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Connectivity of half vertex transitive digraphs
Applied Mathematics and Computation 316 (2018) 25–29
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Connectivity of half vertex transitive digraphsR Laihuan Chen, Jixiang Meng∗, Yingzhi Tian, Xiaodong Liang, Fengxia Liu College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang 830046, PR China
a r t i c l e
i n f o
Keywords: Digraphs Half vertex transitive digraphs Half double coset digraphs
a b s t r a c t A bipartite digraph is said to be a half vertex transitive digraph if its automorphism acts transitively on the sets of its bipartition, respectively. In this paper, bipartite double coset digraphs of groups are defined and it is shown that any half vertex transitive digraph is isomorphic to some half double coset digraph, and we show that the connectivity of any strongly connected half transitive digraph is its minimum degree. © 2017 Elsevier Inc. All rights reserved.
1. Introduction A graph (digraph) is said to be vertex transitive if its automorphism group acts transitively on its vertex set, and is edge (arc) transitive if its automorphism group acts transitively on its edge (arc) set. There is close relation between the graph symmetry and its connectivity. For instance, the edge connectivity of a connected vertex transitive graph is its regular degree [4] and the connectivity of a connected edge transitive graph is its minimum degree [7]. Similar results also hold for some digraphs [1,2]. These motivate us to study the connectivity properties of graphs (digraphs) with k vertex or edge orbits under the action of their automorphism groups. The first step under this direction is to study the connectivity properties of graphs (digraphs) with two orbits. Let D = (V, E ) be a digraph, and F⊆V. Set
N + (F ) = N − (F ) =
{v ∈ V \ F : there exists u ∈ F satisfying (u, v ) ∈ E }, {v ∈ V \ F : there exists u ∈ F satisfying (v, u ) ∈ E }.
Set dD+ (u ) = |N + (u )|, and dD− (u ) = |N − (u )|, which are called out-degree and in-degree of u, respectively. Set δ + (D ) = min{dD+ (u ) : u ∈ V (D )}, δ − (D ) = min{dD− (u ) : u ∈ V (D )}, and δ (D ) = min{δ − (D ), δ + (D )}, which are called the minimum outdegree, the minimum in-degree and the minimum degree of D, respectively. The arc connectivity λ(D) is the minimum cardinality of all arc sets S in D such that D − S is not strongly connected, and the vertex connectivity κ (D), simply connectivity, is the minimum cardinality of all vertex sets T in D such that D − T is not strongly connected. The following result is well known:
κ ( D ) ≤ λ ( D ) ≤ δ ( D ). Thus, if κ (D ) = δ (D ), then λ(D ) = δ (D ). D is said to be strongly connected if κ (D) ≥ 1. For results on the connectivity of vertex transitive or arc transitive digraphs, see the excellent survey [7] and papers [3,5,6]. Notation and definitions not defined here are referred to [8,9]. R ∗
Foundation item: The research is supported by NSFC (Nos. 11531011, 11501478) and the Key Laboratory Project of Xinjiang (2015KL019). Corresponding author. E-mail addresses: [email protected], [email protected] (J. Meng).
http://dx.doi.org/10.1016/j.amc.2017.08.006 0 096-30 03/© 2017 Elsevier Inc. All rights reserved.
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L. Chen et al. / Applied Mathematics and Computation 316 (2018) 25–29
2. Half vertex transitive digraphs and half double coset digraphs Let G be a group and S⊆G{1}, where 1 is the identity element of G. The Cayley digraph D = D(G, S ) = (V (D ), E (D )) is a directed graph with vertex set V (D ) = G and arc set E (D ) = {(x, y ) : there exists s ∈ S satisfying yx−1 = s for x, y ∈ G}. If S is inverse-closed, that is, S = S−1 , then (x, y) ∈ E(D) if and only if (y, x) ∈ E(D). Thus, D corresponds to an undirected graph C(G, S), called a Cayley graph. For a ∈ G, the right multiplication R(a): g → ga, g ∈ G, is clearly an automorphism of any Cayley digraph of G. Automorphism group Aut(D) of D is the set of all automorphism of the digraph D. Let R(G ) = {R(a ) : a ∈ G}. Then R(G) is a subgroup of Aut(D) and it acts transitively on the vertices of D. Thus Cayley digraphs (graphs) are vertex transitive. Definition 1. Let D = (U, W ; E ) be a bipartite digraph with bipartition U ∪ W. If Aut(D) acts transitively on U and W, respectively, then we call D a half vertex transitive digraph. The undirected version of half vertex transitive digraph is due to [9]. Let G be a group, R be a subgroup of G, we use [G: R] to denote the set of all right cosets of R in G, that is [G : R] = {Rg : g ∈ G}. Definition 2. Let G be a group, L and R be two subgroups of G. Assume Ti is a union of some double cosets of the form RgL for i = 1, 2. Define the half double coset digraph D = D(G, L, R; T1 , T2 ) = (V (D ), E (D )) as follows: (i) V (D ) = [G : L] ∪ [G : R], (ii) (Lg, Rg ) ∈ E(D) if and only if Rg = Rt1 g for some t1 ∈ T1 , (iii) (Rg, Lg ) ∈ E(D) if and only if Rg = Rt2 g for some t2 ∈ T2 . For half double coset digraphs, we have the following theorem. Theorem 2.1. Let D = D(G, L, R; T1 , T2 ) be a half double coset digraph. Then (i) R(G) ≤ Aut(D), thus D is half vertex transitive, where R(G ) = {R(a ) : a ∈ G}, (ii) dD+ (Lg) = |[T1 : R]|, dD− (Lg) = |[T2 : R]|, dD+ (Rg) = |[T2−1 : L]| and dD− (Rg) = |[T1−1 : L]|, where |[Ti : R]| and |[Ti−1 : L]| denote the number of right cosets of R in Ti and L in Ti−1 for i = 1, 2, respectively,
(iii) D is strongly connected if and only if G = T2−1 T1 .
