AbstractWe show that the result of Watkins (1990) [19] on constructing vertex-transitive non-Cayley graphs from line graphs yields a simple method that produces infinite families of vertex-transitive non-Cayley graphs from Cayley graphs generated by involutions. We also prove that the graphs arising this way are hamiltonian provided that their valency is at least six

Tomanová, Jana

Elsevier - Publisher Connector 

Discrete Mathematics 310 (2010) 192–195
Contents lists available at ScienceDirect
Discrete Mathematics
journal homepage: www.elsevier.com/locate/disc
Note
A note on vertex-transitive non-Cayley graphs from Cayley graphs
generated by involutions
Jana Tomanová
Comenius University, Bratislava, Slovakia
a r t i c l e i n f o
Article history:
Received 25 August 2008
Received in revised form 15 July 2009
Accepted 28 July 2009
Available online 8 August 2009
Keywords:
Vertex-transitive non-Cayley graph
Line graph
a b s t r a c t
We show that the result of Watkins (1990) [19] on constructing vertex-transitive non-
Cayley graphs from line graphs yields a simple method that produces infinite families of
vertex-transitive non-Cayley graphs from Cayley graphs generated by involutions. We also
prove that the graphs arising this way are hamiltonian provided that their valency is at
least six.
© 2009 Elsevier B.V. All rights reserved.
1. Introduction
The growing interest in constructions of vertex-transitive non-Cayley graphs (for which we will be using the acronym
VTNCGs throughout) goes, to a large extent, back to Marus˘ic˘’s call [12] for a characterization of the set of all possible orders
of VTNCGs. Proving that a particular vertex-transitive graph is not a Cayley graphmay be hard. Two types of generalmethods
for this purpose have crystallized in the past two decades.Directmethods are based on Sabidussi’s characterization of Cayley
graphs [16]. Their essence lies in showing that the automorphism group of a given graph has no subgroup acting regularly
on vertices of the graph, see [5,14,15]. Indirect methods try to identify structural properties of Cayley graphs which they do
not share with vertex-transitive graphs in general. Examples include residue classes (modulo the valency) of the number of
closed walks at a fixed vertex [6], configurations fixed by involutions [13], and relations between a graph and its line graph
[19].
The aim of this note is to show that the ‘line-graph’ method of [19] can be used to produce VTNCGs from Cayley graphs
generated by involutions. The construction (with a number of examples) is presented in Section 2 and further extended in
Section 3. The final section contains a remark on hamiltonicity of the resulting VTNCGs in the case of valency at least six.
2. The construction
Our method of constructing VTNCGs from Cayley graphs generated by involutions is based on a straightforward
application of the following interesting result of Watkins [19] which seems to have been somewhat neglected in the
literature.
Theorem A ([19]). Let Γ be a graph on at least five vertices that is edge-transitive, non-bipartite and whose vertices have odd
valency. Then the line graph of Γ is a VTNCG.
E-mail address: tomanova@fmph.uniba.sk.
0012-365X/$ – see front matter© 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.disc.2009.07.022
J. Tomanová / Discrete Mathematics 310 (2010) 192–195 193
Recall that for a given (finite) group G and a generating set X for G such that 1 6∈ X and X = X−1, the Cayley graph C(G, X)
has vertex set G, and two vertices g, h ∈ G are adjacent if gh−1 ∈ X , which is equivalent to hg−1 ∈ X . In this setting, right
multiplication by any fixed element of G induces an automorphism of the Cayley graph and in this way G acts regularly on
the vertices of C(G, X). The valency of C(G, X) is simply |X |, the number of elements in the generating set. If |X | is odd, then
due to the fact that X is closed under taking inverses, X contains an odd number of involutions. The extreme case when all
elements in X are involutions can be used for constructing VTNCGs using automorphisms of the group G. Any automorphism
ofG that preservesX setwise is a graph automorphismof C(G, X). Moreover, assume thatX is an orbit of a subgroup ofAut(G)
such that |X | is odd. Then, because any group automorphism preserves the order of elements, all the elements of an orbit
