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Perfect codes in power graphs of finite groups

  • Xuanlong Ma EMAIL logo , Ruiqin Fu , Xuefei Lu , Mengxia Guo and Zhiqin Zhao
Published/Copyright: December 9, 2017
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Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

Abstract

The power graph of a finite group is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. The enhanced power graph of a finite group is the graph whose vertex set consists of all elements of the group, in which two vertices are adjacent if they generate a cyclic subgroup. In this paper, we give a complete description of finite groups with enhanced power graphs admitting a perfect code. In addition, we describe all groups in the following two classes of finite groups: the class of groups with power graphs admitting a total perfect code, and the class of groups with enhanced power graphs admitting a total perfect code. Furthermore, we characterize several families of finite groups with power graphs admitting a perfect code, and several other families of finite groups with power graphs which do not admit perfect codes.

Keywords: Power graph; Enhanced power graph; Finite group; Perfect code; Total perfect code
MSC 2010: 05C25; 05C69; 94B99

1 Introduction

Every graph Γ considered in this paper is finite, simple, and undirected with vertex set V(Γ) and edge set E(Γ). A code in Γ is simply a subset of V(Γ). A code C of Γ is called a perfect code [1] if C is an independent set such that every vertex in V(Γ)\ C is adjacent to exactly one vertex in C. A code C is said to be a total perfect code [2] in Γ if every vertex of Γ is adjacent to exactly one vertex in C.

Since the beginning of coding theory in the late 1940s, perfect codes have been important objects of study in information theory; see the surveys [3, 4] on perfect codes and related definitions in the classical setting. It is known in [5] that deciding whether a graph has a total perfect code is NP-complete. Beginning with [1], perfect codes in general graphs have also attracted considerable attention in the community of graph theory (see [6,7,8,9]). In particular, perfect codes in Cayley graphs of groups are especially charming objects of study (see [10,11,12]). For more information on coding applications of algebraic constructions, the readers are referred to [13, §9.1 and §9.2].

Graphs associated with groups and other algebraic structures have been actively investigated, since they have valuable applications (cf. [14,15]) and are related to automata theory (cf. [16,17]). The undirected power graph ΓG of a finite group G has the vertex set G and two distinct elements are adjacent if one is a power of the other. The enhanced power graph ΔG of a finite group G is the graph whose vertex set consists of G, in which two distinct vertices are adjacent if they generate a cyclic subgroup. The concepts of a power graph and an undirected power graph were first introduced by Kelarev and Quinn [18] and Chakrabarty et al. [19], respectively. Since the paper deals only with undirected graphs, we use the term “power graph” to refer to an undirected power graph. In recent years, the study of power graphs has been growing (see [19,20,21,22,23,24,25,26,27,28,29,30]). Also, see [31] for a survey of results and open questions on power graphs. In order to measure how close the power graph is to the commuting graph [32], Aalipour et al. [33] introduced the enhanced power graph which lies in between. See [34] for some properties of the enhanced power graphs.

In this paper, we always use G to denote a finite group with the identity e. Denote by G* the set G \ {e}. For a subset S of G, let ΓG[S] (resp. ΔG[S]) denote the induced subgraph of ΓG (resp. ΔG) by S. If the situation is unambiguous, then we denote ΓG[S] (resp. ΔG[S]) simply by ΓS (resp. ΔS).

The paper is devoted to studying the perfect codes of the power graph of a finite group. We first give sharp lower and upper bounds for the size of a subset of G to be a perfect code in ΓG* and characterizer the groups achieving the bounds (see Theorem 2.2). We also give several families of groups G such that ΓG* admits a perfect code, and several other families of groups G such that ΓG* does not admit perfect codes. Furthermore, we obtain a complete characterization of finite groups whose enhanced power graphs admit a perfect code (see Theorem 2.10). In particular, we characterize the groups G such that ΔG* admits a perfect code with size 1 (see Theorem 2.11), which answers a question posed by Bera and Bhuniya [34]. Also, we classify all nilpotent groups G such that ΔG* admits a perfect code with size 1 (see Proposition 2.12), which extends [34, Theorems 3.2 and 3.3]. In Sect. 3, we describe all groups in the following two classes of finite groups: the class of groups with power graphs admitting a total perfect code (see Corollary 3.4), and the class of groups with enhanced power graphs admitting a total perfect code (see Theorem 3.5).