Proof. For any R(a) ∈ R(G) and (Lg, Rg ) ∈ E(D), there exists t1 ∈ T1 satisfying Rg = Rt1 g. Thus (Lg)R(a ) = Lga, (Rg )R(a ) = Rt1 ga. By Definition 2, ((Lg)R(a) , (Lg )R(a) ) ∈ E(D). (i) follows. To prove (ii), it suffices to note that N + (L ) = {Rt1 : t1 ∈ T1 }, N − (L ) = {Rt2 : t2 ∈ T2 }, N+ (R ) = {Lt2−1 : t2 ∈ T2 } and N− (R ) = {Lt1−1 : t1 ∈ T1 }. Clearly, D is strongly connected if and only if there exists a directed path from L to Lg for any g ∈ G, that is, there exists an integer k and ti(1 ) ∈ T1 and ti(2 ) ∈ T2 for i ∈ {1, 2, . . . , k} satisfying
−1 (1) −1 (1) (2) −1 (1) L → Rt1(1) → L t1(2) t1 → · · · → L tk(2) tk · · · t1 t1 = Lg, the above equation holds if and only if g ∈ T2−1 T1 . (iii) follows.
Theorem 2.2. Let D = (U, W ; E ) be a half vertex transitive digraph. Let G = Aut (D ), L = Gu and R = Gw , where u ∈ U, w ∈ W, and Gu and Gw are the stabilizers of u and w in G, respectively. Set T1 = {g ∈ G : wg ∈ N + (u )} and T2 = {g ∈ G : wg ∈ N − (u )}. Then D∼ =D ( G , L , R ; T 1 , T 2 ) . Proof. We prove a sequences of claims from which the results follow. Claim 1. T1 and T2 are unions of some double cosets of the form RgL. In fact, for any g ∈ T1 , wg ∈ N + (u ), that is, (u, wg ) ∈ E (D ). Then for any l ∈ L, (u, wg )l = (u, wgl ) ∈ E (D ). Since for any r ∈ R, wr = w, thus (u, wrgl ) ∈ E (D ). By definition, rgl ∈ T1 , and therefore RgL⊆T1 . Similarly, if g ∈ T2 , then RgL⊆T2 . Let D1 = D(G, L, R; T1 , T2 ), U = {u1 = u, u2 , . . . , um }, and W = {w1 = w, w2 , . . . , wn }. Since D is half vertex transitive, there (2 ) (1 ) g exists g(i 1 ) , g(j2 ) ∈ G satisfying ugi = ui and w j = w j for i ∈ {1, 2, . . . , m} and j ∈ {1, 2, . . . , n}. Define a map σ from V(D) to V(D1 ) as follows,
σ : ui → Lg(i 1) (2 )
w j → Rg j
Claim 2. σ is bijective.
i = 1, 2, . . . , m, j = 1, 2, . . . , n. (1 )
(1 ) −1 g
For any Lg ∈ [G: L], suppose that ug = ui for i ∈ {1, 2, . . . , m}, then ug = ugi , that is ugi (1 )
= u. Thus g(i 1 ) g−1 ∈ L and
Lg = Lgi = uσi . Similarly, for any Rg ∈ [G: R], if wg = ws for s ∈ {1, 2, . . . , n}, then Rg = Rg(s2 ) = wσs . If uσi = uσj for 1 ≤ i < j ≤ m, (1 ) (1 ) (1 ) g then Lg(i 1 ) = Lg(j1 ) , and so there exists l ∈ L satisfying g(j1 ) = lg(i 1 ) . Thus u j = u j = ulgi = ugi = ui . Similarly, if wσs = wtσ for 1 ≤ s < t ≤ n, then ws = wt .
L. Chen et al. / Applied Mathematics and Computation 316 (2018) 25–29
Claim 3. σ preserves the adjacency.
(1 )
For ui ∈ U, w j ∈ W, (ui , w j ) ∈ E (D ) ⇔ (ugi , w
(2 ) gj
) ∈ E (D ) ⇔ (u, w
E (D1 ) ⇔ (uσi , wσj ) ∈ E (D1 ). Similarly, (w j , ui ) ∈ E (D ) ⇔ (w E ( D1 ). Thus σ is an isomorphism from D to D1 , and D∼ =D 1 .
(2 )
gj
(2 ) (1 ) g j (gi )−1
27
) ∈ E (D ) ⇔ g(j2) (g(i 1) )−1 ∈ T1 ⇔ (Lg(i 1) , Rg(j2) ) ∈
(1 )
, ugi ) ∈ E (D ) ⇔ g(j2 ) (g(i 1 ) )−1 ∈ T2 ⇔ (Rg(j2 ) , Lg(i 1 ) ) = (wσj , uσi ) ∈
In [9], Xu et al. defined the so-called Bi-Cayley graphs. Let G be a group and S⊆G. The Bi-Cayley graph X = BC (G, S ) of G with respect to S is an undirected graph with vertex set V (X ) = G × {0} ∪ G × {1} and edge set E (X ) = {((g1 , 0 ), (g2 , 1 )) : there exists s ∈ S satisfying g2 = sg1 }. Bi-Cayley graphs are half vertex transitive graphs. In fact, it is easy to see that BC(G, S) is isomorphic to the associated graph of the half double coset digraph D(G, {1}, {1}; S, S−1 ). Bi-Cayley graphs are important in constructing semi-symmetric graphs (edge transitive but not vertex transitive regular graphs). Now we can generalize the Bi-Cayley graphs to the Bi-Cayley digraphs. Definition 3. Let G be a group and T1 , T2 ⊆G. Call the half double coset digraph D(G, {1}, {1}; T1 , T2 ) a Bi-Cayley digraph of G with respect to T1 and T2 , denoted by BD(G; T1 , T2 ). The Bi-Cayley digraph is obviously isomorphic to the following defined digraph. Let G be a group and T1 , T2 ⊆G. Define the digraph D = (V (D ), E (D )), where V (D ) = G × {0} ∪ G × {1} and E(D) such that for (g1 , 0) ∈ G × {0}, (g2 , 1) ∈ G × {1}, there is an arc from (g1 , 0) to (g2 , 1) if and only if g2 = t1 g1 for some t1 ∈ T1 and there exists an arc from (g2 , 1) to (g1 , 0) if and only if g2 = t2 g1 for some t2 ∈ T2 . If T2 = T1−1 , then there is an arc from (g1 , 0) to (g2 , 1) if and only if there is an arc from (g2 , 1) to (g1 , 0). Thus, in this case, BD(G; T1 , T2 ) corresponds to an undirected graph, which is just the Bi-Cayley graph BC(G, T1 ). In what follows, we show that bipartite Cayley digraph is a Bi-Cayley digraph. The next theorem characterizes bipartite Cayley digraphs. Theorem 2.3. Let D = D(G, S ) be a strongly connected Cayley digraph. Then D is bipartite if and only if there exists a subgroup N of G satisfying |[G : N]| = 2 and N ∩ S = ∅. Proof. If there exists a subgroup N of G satisfying the specific conditions, then for any g1 , g2 ∈ G, (g1 , g2 ) ∈ E(D) if and only if g2 = sg1 for some s ∈ S. When g1 ∈ N, since N ∩ S = ∅, we have g2 ∈ GN(s ∈ GN). When g1 ∈ GN, since s ∈ GN and |[G : N]| = 2, we have sg1 ∈ N. Thus D is a bipartite digraph with bipartition N ∪ (GN). If D is a bipartite digraph with bipartition U ∪ W, without loss of generality, assume that 1 ∈ U, then g ∈ U if and only if g is a product of even number of elements in S. Thus U U = U and U is a subgroup of G. Since there are no arcs in the subgraph D[U] induced by U, we have U ∩ S = ∅. Since D is vertex transitive, |U | = |W |, it follows that W is a coset of U and |[G : U]| = 2. Theorem 2.4. Bipartite Cayley digraphs are Bi-Cayley digraphs. Proof. Let D = D(G, S ) be a bipartite Cayley digraph. Then by Theorem 2.3, there exists a subgroup N of G with |[G : N]| = 2 and N ∩ S = ∅. Let a ∈ GN, then G \ N = aN = Na−1 . Therefore, we can write S = aT1 = T2 a−1 , where Ti ⊆N for i = 1, 2. Let D1 = (V (D1 ), E (D1 )), where V (D1 ) = N × {0} ∪ N × {1} and E(D1 ) such that, for (n1 , 0), (n2 , 1) ∈ V(D1 ), there is an arc from (n1 , 0) to (n2 , 1) if and only if n2 = t1 n1 for some t1 ∈ T1 , and there is an arc from (n2 , 1) to (n1 , 0) if and only if n2 = t2−1 n1 for some t2 ∈ T2 . Clearly, D1 is a Bi-Cayley digraph of N. We now prove that D∼ =D1 . Define a map σ from G = N ∪ aN to V (D1 ) = N × {0} ∪ N × {1} as follows:
σ : n → (n, 0 ) an → (n, 1 )
n ∈ N.
There is an arc from n1 to an2 if and only if an2 = sn1 for some s ∈ S ⇔ n2 = a−1 sn1 = tn1 for some t ∈ T1 . There is an arc from an2 to n1 if and only if n1 = san2 ⇔ n2 = (sa )−1 n1 ⇔ n2 = t −1 n1 for some t ∈ T2 . 3. Connectivity of half vertex transitive digraphs In this section, we will show that the connectivity of any strongly connected half vertex transitive digraph is its minimum degree. To this end, we first introduce some notion and definitions. Let D = (V (D ), E (D )) be a digraph, and F⊆V(D). Set R+ (F ) = V (D ) \ (F ∪ N + (F )) and R− (F ) = V (D ) \ (F ∪ N − (F )). If |N+ (F )| = κ (D ) and R+ (F ) = ∅, then we say F is a positive fragment. If |N− (F )| = κ (D ) and R− (F ) = ∅, then F is a negative fragment. Clearly, if F is a positive (respectively, negative) fragment, then R+ (F )(R− (F )) is a negative (respectively, positive) fragment. Fragments (both positive and negative) with minimum cardinality are called atoms. The following result is important in investigating the connectivity of digraphs. Theorem 3.1 [7]. Let D = (V, E ) be a strongly connected digraph which is not isomorphic to a complete symmetric digraph and let A be a positive (respectively, negative) atom of D. If B is a positive (respectively, negative) fragment of D with A ∩ B = ∅, then A⊆B.
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We deduce, by the above theorem, distinct positive (respectively, negative) atoms have empty intersection. In what follows we establish a sequence of lemmas and then prove our main result of this section. Lemma 3.2. Let D = D(G, L, R; T1 , T2 ) be a strongly connected half vertex transitive digraph with κ (D) < δ (D), and B be a posp q itive atom of D containing L. Let B = {Lg1 = L, Lg2 , . . . , Lg p } ∪ {Rg 1 , Rg 2 , . . . , Rg q }, AL = ∪i=1 Lgi and AR = ∪ j=1 Rg j . Then AL is a subgroup of G and AR is a left coset of AL . Proof. Since κ (D) < δ (D), |B| ≥ 2. Thus |B ∩ [G: R]| ≥ 1. Without loss of generality, assume that g1 = 1. For any g ∈ AL , since R(g) ∈ Aut(D), Bg is also a positive atom. On the other hand, since Lg ∈ B ∩ Bg, by Theorem 3.1, we have B = Bg. Thus AL g = AL and AR g = AR , and so AL AL = AL and AR AL = AR . The former equality means that AL is a subgroup of G, and the later equality implies that AR is a union of some left coset of AL in G. In what follows we show that |AL | = |AR | and thus AR is a left coset of AL . Let BL = {Lg1 , Lg2 , . . . , Lg p } and BR = {Rg 1 , Rg 2 , . . . , Rg q }. Since R(G) acts transitively on [G: L] and [G: R], respectively, and distinct positive atoms are disjoint, there exist b1 , b2 , . . . , bm satisfying [G : L] = BL b1 ∪ BL b2 ∪ · · · ∪ BL bm and [G: R]⊇BR b1 ∪ BR b2 ∪ ∪ BR bm , where BL bi ∩ BL b j = ∅ and BR bi ∩ BR b j = ∅ for 1 ≤ i = j ≤ m. If [G: R] = BR b1 ∪ BR b2 ∪ ∪ BR bm , let Rg ∈ [G : R] \ (∪m B b ), then Rg is contained in a positive atom of D, say A . Since κ (D) < δ (D), we have |A | ≥ 2 and A ∩ [G: i=1 R i L] = ∅. But then, A ∩ BL bi = ∅ for any i ∈ {1, 2, . . . , m}, this is impossible. Thus [G : R] = BR b1 ∪ BR b2 ∪ · · · ∪ BR bm . Therefore |G| = |L| pm = |R|qm, and so |AL | = |L| p = |R|q = |AR |. In the following, we assume that AR = bAL , where b ∈ AR . By definition, the following lemma is immediate. Lemma 3.3. Under the assumptions of Lemma 3.2, we have
N + (B ) = (∪ip=1 RT1 gi ) \ BR ∪ (∪qj=1 LT2−1 g j ) \ BL , N + (L ) = RT1 = (RT1 \ BR ) ∪ (RT1 ∩ BR ). Lemma 3.4. Under the assumptions of Lemma 3.2, if there exists t2 ∈ T2 satisfying Lt2−1 g j ∈ / BL for some j ∈ {1, 2, . . . , q}. Then Lt2−1 g k ∈ / BL for any k ∈ {1, 2, . . . , q}.
