One-matching bi-Cayley graphs over abelian groups

A bi-Cayley graph is a graph which admits a semiregular group of automorphisms with two orbits (of equal size), and is a onematching bi-Cayley graph if the bipartite graph induced by the edges joining these two orbits is a perfect matching. Typical examples of such graphs are the generalized Petersen graphs. A classification of connected arc-transitive one-matching bi-Cayley graphs over abelian groups is given. This is done without referring to the classification of finite simple groups. Instead, complex irreducible characters of abelian groups are used extensively. © 2008 Elsevier Ltd. All rights reserved. 1. Introductory remarks All groups considered in this paper are finite, and all graphs are finite, connected, simple and undirected. For the group-theoretic and graph-theoretic terminology not defined here we refer the reader to [3,23,25]. A graph is said to be a bi-Cayley graph over a group H if it admits H as a semiregular automorphism group with two orbits (of equal size). (Some authors have used the term semi-Cayley instead [11,12].) Note that every bi-Cayley graph admits the following concrete realization. Let R, S, T be subsets of a group H such that R = R−1, S = S−1 and R ∪ S does not contain the identity element of H . Define the graph BiCay(H; R, S, T ) to have vertex set {0, 1}×H , and with vertices (i, h), (j, g) adjacent if and only if one of the following three possibilities occurs: (1) i = j = 0 and gh−1 ∈ R, (2) i = j = 1 and gh−1 ∈ S, (3) i = 0, j = 1 and gh−1 ∈ T . E-mail address: [email protected] (A. Malnič). 0195-6698/$ – see front matter© 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ejc.2008.06.001 I. Kovács et al. / European Journal of Combinatorics 30 (2009) 602–616 603 Let HR ≤ Sym(H) denote the right regular representation of H . If h ∈ H , then hR is the element in HR acting by the rule ahR = ah, for all a ∈ H . The group HR can be regarded as a group of automorphisms of BiCay(H; R, S, T ) acting on its vertices by the rule (i, a)hR = (i, ah). This justifies the prefix in the notation BiCay(H; R, S, T ). Investigation of bi-Cayley graphs is part of a larger project which aims at obtaining a deeper understanding of various classes of symmetric graphs [13–15,20,21]. In particular, bi-Cayley graphs have received considerable attention in the not so distant past, with a number of papers addressing questions regarding their strong regularity [11,12,17,19], semisymmetry [4,5,16] and hamiltonicity [24]. The aim of this paper is to give a description of connected arc-transitive bi-Cayley graphs over abelian groups, satisfying the additional condition that the two orbits of the group in question are joined by one perfect matching. Unless stated otherwise, abelian groups will be written additively with identity element 0. Since in this case the set T can be chosen to consist of the identity element of H , we will shorten the notation to BiCay(H; R, S, 0). We will refer to such a graph as a onematching bi-Cayley graph. Typical examples of such graphs include the seven arc-transitive generalized Petersen graphs [8] and the hypercubes. In order to state our main theorem, describing connected arc-transitive one-matching bi-Cayley graphs over abelian groups, we need the following additional definitions and notation. Firstly, recall that a Cayley graph Cay(H, C) is normal if HR is normal in Aut(Cay(H, C)). Secondly, given positive integers k and n we let ei (1 ≤ i ≤ n) denote the element (0, . . . , 0, 1, 0, . . . , 0) of Zk having 1 in the i-th position, and 0 elsewhere. We also let 0 = (0, . . . 