2 Perfect codes

We first remark both ΔG and ΓG admit a perfect code {e}. If G is cyclic, then a generator g of G is adjacent to every element of G* \ {g} in ΓG* and ΔG*, and so in this case, both ΓG* and ΔG* admit a perfect code {g}.

In the rest of this section, therefore, we always assume that G is noncyclic. We focus on studying the perfect codes of ΓG* and ΔG*. We first give sharp lower and upper bounds for the size of a subset of G to be a perfect code in ΓG*, and characterizer the groups achieving the bounds. Next, we give some families of finite groups G such that ΓG* admits a perfect code, and give several other families of groups G such that ΓG* does not admit perfect codes. Finally, we give a complete characterization of finite groups G such that ΔG* admits a perfect code.

2.1 Power graphs

The neighborhood of a vertex x in a graph Γ, denoted by NΓ(x), is the set of vertices which have distance one from x. If the situation is unambiguous, then we denote NΓ(x) simply by N(x). A maximal cyclic subgroup of G is a cyclic subgroup, which is not a proper subgroup of some proper cyclic subgroup of G. We remark that a finite group has a unique maximal cyclic subgroup if and only if the group is cyclic of prime power order. Denote by 𝓜G the set of all maximal cyclic subgroups of G. Note that G is always noncyclic. Write

MG={M1,M2,…,Mt} (1)

where t is a positive integer at least 2.

Lemma 2.1

With reference to (1), if C is a perfect code of ΓG*, then for any 1 ≤ i ≤ t, there exists precisely one vertex in C such that it belongs to Mi.

Proof

Let x be a generator of Mi for some 1 ≤ i ≤ t. Suppose that x ∉ C. Since C is a perfect code, there exists an element a in C such that a is adjacent to x. If 〈x〉 ⊆ 〈a〉, since 〈x〉 is maximal cyclic, we have 〈x〉 = 〈a〉, and so a ∈ Mi. If 〈a〉 ⊆ 〈x〉, then it is clear that a ∈ Mi. It follows that there exists at least one vertex in C such that it belongs to Mi.

Now assume, to the contrary, that C contains two distinct elements y and z with y, z ∈ Mi. Since C is an independent set, x ∉ C. Therefore, x is adjacent to exactly one vertex of C. Since y, z ∈ Mi and 〈x〉 = Mi, it follows that y, z ∈ N(x), a contradiction.□

With reference to (1), let MG = ⋃1≤i<j≤t(Mi ∩ Mj). For each 1 ≤ i ≤ t, write

Mi⋆=Mi∖MG. (2)

The generalized quaternion group of order 4n is defined by

Q4n=〈x,y:xn=y2,x2n=1,y−1xy=x−1〉,n≥2. (3)

By verifying (3), y−1 = xny, |xiy| = 4 and (xiy)−1 = x2n−iy for i ∈ {1, …, n − 1}. Now we have the following theorem.

Theorem 2.2

Suppose that ΓG* admits a perfect code C. With reference to (1),

1≤|C|≤t, (4)

the lower bound holds if and only if G ≅ Q2m where Q2m is the generalized quaternion group of order 2m and m ≥ 3, and the upper bound holds if and only if there exists a set {x1, …, xt} ⊆ G* satisfying the conditions:

  1. With reference to (2), xi ∈ Mi⋆ for each 1 ≤ i ≤ t;

  2. N(xi) ∩ N(xj) = ∅ for each two distinct i, j in {1, …, t};

  3. ⋃i=1t N(xi) = G* \ {x1, …, xt}.

Proof

By Lemma 2.1, we have 1 ≤ |C| ≤ t. Clearly, |C| = 1 if and only if there exists an element a in G* such that N(a) = G* \ {a}. Since we are assuming that G is not cyclic, it follows from [21, Proposition 4] that |C| = 1 if and only if G ≅ Q2m, as required.

Now write D = {x1, …, xt}. Suppose that D satisfies conditions (i)–(iii). By (iii), D is an independent set of ΓG*. Also, by (ii) and (iii), it is easy to see that every vertex in G* \ D is adjacent to exactly one vertex of D. This implies that D is a perfect code of ΓG* of size t, as required.