Proof. It suffices to show that if there exists t2 ∈ T2 satisfying Lt2−1 g j ∈ BL for some j ∈ {1, 2, . . . , q}, then Lt2−1 g k ∈ BL for
any k ∈ {1, 2, . . . , q}. In fact, if Lt2−1 g j ∈ BL , assume that Lt2−1 g j = Lgi for some i ∈ {1, 2, . . . , p}, then AL Lt2−1 g j = AL Lgi , that is,
AL t2−1 g j = AL . Hence t2−1 g j ∈ AL and g j ∈ t2 AL = AR . Thus for any k ∈ {1, 2, . . . , q}, since g k ∈ AR = t2 AL , we have t2−1 g k ∈ AL , and so Lt2−1 g k ∈ BL .
Lemma 3.5. Under the assumptions of Lemma 3.2, we have T2 ⊆AR , ∪ j=1 LT2−1 g j ⊆ BL and N + (Rg j ) = LT2−1 g j ⊆ BL for any j ∈ {1, 2, . . . , q}. q
(∪qj=1 LT2−1 g j ) \ BL = ∅, then |(∪qj=1 LT2−1 g j ) \ BL | ≥ q. Since (∪ip=1 RT1 gi ) \ BR ⊇ RT1 \ BR , and q |RT1 ∩ BR | ≤ q, by Lemma 3.3, we have κ (D ) = |N + (B )| ≥ |N + (L )| ≥ δ (D ), this is impossible. Thus ∪ j=1 LT2−1 g j ⊆ BL . Thus for Proof. By
Lemma 3.4,
if
any t2 ∈ T2 , Lt2−1 g j ∈ BL and t2 ∈ g j AL = AR . Hence T2 ⊆AR .
Theorem 3.6. Let D = D(G, L, R; T1 , T2 ) be a strongly connected half vertex transitive digraph. Then κ (D ) = δ (D ). Proof. Let A be an atom of D. Without loss of generality, assume that A is a positive atom. By means of contradiction. If κ (D) < δ (D), then |A| ≥ 2, |A ∩ [G: L]| ≥ 1, and |A ∩ [G: R]| ≥ 1. Since Aut(D) acts transitively on [G: L] and [G: R], respectively, we may assume that L ∈ A. Let A = {Lg1 = L, Lg2 , . . . , Lg p } ∪ {Rg 1 , Rg 2 , . . . , Rg q }, BL = {Lg1 , Lg2 , . . . , Lg p }, BR = {Rg 1 , Rg 2 , . . . , Rg q }, p q and AL = ∪i=1 Lgi and AR = ∪ j=1 Rg j . Then by Lemma 3.2, AL is a subgroup of G and AR is a left coset of AL . By Lemma 3.5, T2 ⊆AR . We proceed to show that T1 ⊆AR . If for some t1 ∈ T1 and some i ∈ {1, 2, . . . , p} satisfying Rt1 gi ∈ BR , let Rt1 gi = Rg j ∈ BR ,
then Rt1 gi ⊆AR . Since gi ∈ AL and AR = bAL , we have t1 ∈ bAL g−1 = bAL = AR and t1 gk ∈ AR for any k ∈ {1, 2, . . . , p}. Hence i p Rt1 gk ∈ BR . Equivalently, if for some t1 ∈ T1 and some i ∈ {1, 2, . . . , p} satisfying Rt1 gi ∈ BR , then (∪i=1 Rt1 gi ) ∩ BR = ∅, thus |N+ (A )| ≥ p. On the other hand, by Lemma 3.5, N+ (Rg i ) ⊆ BL , thus |N+ (Rg i )| ≤ |BL | = p. We have |N+ (A )| ≥ δ (D ), a contradiction. Thus for any t1 ∈ T1 and any i ∈ {1, 2, . . . , p}, Rt1 gi ∈ BR , and so RT1 gi ⊆AR for any i ∈ {1, 2, . . . , p}. Since gi ∈ AL , we have T1 ⊆ RT1 ⊆ AR g−1 = bAL g−1 = bAL = AR . But T2−1 T1 ≤ AL G = T2−1 T1 , a contradiction. i i Since κ (D) ≤ λ(D) ≤ δ (D) for any digraph D, by Theorem 3.6, we have Corollary 3.7. Let D be a strongly connected half vertex transitive digraph. Then κ (D ) = λ(D ) = δ (D ). Since Bi-Cayley digraphs are half vertex transitive digraphs, we have Corollary 3.8. Let D = D(G; T1 , T2 ) be a strongly connected Bi-Cayley digraph. Then κ (D ) = λ(D ) = δ (D ). Since bipartite Cayley digraphs are Bi-Cayley digraphs, by the above corollary, we deduce the following result. Corollary 3.9. Let D = D(G, S ) be a strongly connected bipartite Cayley digraph. Then κ (D ) = λ(D ) = δ (D ).