0) ∈ Zk . We warn the reader that the same symbols ei will be used for different values of k and n simultaneously. This should cause no confusion. Theorem 1.1. Let X be a connected one-matching bi-Cayley graph over an abelian group H. If X is arctransitive then it is isomorphic to BiCay(H; R, S, 0) where one of the following possibilities occurs: (i) H = Zn, R = {1,−1}, S = {k,−k}, where (n, k) is one of the following seven ordered pairs: (4, 1), (5, 2), (8, 3), (10, 2), (10, 3), (12, 5), (24, 5). In this case X is isomorphic to the generalized Petersen graph GP(n, k). (ii) H = Z10×Z2, R = {(1, 0), (−1, 0)}, S = {(3, 1), (−3, 1)}. In this case X is the unique arc-transitive cubic graph on 40 points, denoted by F040A in the Foster Census [2,22]. (iii) H = Z2, where R = S = ∅ for n = 0, and R = S = {ei | 1 ≤ i ≤ n} for n ≥ 1. In this case X is isomorphic to the hypercube Qn+1. (iv) H = Z8 × Z 2 2, where R = {e1} and S = {e2} for n = 0, and R = {±(e1, 0), . . . ,±(en, 0), (0, e1)} and S = {±(e1, 0), . . . ,±(en, 0), (0, e2)} for n ≥ 1. In this case X is isomorphic to the direct product C 8 . (v) H = Z8×Z m 4 ×Z2, l ≥ 1, m ≥ 0, R∩S contains no involutions, its cardinality equals |R|−1 = |S|−1 (consequently R \ S and S \ R consist of one involution each), and 〈R ∩ S〉 = Z8 × Z m 4 . In this case X is isomorphic to a normal Cayley graph Cay(H̃, (R ∩ S) ∪ {a,−a}), where H̃ = 〈(R ∩ S) ∪ {a}〉 with a an element of order 8. Moreover, H̃ is isomorphic to Z8 × Z m+1 4 . Remark 1.2. Note that the description of hypercubes in case (iii) is a special case of a more general bi-Cayley representation. Let ` and m be non-negative integers such that 2` + m = n, and let H = Z`4 × Z m 2 . Let R1 and R2 be subsets of H defined as follows. If ` = 0 then let R1 = ∅, and if ` ≥ 1 let R1 = {±(ei, 0) | 1 ≤ i ≤ `}. Ifm = 0, let R2 = ∅, and ifm ≥ 1 let R2 = {(0, ei) | 1 ≤ i ≤ m}. Then Qn+1 is isomorphic to BiCay(H; R, R, 0), where R = R1 ∪ R2. Remark 1.3. Regarding graphs in case (v), this paper is short of a complete classification, which will be the content of a sequel to this paper. We here only show that the class of these graphs is nonempty by giving a construction of an infinite family. For positive integer n we let X(n) = Cay(H, C), where H = Z8 × Z4, and C = {±(1, 0),±(1, e1), . . . ,±(1, en)}. 604 I. Kovács et al. / European Journal of Combinatorics 30 (2009) 602–616 We show that X(n) is a connected, arc-transitive bi-Cayley graph over Z8 × Z 4 × Z2, the orbits of which are joined by a perfect matching. Clearly, X(n) is connected since 〈C〉 = H . Let AutC (H) be the group of all α ∈ Aut(H) such that C = C . An automorphism α of H is determined uniquely by the images (1, 0) and (0, ei) , 1 ≤ i ≤ n, and hence it can be identified with the (n + 1)-tuple [(1, 0), (0, e1), . . . , (0, en)]. Observe that the following (n+ 1)-tuples are in fact automorphisms of X(n) fixing C setwise, and are therefore in AutC (H): [−(1, 0),−(0, e1), . . . ,−(0, en)], [(1, 0), (0, eπ(1)), . . . , (0, eπ(n))], and [(1, e1), (−2,−e1), (0, e2 − e1), . . . , (0, en − e1)], where π ranges over the symmetric group Sn. In fact, the set C coincides with the orbit (1, 0)AutC (H), thus showing that X(n) is arc-transitive. Further, it may be seen that Aut(X(n)) = HRAutC (H), implying that X(n) is a normal Cayley graph of H . To show that X(n) is bi-Cayley, let H̄ = 〈(1, 0), (0, e1), . . . , (0, en−1)〉 and let α be the automorphism [(1, 0), (0, e1), . . . , (0, en−1), (−2,−en)]. We can see that α ∈ AutC (H) and that it fixes H̄ pointwise. Hence α centralizes H̄R . Let τ = α (3,−en)R , and N = 〈H̄R, τ 〉. Then N is abelian of order 2 |H| and it has two orbits H̄ ∪ (H̄ − (0, en)) and (H̄ + (0, en)) ∪ (H̄ + (2, 2en)) on X(n). It is easy to check that these two orbits are joined by a perfect matching. The proof of Theorem 1.1 will be carried out in Section 3 following a series of lemmas, given in Section 2, containing various algebraic and combinatorial ingredients. 2. Preliminary lemmas This section, in which we develop essential tools needed in the proof of Theorem 1.1, has three subsections. In the first we analyze 4-cycles in X = BiCay(H; R, S, 0) in order to obtain certain restrictions on the cardinality of R ∩ S. In the second subsection we consider Aut(X) acting on the right eigenspaces of X . This leads us to a subgroup N of H such that NR is normal in Aut(X), and that the quotient X/NR is a connected arc-transitive one-matching bi-Cayley graph over the group H/N . In the third subsection we recall certain results on normal Cayley graphs. 