Let C = {x1, …, xt} be a perfect code of ΓG* of size t. With reference to (1), if xi ∈ Mj ∩ Ml and j ≠ l for some i, j, l in {1, …, t}, by the Dirichlet principle, there exist two elements in C such that they both belong to same maximal cyclic subgroup of G, contrary to Lemma 2.1. We conclude that (i) holds. Also, since C is a perfect code, it is clear that C satisfies (ii) and (iii).□

Now we give a family of groups G such that ΓG* admits a perfect code C, where |C| satisfies the upper bound of (4).

Example 2.3

Let G = ℤp × Zqn, where p and q are two distinct primes and n is a positive integer at least 2. Then ΓG* admits a perfect code of size qn−1q−1. In particular, every perfect code of ΓG* has size |𝓜G|.

Proof

Clearly, G has a unique subgroup of order p, say P. Let t = qn−1q−1. Then it is not hard to see that G has t maximal cyclic subgroups M1, …, Mt and each of them is isomorphic to ℤpq. Furthermore, the intersection of each two distinct maximal cyclic subgroups is P. Suppose that C is a perfect code of ΓG*. If there exists x in P* such that x ∈ C, then by Lemma 2.1 |C| = 1, and hence we have a contradiction by Theorem 2.2, since G is neither cyclic nor generalized quaternion. It follows that if C has an element y ∈ Mi for some i ∈ {1, …, t}, then y ∈ Mi⋆ , which implies that C has size t by Lemma 2.1.

Now take elements x1, …, xt in ⋃i=1tMi⋆ with xi ∈ Mi⋆ for each i ∈ {1, …, t}, |xt| = pq, and |xj| = q for each 1 ≤ j ≤ t − 1. By Theorem 2.2, it is easy to check that {x1, …, xt} is a perfect code, as required.□

The following example shows that, although ΓG* admits a perfect code, the size of each perfect code of ΓG* dose not achieve the upper bound of (4).

Example 2.4

Let G = Z2n × ℤ12, where n is a positive integer at least 2. Then ΓG* admits a perfect code of size 2n+1−1. In particular, ΓG* does not admit a perfect code of size |𝓜G|.

Proof

It is evident that G has a unique subgroup P of order 3, and has 2n and 2n+1-2 maximal cyclic subgroups of order 12 and 6, respectively. Moreover, the intersection of each two maximal cyclic subgroups of order 12 has order 6. Write

MG={M1,…,M2n,K1,…,K2n+1−2},

where |Mi| = 12 and |Kj| = 6 for all 1 ≤ i ≤ 2n and 1 ≤ j ≤ 2n+1-2, ⋂i=12nMi≅Z6 and ⋂i=12n+1−2Ki=P.

Suppose that C is a perfect code of ΓG* of size |𝓜G|. By Theorem 2.2, there exist x ∈ M1 \ M2 and y ∈ M2 \ M1 such that x, y ∈ C. Take a ∈ M1 ∩ M2 with |a| = 2. It follows that a ∉ C and a ∈ N(x) ∩ N(y), a contradiction. We conclude that ΓG* does not admit perfect codes of size |𝓜G|.

Take x1 in M1, x0 ∈ K1 and xi ∈ Ki for all i ∈ {2, …, 2n+1−2}, so that |x0| = 6 and |xj| = 2 for each 1 ≤ j ≤ 2n+1 − 2. Then it is easy to check that {x0, x1, …, x2n+1−2} is a perfect code of ΓG*, as required.□

Finally, we remark that many power graphs do not admit perfect codes. In the following we give two families of examples.

Proposition 2.5

With reference to (1), suppose that G satisfies

Mi≅Z4p,⋂j=1tMj≅Z2p

for some odd prime p and all 1 ≤ i ≤ t. Then ΓG* dose not admit perfect codes.

Proof

Let ⋂j=1t Mj = P. Suppose for a contradiction that C is a perfect code of ΓG*. Note that G is not isomorphic to a generalized quaternion 2-group. It follows from Theorem 2.2 that |C| ≥ 2. Also, by Lemma 2.1 we deduce C ∩ P* = ∅, and so there exist x ∈ M1\ P and y ∈ M2\ P such that x, y ∈ C. We conclude that |x|, |y| ∈ {4p, 4}. Take z in P with |z| = 2. Then z ∈ N(x) ∩ N(y), which is a contradiction since C is a perfect code.□

The following result follows from Proposition 3.3.

