, and so by (3.1), we have
P
[
ϕ∈G
Eϕ
≤ |G|2F−|L|2 < qn22−qn
−1−qn−2−1
2 <2−qn
−1 3 ,
forq≥5,n≥6.
Computations and estimates similar to the ones presented in the proof of Lemma3.3 have been proved useful in a variety of problems, see for instance [1,8] and [12, Sec- tion 6.4].
Proof of Theorem1.1. Givenk∈Nwithk≥4, pick a prime numberqwithk < q <2k.
Forn ≥ 6, consider the random graphGn,S of the groupFnq as constructed above. By Lemmas3.1,3.2and3.3, with positive probability, the graphGn,Ssatisfies the statements of the lemmas, and hence satisfies the conclusions of Theorem1.1.
4 Concluding remarks
• We observe that, forSchosen randomly as in the proof of our result, the distinguish- ing chromatic number ofGn,S isq+ 1with high probability. Indeed, consider the q-coloringC described in Lemma3.1. Re-color the vertex0using an additional color. Then the coloring described by the partitionC0 =C∪ {0}is a proper, dis- tinguishing coloring ofGn,S withq+ 1colors. In fact,C0 is clearly proper, and to show that it is distinguishing, considerϕ∈Aut(Gn,S) =Fnq o F∗q (by Lemma3.3) that fixes every color class. Writeϕ(x) =λx+bwithλ∈F∗q, b∈Fnq. Sinceϕfixes the color class containing0, we haveb=0. Also,xandλxcannot be in same color class unlessλ= 1. Thereforeϕis the identity automorphism.
It is interesting to determine if one can obtain families of vertex-transitive graphs withχD(G)> χ(G) + 1, with ‘small’ automorphism groups and withχ(G)being arbitrarily large. In fact, fork ∈ N, there is no known family of vertex-transitive graphs for whichχD(G)> χ(G) + 1> kand|Aut(G)| =O(|V(G)|O(1)). It is plausible that Cayley graphs over certain groups may provide the correct construc- tions.
• Theorem 1.1establishes, for any fixed k, the existence of vertex-transitive graphs Gn = (Vn, En)withχD(Gn) > χ(Gn) > k and with|Aut(Gn)| < 2k|Vn|. It would be interesting to obtain a similar family of graphs that satisfy withχD(Gn)>
χ(Gn)> kand with|Aut(Gn)| ≤C|Vn|, for some absolute constantC.
References
[1] N. Balachandran and S. Padinhatteeri,χD(G),|Aut(G)|and a variant of the Motion Lemma, Ars Math. Contemp.12(2017), 89–109, doi:10.26493/1855-3974.848.669.
[2] Z. Che and K. L. Collins, The distinguishing chromatic number of Kneser graphs,Electron.
J. Combin. 20 (2013), #P23 (12 pages), https://www.combinatorics.org/ojs/
index.php/eljc/article/view/v20i1p23.
[3] J. O. Choi, S. G. Hartke and H. Kaul, Distinguishing chromatic number of Cartesian products of graphs,SIAM J. Discrete Math.24(2010), 82–100, doi:10.1137/060651392.
[4] K. L. Collins, M. Hovey and A. N. Trenk, Bounds on the distinguishing chromatic number, Electron. J. Combin.16(2009), #R88 (14 pages),https://www.combinatorics.org/
ojs/index.php/eljc/article/view/v16i1r88.
[5] K. L. Collins and A. N. Trenk, The distinguishing chromatic number,Electron. J. Combin.
13(2006), #R16 (19 pages),https://www.combinatorics.org/ojs/index.php/
eljc/article/view/v13i1r16.
[6] E. Dobson, P. Spiga and G. Verret, Cayley graphs on abelian groups,Combinatorica36(2016), 371–393, doi:10.1007/s00493-015-3136-5.
[7] K. Doerk and T. O. Hawkes,Finite Soluble Groups, volume 4 ofDe Gruyter Expositions in Mathematics, De Gruyter, Berlin, 1992, doi:10.1515/9783110870138.
[8] S. Guest and P. Spiga, Finite primitive groups and regular orbits of group elements,Trans.
Amer. Math. Soc.369(2017), 997–1024, doi:10.1090/tran6678.
[9] E. Jabara and P. Spiga, Abelian Carter subgroups in finite permutation groups, Arch. Math.
(Basel)101(2013), 301–307, doi:10.1007/s00013-013-0558-4.
[10] S. Janson, T. Łuczak and A. Ruci´nski,Random Graphs, Wiley-Interscience Series in Dis- crete Mathematics and Optimization, John Wiley & Sons, New York, 2000, doi:10.1002/
9781118032718.
[11] C. H. Li and H. Zhang, The finite primitive groups with soluble stabilizers, and the edge- primitives-arc transitive graphs,Proc. Lond. Math. Soc.103(2011), 441–472, doi:10.1112/
plms/pdr004.
[12] P. Potoˇcnik, P. Spiga and G. Verret, Asymptotic enumeration of vertex-transitive graphs of fixed valency,J. Comb. Theory Ser. B122(2017), 221–240, doi:10.1016/j.jctb.2016.06.002.
[13] L. Strome and P. Sziklai, Linear point sets and R´edei typek-blocking setsPG(n, q),J. Alge- braic Combin.14(2001), 221–228, doi:10.1023/a:1012724219499.
[14] G. V. Voorde,Blocking Sets in Finite Projective Spaces and Coding Theory, Ph.D. thesis, Ghent University, Belgium, 2010.