must be of the same order, and since |X | is odd, and as such has to contain at least one involution, all the elements in X have
to be involutions. This is the reason why the all-involution assumption has been included in the statement of the following
consequence of Theorem A.
Corollary 1. Let C(G, X) be a non-bipartite Cayley graph of order at least five and let X consist of an odd number of involutions.
If X is an orbit of some subgroup of Aut(G), then the line graph of C(G, X) is a VTNCG.
Proof. LetH be a subgroup of Aut(G) such that X is an orbit ofH . The regular action ofG on C(G, X) combinedwith the action
of H guarantees that the group Aut(C(G, X)) acts transitively on the arcs (and hence on the edges) of C(G, X). By Theorem A,
the line graph of C(G, X) is a VTNCG. 
The simplest way to construct a set X satisfying the assumption of our Corollary is to take a conjugacy class of a single
involution x by an element y of odd order. To make sure that the graph C(G, X) is not bipartite, it is sufficient to consider
groups G containing no subgroups of index two. Indeed, if C(G, X) is bipartite, then the subgroup H generated by all the
products abwhere a, b ∈ X is a subgroup of index two in G. This leads us to the following application of Corollary 1.
Corollary 2. Let G be a group with generating set {x, y} where x is an involution and y is of odd order d ≥ 3, and let X =
{yixy−i : 0 ≤ i < d}. If G does not contain proper normal subgroups of index less than or equal to d, then the line graph of
C(G, X) is a connected VTNCG.
Proof. In view of the facts listed above it is sufficient to show that X is a generating set for G. Observe that the subgroup
〈X〉 of G = 〈x, y〉 is invariant under conjugation by both y and x and hence is normal in G. We show that G/〈X〉 is a cyclic
group of order a divisor of d, and hence the index of 〈X〉 in G is a divisor of d as well. Our assumption G = 〈x, y〉 ensures
that G/〈X〉 = 〈〈X〉, y〈X〉〉. It is now clear that if y belongs to 〈X〉 then G = 〈X〉 and we are done. Assume that y 6∈ 〈X〉. Then
G/〈X〉 is a cyclic group whose order is the least integer m such that ym ∈ 〈X〉 and 1 < m < d. But then 1 = (ym)s = (ys)m,
s being the order of ym in G, which implies thatm is a divisor of d. Thus the index of 〈X〉 in G is a divisor of d as well. By our
assumptions, however, G has no such proper normal subgroup and therefore 〈X〉 = G. The rest follows from Corollary 1. 
We note that the assumption that G does not contain normal subgroups of index at most d can be replaced with the
assumption ‘G has no subgroup of index 2 and x is not contained in any proper normal subgroup of G’.
Since simple groups automatically satisfy the last assumption of Corollary 2 (as well as its version described above), they
are prime candidates for a possible application of this result. The hard part here is to find a generating set consisting of an
involution and an element of odd order. Herewe invoke a consequence of [8,9] bywhich all finite non-abelian simple groups
except Suzuki groups, the symplectic groups PSp4(2k) and PSp4(3k), and a finite number of other exceptions are generated
by an involution and an element of order 3. Corollary 2 then implies that for all such simple groups G there exists a Cayley
graph C(G, X) of valency 3 such that its line graph is a connected VTNCG.
Anotherway to satisfy the assumptions of Corollary 1 is to take conjugacy classes that cannot be generated by conjugation
by powers of a single element.
Example 1. Consider the alternating group G = An, where n ≥ 5 and its subset X = {(ij)(kl) : |{i, j, k, l}| = 4 and 1 ≤
i, j, k, l ≤ n}. It is well known that An is, for all n ≥ 3, generated by the set of all 3-cycles. Since any 3-cycle (ijk) is a product
((ik)(i′k′))((i′k′)(kj)) of elements in X , where all i, j, k, i′, k′ are pairwise distinct (note that we have assumed n ≥ 5), it
follows that X generates G. It is clear that |X | = 3 ( n4 ) and therefore, |X | is odd if and only if n ≡ 4, 5, 6, 7(mod 8).
For application of Corollary 1 we take the symmetric group Sn as the subgroup of Aut(G); as an aside, we mention the
well known fact that Sn is isomorphic to Aut(An) for all n except 2 and 6. Two permutations are conjugate in Sn if and only
if they have the same cycle type. This together with the fact that the set X comprises all permutations of a given cycle type