Example 2.6

Let G = Q8 × ℤp for some odd prime p. Then ΓG* dose not admit perfect codes.

Proposition 2.7

Suppose that G satisfies

MG={M11,M12,M21,M22,…,Mk1,Mk2},

where k is a positive integer at least 2, ⋂i=1k Mi1 ≅ ℤp, Mi1 ∩ Mi2 ≅ ℤ2p and Mij ≅ ℤ4p for some odd prime p, all 1 ≤ i ≤ k and all 1 ≤ j ≤ 2. Then ΓG* does not admit perfect codes.

Proof

It is clear that G has a unique subgroup of order p, say P. Suppose that C is a perfect code of ΓG*, we work to obtain a contradiction. Clearly, G is not a generalized quaternion 2-group. By Theorem 2.2, |C| ≥ 2, and it follows from Lemma 2.1 that P* ∩ C = ∅. Also, Lemma 2.1 implies that there exist x in M11 and y in M12 such that x, y ∈ C. If x ≠ y, then x ∈ M11 \ M12 and y ∈ M12 \ M11, and so |x|, |y| ∈ {4p, 4}, which implies that for the involution z of M11∩ M12, we have z ∈ N(x)∩ N(y), a contradiction. We deduce x = y, and hence x ∈ (M11∩ M12) \ P, which implies that |x| = 2 or 2p. Since C is a perfect code, the element of order 4 of M11 is adjacent to x in ΓG*, and thus |x| = 2. Similarly, considering Mi1 and Mi2 for each 2 ≤ i ≤ k, we conclude that every element of C is an involution. It follows that there does not exist an element in C such that it is adjacent to a generator of P, which is a contradiction.□

By Proposition 2.7, we have the following example.

Example 2.8

Γ(ℤ4 × ℤ12)* dose not admit perfect codes.

2.2 Enhanced power graphs

In this subsection, we give a complete characterization of finite groups whose enhanced power graphs admit a perfect code. We begin with the following lemma. Since ΓG∗ is a subgraph of ΔG*, the proof of Lemma 2.9 is similar to the proof of Lemma 2.1.

Lemma 2.9

With reference to (1), if C is a perfect code of ΔG*, then for any 1 ≤ i ≤ t, there exists precisely one element in C such that it belongs to Mi.

Suppose that G is a group with the property that each two maximal cyclic subgroups of G have trivial intersection, or that if

⋂i=1lMi≠{e}

for some maximal cyclic subgroups M1, …, Ml of G (here l may be 1), and there exists Mk in 𝓜G \ {M1, …, Ml} such that Mk ∩ Mm is nontrivial for some m ∈ {1, 2, …, l}, then

(⋂i=1lMi)∩Mk≠{e}.

Then G is said to satisfy the intersection property. For example, both the elementary abelian p-group Zpn and Zp×Zqn satisfy the intersection property, where p and q are distinct primes.

Theorem 2.10

ΔG* admits a perfect code if and only if G satisfies the intersection property.

Proof

First, assume that G satisfies the intersection property. Write

MG={M11,M12,…,M1l1,M21,M22,…,M2l2,…,Mm1,Mm2,…,Mmlm,Mm+1,Mm+2,…,Mm+n},

where li ≥ 2 and ⋂j=1liMij is nontrivial for each 1 ≤ i ≤ m, and every two of the rest have trivial intersection. For each 1 ≤ i ≤ m and 1 ≤ k ≤ n, take ai ∈ ⋂j=1liMij∗ and bk∈Mm+k∗. Let C = {a1, a2, …, am, b1, b2, …, bn}.

Suppose that a1 and a2 are adjacent in ΔG*. Then 〈a1, a2〉 is cyclic, and so there exists a maximal cyclic subgroup M of G such that 〈a1, a2〉 ⊆ M. Since G satisfies the intersection property, a1 must belong to one of {M11, M12, …, M1l1} and a2 must belong to one of {M21, M22, …, M2l2}, which implies that

M∈{M11,M12,…,M1l1}∩{M21,M22,…,M2l2},

a contradiction.