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Connectivity of half vertex transitive digraphsR Laihuan Chen, Jixiang Meng∗, Yingzhi Tian, Xiaodong Liang, Fengxia Liu College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang 830046, PR China
a r t i c l e
i n f o
Keywords: Digraphs Half vertex transitive digraphs Half double coset digraphs
a b s t r a c t A bipartite digraph is said to be a half vertex transitive digraph if its automorphism acts transitively on the sets of its bipartition, respectively. In this paper, bipartite double coset digraphs of groups are defined and it is shown that any half vertex transitive digraph is isomorphic to some half double coset digraph, and we show that the connectivity of any strongly connected half transitive digraph is its minimum degree. © 2017 Elsevier Inc. All rights reserved.
1. Introduction A graph (digraph) is said to be vertex transitive if its automorphism group acts transitively on its vertex set, and is edge (arc) transitive if its automorphism group acts transitively on its edge (arc) set. There is close relation between the graph symmetry and its connectivity. For instance, the edge connectivity of a connected vertex transitive graph is its regular degree [4] and the connectivity of a connected edge transitive graph is its minimum degree [7]. Similar results also hold for some digraphs [1,2]. These motivate us to study the connectivity properties of graphs (digraphs) with k vertex or edge orbits under the action of their automorphism groups. The first step under this direction is to study the connectivity properties of graphs (digraphs) with two orbits. Let D = (V, E ) be a digraph, and F⊆V. Set
N + (F ) = N − (F ) =
{v ∈ V \ F : there exists u ∈ F satisfying (u, v ) ∈ E }, {v ∈ V \ F : there exists u ∈ F satisfying (v, u ) ∈ E }.
Set dD+ (u ) = |N + (u )|, and dD− (u ) = |N − (u )|, which are called out-degree and in-degree of u, respectively. Set δ + (D ) = min{dD+ (u ) : u ∈ V (D )}, δ − (D ) = min{dD− (u ) : u ∈ V (D )}, and δ (D ) = min{δ − (D ), δ + (D )}, which are called the minimum outdegree, the minimum in-degree and the minimum degree of D, respectively. The arc connectivity λ(D) is the minimum cardinality of all arc sets S in D such that D − S is not strongly connected, and the vertex connectivity κ (D), simply connectivity, is the minimum cardinality of all vertex sets T in D such that D − T is not strongly connected. The following result is well known:
κ ( D ) ≤ λ ( D ) ≤ δ ( D ). Thus, if κ (D ) = δ (D ), then λ(D ) = δ (D ). D is said to be strongly connected if κ (D) ≥ 1. For results on the connectivity of vertex transitive or arc transitive digraphs, see the excellent survey [7] and papers [3,5,6]. Notation and definitions not defined here are referred to [8,9]. R ∗
Foundation item: The research is supported by NSFC (Nos. 11531011, 11501478) and the Key Laboratory Project of Xinjiang (2015KL019). Corresponding author. E-mail addresses: [email protected], [email protected] (J. Meng).
http://dx.doi.org/10.1016/j.amc.2017.08.006 0 096-30 03/© 2017 Elsevier Inc. All rights reserved.
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2. Half vertex transitive digraphs and half double coset digraphs Let G be a group and S⊆G{1}, where 1 is the identity element of G. The Cayley digraph D = D(G, S ) = (V (D ), E (D )) is a directed graph with vertex set V (D ) = G and arc set E (D ) = {(x, y ) : there exists s ∈ S satisfying yx−1 = s for x, y ∈ G}. If S is inverse-closed, that is, S = S−1 , then (x, y) ∈ E(D) if and only if (y, x) ∈ E(D). Thus, D corresponds to an undirected graph C(G, S), called a Cayley graph. For a ∈ G, the right multiplication R(a): g → ga, g ∈ G, is clearly an automorphism of any Cayley digraph of G. Automorphism group Aut(D) of D is the set of all automorphism of the digraph D. Let R(G ) = {R(a ) : a ∈ G}. Then R(G) is a subgroup of Aut(D) and it acts transitively on the vertices of D. Thus Cayley digraphs (graphs) are vertex transitive. Definition 1. Let D = (U, W ; E ) be a bipartite digraph with bipartition U ∪ W. If Aut(D) acts transitively on U and W, respectively, then we call D a half vertex transitive digraph. The undirected version of half vertex transitive digraph is due to [9]. Let G be a group, R be a subgroup of G, we use [G: R] to denote the set of all right cosets of R in G, that is [G : R] = {Rg : g ∈ G}. Definition 2. Let G be a group, L and R be two subgroups of G. Assume Ti is a union of some double cosets of the form RgL for i = 1, 2. Define the half double coset digraph D = D(G, L, R; T1 , T2 ) = (V (D ), E (D )) as follows: (i) V (D ) = [G : L] ∪ [G : R], (ii) (Lg, Rg ) ∈ E(D) if and only if Rg = Rt1 g for some t1 ∈ T1 , (iii) (Rg, Lg ) ∈ E(D) if and only if Rg = Rt2 g for some t2 ∈ T2 . For half double coset digraphs, we have the following theorem. Theorem 2.1. Let D = D(G, L, R; T1 , T2 ) be a half double coset digraph. Then (i) R(G) ≤ Aut(D), thus D is half vertex transitive, where R(G ) = {R(a ) : a ∈ G}, (ii) dD+ (Lg) = |[T1 : R]|, dD− (Lg) = |[T2 : R]|, dD+ (Rg) = |[T2−1 : L]| and dD− (Rg) = |[T1−1 : L]|, where |[Ti : R]| and |[Ti−1 : L]| denote the number of right cosets of R in Ti and L in Ti−1 for i = 1, 2, respectively,
(iii) D is strongly connected if and only if G = T2−1 T1 .