implies that the symmetric group Sn fulfills the assumption of Corollary 1 including non-bipartiteness since An is simple for
n ≥ 5. It follows that the line graph of C(G, X) is a VTNCG.
Our next example is an illustration of a situation when one can take the generating set to be the set of all involutions of
a group.
Example 2. Consider the projective special linear group G = PSL(2, q), where q > 3 is a prime power, and the subset X
consisting of all involutions in G. We will show that the line graph of C(G, X) is a VTNCG. Suppose first that q is an odd
number. Observe that A ∈ G is an involution if and only if A =
(
a b
c −a
)
where a2 + bc = −1 and therefore, it is sufficient
194 J. Tomanová / Discrete Mathematics 310 (2010) 192–195
to determine the number of triples (a, b, c) such that a2 + bc = −1. Recalling that the multiplicative group of GF(q) is
a cyclic group of order q − 1, it follows that −1 is a square in GF(q) if and only if q ≡ 1(mod 4). It is now clear that if
q ≡ 1(mod 4) then there are q − 2 choices for a such that −1 − a2 6= 0 and for each non-zero b we have a unique c such
that bc = −1 − a2 6= 0. This gives (q − 2)(q − 1) triples. Similarly, there are 2 choices for a such that −1 − a2 = 0 and
2q− 1 choices for b, c such that bc = 0. Since A = −A, we have 12 ((q− 2) (q− 1)+ 2 (2q− 1)) = q
( q+1
2
)
involutions in
G. If q ≡ 3(mod 4) then there are q choices for a such that −1 − a2 6= 0 and for each non-zero b we have a unique c . This
gives q
( q−1
2
)
involutions. Obviously, the cardinality of X is odd in the both cases q ≡ 1, 3(mod 4). If q is a power of two,
then a similar but easier counting shows that PSL(2, q) = SL(2, q) contains exactly q2 − 1 elements of order two, an odd
number again. For any two elements g ∈ G and x ∈ X we have gxg−1 ∈ X and therefore, 〈X〉 is normal in G. By simplicity
of G it follows that 〈X〉 = G and C(G, X) is a non-bipartite graph. Since any two involutions of PSL(2, q) are conjugate in the
group PGL(2, q) < Aut(PSL(2, q)) [4], the assertion follows from Corollary 1.
3. Extension to categorical product
Let us now consider the categorical product of graphs to generate VTNCGs. Recall that for given two graphs Γ1 = (V1, E1)
and Γ2 = (V2, E2) the categorical product Γ1 × Γ2 of Γ1 and Γ2 is the graph with the vertex set V1 × V2, and two vertices
(u1, u2) and (v1, v2) are joined by an edge if and only if {u1, v1} ∈ E1 and {u2, v2} ∈ E2. Note, that if Γ1 = C(G1, X1) and
Γ2 = C(G2, X2) then by definition of the adjacency relation in Cayley graphs, their categorical product is the Cayley graph
on the group G1×G2 with respect to the generating set X1×X2. Assume that we are given two non-bipartite, edge-transitive
graphs Γ1 and Γ2 of odd valency d1 and d2, respectively. Then immediately from the definition of the categorical product
follows that the graph Γ1×Γ2 is edge-transitive and of odd valency d = d1d2. To see that the graph Γ1×Γ2 is not bipartite,
it is sufficient to observe that to any given odd cycles C1 and C2 of Γ1 and Γ2, respectively, there corresponds a closed walk
of length equal to lcm(|C1|, |C2|) in Γ1 × Γ2.
We have proved that Γ1 × Γ2 fulfills the assumptions of Theorem A and we may conclude that the line graph of Γ1 × Γ2
is a VTNCG. Obviously, the above can be extended by induction to the categorical product of any finite family of appropriate
graphs.
Theorem 1. Let Γ1, . . . ,Γn, n ≥ 2, be edge-transitive, non-bipartite and odd valency graphs. Then the line graph of Γ1×· · ·×Γn
is a VTNCG. 
We apply this construction to show that there are infinite families of simple orbit graphs that are VTNCGs. The simple orbit
graph O(G, X) associatedwith a (finite) group G and a non-empty subset X of G such that 〈x〉∩〈x′〉 = {1} for any two distinct
elements x, x′ ∈ X is defined as follows [18]. The vertex set ofO(G, X) is the union, over all x ∈ X , of orbits of left translations
induced by x, and two distinct vertices u and v are joined by an edge if and only if the orbits they represent have a (unique)
element in common. In the same paper it was shown that any simple orbit graphO(G, X) admits a representation as a colour
clique graph of a Cayley graph on the group G. This gives rise a natural question of characterizing simple orbit graphs that
are VTNCGs, maybe, in terms of Cayley graphs. Since the line graph of C(G, X) is the simple orbit graph O(G, X) whenever