We conclude that a1 and a2 are nonadjacent in ΔG*. Similarly, we can obtain that C is an independent set of ΔG*. Let y be an arbitrary element of G* \ C. Without loss of generality, let y ∈ M11∗ \ C. Clearly, a1 and y are adjacent in ΔG*. Suppose that there exists z ∈ C \ {a1} such that y is adjacent to z. If z = ai for some 2 ≤ i ≤ m, then 〈z, y〉 is contained in one of {Mi1, Mi2, …, Mili}, and by the intersection property of G, it follows that M11 ∩ Mi1 is nontrivial, contrary to the hypotheses of 𝓜G. Similarly, we can show that z ≠ bi for some 1 ≤ i ≤ n. We conclude, therefore, that y is adjacent to exactly one vertex in C. It follows that C is a perfect code, as required.

Conversely, assume that ΔG* admits a perfect code D. Suppose for a contradiction that G dose not satisfy the intersection property. In other words, there exist 3 distinct maximal cyclic subgroups M1, M2, M3 of G such that

|M1∩M2|≠1,|M1∩M3|≠1,|M1∩M2∩M3|=1.

If D ∩ M1 ∩ M2 is trivial, then by Lemma 2.9, there exist m1 ∈ M1 \ M2 and m2 ∈ M2 \ M1 such that m1, m2 ∈ D, and hence x ∈ N(m1) ∩ N(m2) for some x ∈ M1∗∩M2∗ , a contradiction. We conclude that there exists u ∈ M1∗∩M2∗ such that u ∈ D. By Lemma 2.9 again, there exists v ∈ M3 ∩ D such that both u and v belong to same maximal cyclic subgroup. Take w ∈ M1∗∩M3∗ . Clearly, w ≠ v, so we deduce w ∈ N(v)∩ N(u), and this contradiction completes the proof.□

A graph Γ is said to satisfy the cone property if Γ has a vertex which is adjacent to every vertex except itself. Bera and Bhuniya [34] posed the question: Characterize all finite non-abelian groups G such that ΔG* satisfies the cone property. Now we give an answer to the question.

Theorem 2.11

The following are equivalent for any group G with

MG={M1,M2,…,Mt}.

  1. ΔG* satisfies the cone property.

  2. ⋂i=1t Mi is nontrivial.

  3. ΔG* admits a perfect code of size 1.

Proof

First, assume (i), and let x be a vertex with |N(x)| = |G| − 2 in ΔG*. Without loss of generality, let x ∈ M1. Choose a generator x2 of M2. Since x is adjacent to x2, 〈x, x2〉 = 〈x2〉, which implies x ∈ M2. Similarly, we deduce that x belongs to Mi for each 3 ≤ i ≤ t. It follows that x ∈ ⋂i=1t Mi, proving (ii).

Now assume (ii), so that y ∈ ⋂i=1tMi∗. For some z ∈ G* \ {y}, let z ∈ Mi for some 1 ≤ i ≤ t. Then y, z ∈ Mi, and thus 〈y, z〉 is cyclic, and it follows that y and z are adjacent in ΔG*. This implies that {y} is a perfect code of ΔG*, and (iii) follows.

Finally, since it is obvious that (iii) implies (i), the proof is complete.□

Next, we classify all finite nilpotent groups G such that ΔG* admits a perfect code of size 1, which extends [34, Theorems 3.2 and 3.3].

Proposition 2.12

Let G be a nilpotent group. Then ΔG* admits a perfect code of size 1 if and only if

G≅Q2n×HorZpm×K,

where n ≥ 3, m ≥ 1, p is a prime, and both H and K are nilpotent with 2 ∤ |H| and p ∤ |K|.

Proof

First, assume that ΔG* admits a perfect code {x}. Let q be a prime divisor of |x| and let y be an element of G with |y| = p. If y ∉ 〈x〉, since x and y are adjacent, 〈x, y〉 is cyclic, and so 〈x, y〉 has two distinct subgroups of order p, a contradiction. It follows that G has a unique subgroup of order p. Now let P be the unique Sylow p-subgroup of G. By [35, Theorem 5.4.10], a p-group having a unique subgroup of order p is either cyclic or a generalized quaternion, so it follows that P is either cyclic or a generalized quaternion. Since G is nilpotent, the desired result follows.