Proof. For any R(a) ∈ R(G) and (Lg, Rg ) ∈ E(D), there exists t1 ∈ T1 satisfying Rg = Rt1 g. Thus (Lg)R(a ) = Lga, (Rg )R(a ) = Rt1 ga. By Definition 2, ((Lg)R(a) , (Lg )R(a) ) ∈ E(D). (i) follows. To prove (ii), it suffices to note that N + (L ) = {Rt1 : t1 ∈ T1 }, N − (L ) = {Rt2 : t2 ∈ T2 }, N+ (R ) = {Lt2−1 : t2 ∈ T2 } and N− (R ) = {Lt1−1 : t1 ∈ T1 }. Clearly, D is strongly connected if and only if there exists a directed path from L to Lg for any g ∈ G, that is, there exists an integer k and ti(1 ) ∈ T1 and ti(2 ) ∈ T2 for i ∈ {1, 2, . . . , k} satisfying
−1 (1) −1 (1) (2) −1 (1) L → Rt1(1) → L t1(2) t1 → · · · → L tk(2) tk · · · t1 t1 = Lg, the above equation holds if and only if g ∈ T2−1 T1 . (iii) follows.
Theorem 2.2. Let D = (U, W ; E ) be a half vertex transitive digraph. Let G = Aut (D ), L = Gu and R = Gw , where u ∈ U, w ∈ W, and Gu and Gw are the stabilizers of u and w in G, respectively. Set T1 = {g ∈ G : wg ∈ N + (u )} and T2 = {g ∈ G : wg ∈ N − (u )}. Then D∼ =D ( G , L , R ; T 1 , T 2 ) . Proof. We prove a sequences of claims from which the results follow. Claim 1. T1 and T2 are unions of some double cosets of the form RgL. In fact, for any g ∈ T1 , wg ∈ N + (u ), that is, (u, wg ) ∈ E (D ). Then for any l ∈ L, (u, wg )l = (u, wgl ) ∈ E (D ). Since for any r ∈ R, wr = w, thus (u, wrgl ) ∈ E (D ). By definition, rgl ∈ T1 , and therefore RgL⊆T1 . Similarly, if g ∈ T2 , then RgL⊆T2 . Let D1 = D(G, L, R; T1 , T2 ), U = {u1 = u, u2 , . . . , um }, and W = {w1 = w, w2 , . . . , wn }. Since D is half vertex transitive, there (2 ) (1 ) g exists g(i 1 ) , g(j2 ) ∈ G satisfying ugi = ui and w j = w j for i ∈ {1, 2, . . . , m} and j ∈ {1, 2, . . . , n}. Define a map σ from V(D) to V(D1 ) as follows,
σ : ui → Lg(i 1) (2 )
w j → Rg j
Claim 2. σ is bijective.
i = 1, 2, . . . , m, j = 1, 2, . . . , n. (1 )
(1 ) −1 g
For any Lg ∈ [G: L], suppose that ug = ui for i ∈ {1, 2, . . . , m}, then ug = ugi , that is ugi (1 )
= u. Thus g(i 1 ) g−1 ∈ L and
Lg = Lgi = uσi . Similarly, for any Rg ∈ [G: R], if wg = ws for s ∈ {1, 2, . . . , n}, then Rg = Rg(s2 ) = wσs . If uσi = uσj for 1 ≤ i < j ≤ m, (1 ) (1 ) (1 ) g then Lg(i 1 ) = Lg(j1 ) , and so there exists l ∈ L satisfying g(j1 ) = lg(i 1 ) . Thus u j = u j = ulgi = ugi = ui . Similarly, if wσs = wtσ for 1 ≤ s < t ≤ n, then ws = wt .
L. Chen et al. / Applied Mathematics and Computation 316 (2018) 25–29
Claim 3. σ preserves the adjacency.
(1 )
For ui ∈ U, w j ∈ W, (ui , w j ) ∈ E (D ) ⇔ (ugi , w
(2 ) gj
) ∈ E (D ) ⇔ (u, w
E (D1 ) ⇔ (uσi , wσj ) ∈ E (D1 ). Similarly, (w j , ui ) ∈ E (D ) ⇔ (w E ( D1 ). Thus σ is an isomorphism from D to D1 , and D∼ =D 1 .
(2 )
gj
(2 ) (1 ) g j (gi )−1
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) ∈ E (D ) ⇔ g(j2) (g(i 1) )−1 ∈ T1 ⇔ (Lg(i 1) , Rg(j2) ) ∈
(1 )
, ugi ) ∈ E (D ) ⇔ g(j2 ) (g(i 1 ) )−1 ∈ T2 ⇔ (Rg(j2 ) , Lg(i 1 ) ) = (wσj , uσi ) ∈
In [9], Xu et al. defined the so-called Bi-Cayley graphs. Let G be a group and S⊆G. The Bi-Cayley graph X = BC (G, S ) of G with respect to S is an undirected graph with vertex set V (X ) = G × {0} ∪ G × {1} and edge set E (X ) = {((g1 , 0 ), (g2 , 1 )) : there exists s ∈ S satisfying g2 = sg1 }. Bi-Cayley graphs are half vertex transitive graphs. In fact, it is easy to see that BC(G, S) is isomorphic to the associated graph of the half double coset digraph D(G, {1}, {1}; S, S−1 ). Bi-Cayley graphs are important in constructing semi-symmetric graphs (edge transitive but not vertex transitive regular graphs). Now we can generalize the Bi-Cayley graphs to the Bi-Cayley digraphs. Definition 3. Let G be a group and T1 , T2 ⊆G. Call the half double coset digraph D(G, {1}, {1}; T1 , T2 ) a Bi-Cayley digraph of G with respect to T1 and T2 , denoted by BD(G; T1 , T2 ). The Bi-Cayley digraph is obviously isomorphic to the following defined digraph. Let G be a group and T1 , T2 ⊆G. Define the digraph D = (V (D ), E (D )), where V (D ) = G × {0} ∪ G × {1} and E(D) such that for (g1 , 0) ∈ G × {0}, (g2 , 1) ∈ G × {1}, there is an arc from (g1 , 0) to (g2 , 1) if and only if g2 = t1 g1 for some t1 ∈ T1 and there exists an arc from (g2 , 1) to (g1 , 0) if and only if g2 = t2 g1 for some t2 ∈ T2 . If T2 = T1−1 , then there is an arc from (g1 , 0) to (g2 , 1) if and only if there is an arc from (g2 , 1) to (g1 , 0). Thus, in this case, BD(G; T1 , T2 ) corresponds to an undirected graph, which is just the Bi-Cayley graph BC(G, T1 ). In what follows, we show that bipartite Cayley digraph is a Bi-Cayley digraph. The next theorem characterizes bipartite Cayley digraphs. Theorem 2.3. Let D = D(G, S ) be a strongly connected Cayley digraph. Then D is bipartite if and only if there exists a subgroup N of G satisfying |[G : N]| = 2 and N ∩ S = ∅. Proof. If there exists a subgroup N of G satisfying the specific conditions, then for any g1 , g2 ∈ G, (g1 , g2 ) ∈ E(D) if and only if g2 = sg1 for some s ∈ S. When g1 ∈ N, since N ∩ S = ∅, we