2.1. Analysis of 4-cycles Lemma 2.1. Let X = BiCay(H; R, S, 0) be a connected regular edge-transitive bi-Cayley graph over an abelian group H, and denote by k = |R| + 1 = |S| + 1 the valency of X. Then one of the following possibilities occurs. (i) R ∩ S = ∅, k ≤ 3, and R ∪ S contains no involutions when k = 3; (ii) R = S 6= ∅ consists of elements of orders 2 or 4; (iii) R ∩ S has cardinality k− 2, k > 2, and contains no involutions. Moreover, if c denotes the number of 4-cycles of X containing a fixed edge, then c = 0 in the case (i), c = k− 1 in the case (ii), and c = k− 2 in the case (iii). Proof. Since X is edge-transitive, the number c of 4-cycles of X containing a given edge does not depend on the choice of this edge. To simplify notation we define ug = (0, g) and vg = (1, g) for all g ∈ H . Observe that e = {u0, v0} is contained in exactly |R ∩ S| 4-cycles, implying that c = |R ∩ S|. (This already proves the statement about the number of 4-cycles containing a fixed edge.) On the other hand, note that for any r ∈ R, the edge {u0, ur} is contained in 4-cycles of the form (u0, ur , ur+r ′ , ur ′ , u0), where r ′ ∈ R\{r,−r}. Similarly, for any s ∈ S, the edge {v0, vs} is contained in 4cycles of the form (v0, vs, vs+s′ , vs′ , v0), where s ∈ S\{s,−s}. Therefore, c = |R∩S| ≥ |R|−1 = |S|−1 if R ∪ S contains an involution, and c = |R ∩ S| ≥ |R| − 2 = |S| − 2 otherwise. Assume first that R ∩ S = ∅, implying c = 0. Then k = |R| + 1 ≤ 3. Moreover, if R ∪ S contains an involution, then k = |R| + 1 ≤ 2. This proves (i). I. Kovács et al. / European Journal of Combinatorics 30 (2009) 602–616 605 Assume next that R∩S 6= ∅ and choose r ∈ R∩S. Then the edge {u0, ur} is contained in the 4-cycle (u0, ur , vr , v0, u0), implying c = |R∩S| ≥ |R| if r is an involution, and c = |R∩S| ≥ |R|−1 otherwise. But now either R = S, or R∩ S has cardinality k− 2 = |R| − 1 = |S| − 1 and contains no involutions. This proves (iii). Finally, in order to prove (ii) we show that if R = S, then every element of R is of order 2 or 4. Choose r ∈ R and observe that 2-paths (v0, u0, ur) and (u0, v0, vr) are contained in exactly one 4-cycle, namely the 4-cycle (u0, v0, vr , ur , u0). Now choose any 4-cycle (ug1 , ug2 , ug3 , ug4 , ug1), g1, g2, g3, g4 ∈ H . Since X is edge-transitive, there exists an automorphism of X , mapping the edge {ug1 , ug2} to the edge {u0, v0}. This implies that the 2-path (ug1 , ug2 , ug3) is contained in exactly one 4cycle, namely the 4-cycle (ug1 , ug2 , ug3 , ug4 , ug1). Assume that r is not an involution. Then there exists a unique 4-cycle, say (u0, ur , ut , u−r , u0) for t ∈ H , which contains the 2-path (u−r , u0, ur). Since ur and ut are adjacent, there exists r1 ∈ R such that r + r1 = t . But then ur1 is joined to both u0 and ut . Observe that r1 6= −r since t 6= 0. Now if r1 6= r , then the 2-path (u0, ur , ut) is contained in two different 4-cycles, namely the 4-cycles (u0, ur , ut , u−r , u0) and (u0, ur , ut , ur1 , u0), contradicting the above property of 2-paths. Hence r1 = r , implying t = 2r . Similarly, since u−r and ut are adjacent, there exists r2 ∈ R such that−r + r2 = t . Using the same argument as above we obtain that r2 = −r , and hence t = −2r . But this implies t = 2r = −2r , and so r has order 4, completing the proof of the lemma. 2.2. The normal subgroup NR and the quotient X/NR Let X = BiCay(H; R, S, 0) be a connected regular bi-Cayley graph over an abelian group H of order |H| = n. A fixed chosen ordering h1, . . . , hn of elements of H such that h1 = 0 naturally determines the following induced ordering: (0, h1), (0, h2), . . . , (0, hn), (1, h1), (1, h2), . . . , (1, hn) of vertices of X . Let A ∈ M2n×2n(C) be the (0, 1)-adjacency matrix of X (note that X is a simple graph) relative to this ordering. Observe that A has the following block form: [ BR I I BS ]