Conversely, assume that G ≅ Q2n × H or ℤpm × K, where n ≥ 3, m ≥ 1, p is a prime, and both H and K are nilpotent with 2 ∤ |H| and p ∤ |K|. Then G has a unique subgroup P of order 2 or p, and so we conclude that every maximal cyclic subgroup of G contains P. It follows that the intersection of all maximal cyclic subgroups is nontrivial. Now the desired result follows from Theorem 2.11.□

2.3 Examples

In this subsection, we give some families of finite groups whose power graphs or enhanced power graphs admit a perfect code.

Proposition 2.13

If every two maximal cyclic subgroups of G have trivial intersection, then both ΔG* and ΓG* admit a perfect code.

Proof

By Theorem 2.10, ΔG* admits a perfect code. In the following we prove that ΓG* admits a perfect code. With reference to (1), let 〈xi〉 = Mi for all 1 ≤ i ≤ t and let C = {x1, x2, …, xt}. Since Mi is maximal cyclic, C is an independent set of ΓG*. Let x be an arbitrary element of G* \ C. Without loss of generality, say x ∈ M1. Clearly, x1 and x are adjacent. Since |M1 ∩ Mj| = 1 for each 2 ≤ j ≤ t, x is not adjacent to each of C \ {x1}. So C is a perfect code of ΓG*, as required.□

A finite group is called a P-group [36] if every nontrivial element of the group has prime order. For example, Zpm is a P-group for some prime p. Also, it is clear that every two maximal cyclic subgroups of a P-group have trivial intersection. A finite group is called a CP-group [37] if every nontrivial element of the group has prime power order. For example, every p-group is a CP-group. Certainly, a P-group is also a CP-group.

By Proposition 2.13 we see that both ΔH* and ΓH* admit a perfect code for each P-group H. What is more, here we prove that both ΔG* and ΓG* admit a perfect code for each CP-group G.

Theorem 2.14

Let G be a CP-group. Then both ΔG* and ΓG* admit a perfect code.

Proof

For two distinct elements x, y of G*, if x is a power of y, or y is a power of x, then 〈x, y〉 is cyclic. Also, if 〈x, y〉 is cyclic, since G is a CP-group, 〈x, y〉 is isomorphic to a cyclic group of prime power order, which implies that one of {x, y} is a power of the other. It follows that ΔG* and ΓG* are the same, and hence it suffices to prove ΔG* admits a perfect code.

As refer to (1), assume that there exist two distinct indices i, j in {1, …, t} such that Mi ∩ Mj is nontrivial. Let x ∈ Mi ∩ Mj with |x| = p, where p is a prime. If there exists Mt in 𝓜G \ {Mi, Mj} such that Mt ∩ Mi is nontrivial, then x ∈ Mt, and so we deduce Mi ∩ Mj ∩ Mt is nontrivial. This implies that G satisfies the intersection property. It follows from Theorem 2.10 that ΔG* admits a perfect code.□

For n ≥ 3, denote by D2n the dihedral group of order 2n, where

D2n=〈a,b:an=b2=e,bab=a−1〉.

It is not hard to see that 𝓜D2n = {〈a〉, 〈ab〉, 〈a2b〉, …, 〈anb〉} and |aib| = 2 for each 1 ≤ i ≤ n. It follows that every two maximal cyclic subgroups of D2n have trivial intersection. The following result is immediate by Proposition 2.13.

Example 2.15

Both ΓD2n∗andΔD2n∗ admit a perfect code.

For the generalized quaternion group Q4n, by (3) we have

V(ΓQ4n)={e,x,…,x2n−1}∪(⋃i=0n−1{xiy,(xiy)−1}),E(ΓQ4n)=E(Γ〈x〉)∪⋃i=0n−1E(Γ〈xiy〉).

The structure of ΓQ4n is shown in Figure 1. Now we study the perfect codes of ΓD4n∗andΔD4n∗ .