have g2 ∈ GN(s ∈ GN). When g1 ∈ GN, since s ∈ GN and |[G : N]| = 2, we have sg1 ∈ N. Thus D is a bipartite digraph with bipartition N ∪ (GN). If D is a bipartite digraph with bipartition U ∪ W, without loss of generality, assume that 1 ∈ U, then g ∈ U if and only if g is a product of even number of elements in S. Thus U U = U and U is a subgroup of G. Since there are no arcs in the subgraph D[U] induced by U, we have U ∩ S = ∅. Since D is vertex transitive, |U | = |W |, it follows that W is a coset of U and |[G : U]| = 2. Theorem 2.4. Bipartite Cayley digraphs are Bi-Cayley digraphs. Proof. Let D = D(G, S ) be a bipartite Cayley digraph. Then by Theorem 2.3, there exists a subgroup N of G with |[G : N]| = 2 and N ∩ S = ∅. Let a ∈ GN, then G \ N = aN = Na−1 . Therefore, we can write S = aT1 = T2 a−1 , where Ti ⊆N for i = 1, 2. Let D1 = (V (D1 ), E (D1 )), where V (D1 ) = N × {0} ∪ N × {1} and E(D1 ) such that, for (n1 , 0), (n2 , 1) ∈ V(D1 ), there is an arc from (n1 , 0) to (n2 , 1) if and only if n2 = t1 n1 for some t1 ∈ T1 , and there is an arc from (n2 , 1) to (n1 , 0) if and only if n2 = t2−1 n1 for some t2 ∈ T2 . Clearly, D1 is a Bi-Cayley digraph of N. We now prove that D∼ =D1 . Define a map σ from G = N ∪ aN to V (D1 ) = N × {0} ∪ N × {1} as follows:
σ : n → (n, 0 ) an → (n, 1 )
n ∈ N.
There is an arc from n1 to an2 if and only if an2 = sn1 for some s ∈ S ⇔ n2 = a−1 sn1 = tn1 for some t ∈ T1 . There is an arc from an2 to n1 if and only if n1 = san2 ⇔ n2 = (sa )−1 n1 ⇔ n2 = t −1 n1 for some t ∈ T2 . 3. Connectivity of half vertex transitive digraphs In this section, we will show that the connectivity of any strongly connected half vertex transitive digraph is its minimum degree. To this end, we first introduce some notion and definitions. Let D = (V (D ), E (D )) be a digraph, and F⊆V(D). Set R+ (F ) = V (D ) \ (F ∪ N + (F )) and R− (F ) = V (D ) \ (F ∪ N − (F )). If |N+ (F )| = κ (D ) and R+ (F ) = ∅, then we say F is a positive fragment. If |N− (F )| = κ (D ) and R− (F ) = ∅, then F is a negative fragment. Clearly, if F is a positive (respectively, negative) fragment, then R+ (F )(R− (F )) is a negative (respectively, positive) fragment. Fragments (both positive and negative) with minimum cardinality are called atoms. The following result is important in investigating the connectivity of digraphs. Theorem 3.1 [7]. Let D = (V, E ) be a strongly connected digraph which is not isomorphic to a complete symmetric digraph and let A be a positive (respectively, negative) atom of D. If B is a positive (respectively, negative) fragment of D with A ∩ B = ∅, then A⊆B.
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We deduce, by the above theorem, distinct positive (respectively, negative) atoms have empty intersection. In what follows we establish a sequence of lemmas and then prove our main result of this section. Lemma 3.2. Let D = D(G, L, R; T1 , T2 ) be a strongly connected half vertex transitive digraph with κ (D) < δ (D), and B be a posp q itive atom of D containing L. Let B = {Lg1 = L, Lg2 , . . . , Lg p } ∪ {Rg 1 , Rg 2 , . . . , Rg q }, AL = ∪i=1 Lgi and AR = ∪ j=1 Rg j . Then AL is a subgroup of G and AR is a left coset of AL . Proof. Since κ (D) < δ (D), |B| ≥ 2. Thus |B ∩ [G: R]| ≥ 1. Without loss of generality, assume that g1 = 1. For any g ∈ AL , since R(g) ∈ Aut(D), Bg is also a positive atom. On the other hand, since Lg ∈ B ∩ Bg, by Theorem 3.1, we have B = Bg. Thus AL g = AL and AR g = AR , and so AL AL = AL and AR AL = AR . The former equality means that AL is a subgroup of G, and the later equality implies that AR is a union of some left coset of AL in G. In what follows we show that |AL | = |AR | and thus AR is a left coset of AL . Let BL = {Lg1 , Lg2 , . . . , Lg p } and BR = {Rg 1 , Rg 2 , . . . , Rg q }. Since R(G) acts transitively on [G: L] and [G: R], respectively, and distinct positive atoms are disjoint, there exist b1 , b2 , . . . , bm satisfying [G : L] = BL b1 ∪ BL b2 ∪ · · · ∪ BL bm and [G: R]⊇BR b1 ∪ BR b2 ∪ ∪ BR bm , where BL bi ∩ BL b j = ∅ and BR bi ∩ BR b j = ∅ for 1 ≤ i = j ≤ m. If [G: R] = BR b1 ∪ BR b2 ∪ ∪ BR bm , let Rg ∈ [G : R] \ (∪m B b ), then Rg is contained in a positive atom of D, say A . Since κ (D) < δ (D), we have |A | ≥ 2 and A ∩ [G: i=1 R i L] = ∅. But then, A ∩ BL bi = ∅ for any i ∈ {1, 2, . . . , m}, this is impossible. Thus [G : R] = BR b1 ∪ BR b2 ∪ · · · ∪ BR bm . Therefore |G| = |L| pm = |R|qm, and so |AL | = |L| p = |R|q = |AR |. In the following, we assume that AR = bAL , where b ∈ AR . By definition, the following lemma is immediate. Lemma 3.3. Under the assumptions of Lemma 3.2, we have
N + (B ) = (∪ip=1 RT1 gi ) \ BR ∪ (∪qj=1 LT2−1 g j ) \ BL , N + (L ) = RT1 = (RT1 \ BR ) ∪ (RT1 ∩ BR ). Lemma 3.4. Under the assumptions of Lemma 3.2, if there exists t2 ∈ T2 satisfying Lt2−1 g j ∈ / BL for some j ∈ {1, 2, . . . , q}. Then Lt2−1 g k ∈ / BL for any k ∈ {1, 2, . . . , q}.