X consists of involutions [18], our examples together with their extensions in the spirit of Theorem 1 give a partial answer
to the problem. We remark that (simple) orbit graphs are a special case of G-graphs introduced as a potential tool for group
isomorphism testing in [1]. Also, simple orbit graphs can be regarded as a generalization of double-coset graphs introduced
in [3] and studied in connection with constructions of edge- but not vertex-transitive graphs [7,11].
4. A remark on hamiltonicity
We conclude with addressing the question of whether or not every VTNCG that arises by our constructions contains a
Hamiltonian cycle. First we recall that a connected, vertex-transitive graph of valency k is k-edge-connected [10], and the
line graph of a k-edge-connected graph is k-connected [2]. From this it follows that the line graph of a Cayley graph C(G, X)
is |X |-connected.
A well-known conjecture of Thomassen states that every 4-connected line graph is hamiltonian. Moreover, it was
observed in [17] that if Γ is a counterexample to this conjecture, then Γ is the line graph of a graph that contains a vertex
of valency at most 3. Combining all the above facts we obtain:
Theorem 2. Let Γ = C(G, X) be a connected Cayley graph. If |X | > 3, then the line graph of Γ is hamiltonian. 
From this it immediately follows that the VTNCGswehave considered are hamiltonianwhenever their root Cayley graphs
are of valency greater than three, in which case our VTNCGs have valency at least six.
Acknowledgment
I would like to thank J. Širáň for helpful discussions during the preparation of this Note.
J. Tomanová / Discrete Mathematics 310 (2010) 192–195 195
References
[1] A. Bretto, A. Faisant, Another way for associating a graph to a group, Math. Slovaca 55 (1) (2005) 1–8.
[2] G. Chartrand, M.J. Stewart, The connnectivity of line-graphs, Math. Ann. 182 (1969) 170–174.
[3] D. Goldschmidt, Automorphisms of trivalent graphs, Ann. of Math. 111 (1980) 377–406.
[4] B. Huppert, Endliche Gruppen I, Springer, Berlin, 1967.
[5] M.A. Iranmanesh, Ch. Praeger, On non-Cayley vertex-transitive graphs of order a product of three primes, J. Combin. Theory (B) 81 (2001) 1–10.
[6] R. Jajcay, J. Širáň, More constructions of vertex-transitive non-Cayley graphs based on counting closed walks, Australas. J. Combin. 14 (1996) 121–132.
[7] J. Lauri, Constructing graphs with several pseudosimilar vertices or edges, Discrete Math. 267 (2003) 197–211.
[8] M.W. Liebeck, A. Shalev, Classical groups, probabilistic methods, and the (2, 3)-generation problem, Ann. of Math. 144 (1996) 77–125.
[9] F. Lübeck, G. Malle, (2, 3)-generation of exceptional groups, J. London Math. Soc. (2) 59 (1) (1999) 109–122.
[10] W. Mader, Eine Eigenschaft der Atome endlicher Graphen, Arch. Math. 22 (1971) 333–336.
[11] A. Malnic˘, D. Marus˘ic˘, Ch. Wang, Cubic edge-transitive graphs of order 2p3 , Discrete Math. 274 (2004) 187–198.
[12] D. Marus˘ic˘, Cayley properties of vertex-symmetric graphs, Ars Combin. 16 B (1983) 297–302.
[13] B.D. McKay, M. Miller, J. Širáň, A note on large graphs of diameter two and given maximum degree, J. Combin. Theory Ser. B 74 (1) (1998) 110–118.
[14] B.D. McKay, Ch.E. Praeger, Vertex-transitive graphs which are not Cayley graphs I, J. Aust. Math. Soc. (A) 56 (1994) 53–63.
[15] B.D. McKay, Ch.E. Praeger, Vertex-transitive graphs which are not Cayley graphs II, J. Graph Theory 22 (1996) 321–334.
[16] G. Sabidussi, A class of fixed-point-free graphs, Proc. Amer. Math. Soc. 9 (1958) 800–804.
[17] F.B. Shepherd, Hamiltonicity in claw-free graphs, J. Combin. Theory Ser. B 53 (1991) 173–194.
[18] J. Tomanová, A note on orbit graphs of finite groups and colour-clique graphs of Cayley graphs, Australas. J. Combin. 44 (2009) 57–62.
[19] M.E. Watkins, Vertex-transitive graphs that are not Cayley graphs, in: G. Hahn, et al. (Eds.), Cycles and Rays, Kluwer, Netherlands, 1990, pp. 243–256.


A note on vertex-transitive non-Cayley graphs from Cayley graphs generated by involutions 

file:///data/core-remote/dit/data/elsevier/pdf/208/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMDAxMjM2NXgwOTAwMzg5Ng==.pdf

A note on vertex-transitive non-Cayley graphs from Cayley graphs generated by involutions

Abstract

Similar works

Full text

Available Versions

Elsevier - Publisher Connector