Proof. It suffices to show that if there exists t2 ∈ T2 satisfying Lt2−1 g j ∈ BL for some j ∈ {1, 2, . . . , q}, then Lt2−1 g k ∈ BL for
any k ∈ {1, 2, . . . , q}. In fact, if Lt2−1 g j ∈ BL , assume that Lt2−1 g j = Lgi for some i ∈ {1, 2, . . . , p}, then AL Lt2−1 g j = AL Lgi , that is,
AL t2−1 g j = AL . Hence t2−1 g j ∈ AL and g j ∈ t2 AL = AR . Thus for any k ∈ {1, 2, . . . , q}, since g k ∈ AR = t2 AL , we have t2−1 g k ∈ AL , and so Lt2−1 g k ∈ BL .
Lemma 3.5. Under the assumptions of Lemma 3.2, we have T2 ⊆AR , ∪ j=1 LT2−1 g j ⊆ BL and N + (Rg j ) = LT2−1 g j ⊆ BL for any j ∈ {1, 2, . . . , q}. q
(∪qj=1 LT2−1 g j ) \ BL = ∅, then |(∪qj=1 LT2−1 g j ) \ BL | ≥ q. Since (∪ip=1 RT1 gi ) \ BR ⊇ RT1 \ BR , and q |RT1 ∩ BR | ≤ q, by Lemma 3.3, we have κ (D ) = |N + (B )| ≥ |N + (L )| ≥ δ (D ), this is impossible. Thus ∪ j=1 LT2−1 g j ⊆ BL . Thus for Proof. By
Lemma 3.4,
if
any t2 ∈ T2 , Lt2−1 g j ∈ BL and t2 ∈ g j AL = AR . Hence T2 ⊆AR .
Theorem 3.6. Let D = D(G, L, R; T1 , T2 ) be a strongly connected half vertex transitive digraph. Then κ (D ) = δ (D ). Proof. Let A be an atom of D. Without loss of generality, assume that A is a positive atom. By means of contradiction. If κ (D) < δ (D), then |A| ≥ 2, |A ∩ [G: L]| ≥ 1, and |A ∩ [G: R]| ≥ 1. Since Aut(D) acts transitively on [G: L] and [G: R], respectively, we may assume that L ∈ A. Let A = {Lg1 = L, Lg2 , . . . , Lg p } ∪ {Rg 1 , Rg 2 , . . . , Rg q }, BL = {Lg1 , Lg2 , . . . , Lg p }, BR = {Rg 1 , Rg 2 , . . . , Rg q }, p q and AL = ∪i=1 Lgi and AR = ∪ j=1 Rg j . Then by Lemma 3.2, AL is a subgroup of G and AR is a left coset of AL . By Lemma 3.5, T2 ⊆AR . We proceed to show that T1 ⊆AR . If for some t1 ∈ T1 and some i ∈ {1, 2, . . . , p} satisfying Rt1 gi ∈ BR , let Rt1 gi = Rg j ∈ BR ,
then Rt1 gi ⊆AR . Since gi ∈ AL and AR = bAL , we have t1 ∈ bAL g−1 = bAL = AR and t1 gk ∈ AR for any k ∈ {1, 2, . . . , p}. Hence i p Rt1 gk ∈ BR . Equivalently, if for some t1 ∈ T1 and some i ∈ {1, 2, . . . , p} satisfying Rt1 gi ∈ BR , then (∪i=1 Rt1 gi ) ∩ BR = ∅, thus |N+ (A )| ≥ p. On the other hand, by Lemma 3.5, N+ (Rg i ) ⊆ BL , thus |N+ (Rg i )| ≤ |BL | = p. We have |N+ (A )| ≥ δ (D ), a contradiction. Thus for any t1 ∈ T1 and any i ∈ {1, 2, . . . , p}, Rt1 gi ∈ BR , and so RT1 gi ⊆AR for any i ∈ {1, 2, . . . , p}. Since gi ∈ AL , we have T1 ⊆ RT1 ⊆ AR g−1 = bAL g−1 = bAL = AR . But T2−1 T1 ≤ AL G = T2−1 T1 , a contradiction. i i Since κ (D) ≤ λ(D) ≤ δ (D) for any digraph D, by Theorem 3.6, we have Corollary 3.7. Let D be a strongly connected half vertex transitive digraph. Then κ (D ) = λ(D ) = δ (D ). Since Bi-Cayley digraphs are half vertex transitive digraphs, we have Corollary 3.8. Let D = D(G; T1 , T2 ) be a strongly connected Bi-Cayley digraph. Then κ (D ) = λ(D ) = δ (D ). Since bipartite Cayley digraphs are Bi-Cayley digraphs, by the above corollary, we deduce the following result. Corollary 3.9. Let D = D(G, S ) be a strongly connected bipartite Cayley digraph. Then κ (D ) = λ(D ) = δ (D ).
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