Figure 1 ΓQ4n
Figure 1

ΓQ4n

Example 2.16

  1. For any n ≥ 2, ΓQ4n∗ admits a perfect code if and only if n is a power of 2.

  2. For any n ≥ 2, ΔQ4n∗ admits a perfect code.

Proof

  1. If n is a power of 2, then Q4n is a CP-group, and so ΓQ4n∗ admits a perfect code by Theorem 2.14. Conversely, let C be a perfect code of ΓQ4n∗ . Suppose that n is not a power of 2, we work to obtain a contradiction. With reference to (3), we deduce that

    MQ4n={〈x〉,〈xy〉,〈x2y〉,…,〈xny〉},xn∈⋂i=1n〈xiy〉.

    If xn ∈ C, since xn ∈ 〈x〉, C = {xn} by Lemma 2.1, contrary to Theorem 2.2. We conclude that xn ∉ C. By Lemma 2.1 again, there exist b, c in C such that b ∈ 〈 xy 〉 and c ∈ 〈 x2y〉. Since |xy| = |x2y| = 4 and (xy)2 = (x2y)2 = xn ∉ C, it follows that xn is adjacent to both b and c. This is a contradiction since C is a perfect code.

  2. With reference to (3), xn belongs to each maximal cyclic subgroup of Q4n. The desired result follows from Theorem 2.11.

□

3 Total perfect codes

In this section we characterize all finite groups whose power graphs or enhanced power graphs admit a total perfect code. The following observation follows easily from the definition of a total perfect code.

Observation 3.1

Let Γ be a graph. A code C of Γ is a total perfect code if and only if the subgraph of Γ induced by C is a matching and {N(u) \ C : u ∈ C} is a partition of V(Γ)\ C.

Theorem 3.2

Suppose that Γ is a graph containing a vertex X of degree n − 1, where n = |V(Γ)|. Then Γ admits a total perfect code if and only if Γ has a leaf. In particular, C is a total perfect code of Γ if and only if C = {a, b} for some a, b ∈ V(Γ) with |N(a)| = n − 1 and |N(b)| = 1.

Proof

If Γ has a leaf y, then by Observation 3.1 {x, y} is a total perfect code, as desired. Now suppose that Γ admits a total perfect code C. Since |N(x)| = n − 1, x ∈ C. Also, since the subgraph of Γ induced by C is a matching, we may assume that C = {x, z} for some z ∈ V(Γ). It follows that {N(x) \{z}, N(z) \ {x}} is a partition of V(Γ)\ C, and hence N(z) \ {x} = ∅. This implies that z is a leaf, as required.□

An involution x of G is maximal if the only cyclic subgroup containing x is 〈x〉. For example, each involution of D2m is maximal for some odd number m. We remark that ℤn has a maximal involution if and only if n = 2. The proof of the following result is straightforward.

Proposition 3.3

The following are equivalent.

  1. ΓG has a leaf.

  2. ΔG has a leaf.

  3. G has a maximal involution.

Since in ΓG and ΔG, e has degree |G| − 1. The following result is immediate by Theorem 3.2 and Proposition 3.3.

Corollary 3.4

The following are equivalent.

  1. ΓG admits a total perfect code.

  2. ΔG admits a total perfect code.

  3. G has a maximal involution.

The exponent of G is the least common multiple of the orders of the elements of G. Next, we characterize all finite groups G such that ΓG* or ΔG* admits a total perfect code.

Theorem 3.5

The following are equivalent.

  1. ΓG* admits a total perfect code.

  2. G is a finite group of exponent 3.

  3. ΔG* admits a total perfect code.

Proof

First, suppose that ΓG* admits a total perfect code C. Let x, y ∈ C with {x, y} ∈ E(ΓG*). Assume that one of {x, y} has order at least 4, without loss of generality, let |x| ≥ 4. If y ≠ x−1, by Observation 3.1 x−1 ∈ V(ΓG*) \ C, and thus we deduce x−1 ∈ N(x)∩ N(y), a contradiction. It follows that y = x−1. Also, it is clear x2 ∉ C, and so x2 ∈ N(x) ∩ N(y), a contradiction.

We conclude that both x and y have order at most 3. Observe that |x| = 3 and y = x−1. It follows that every element of C has order 3 and C is inverse-closed. If V(ΓG*) \ C has an element u, then there exists z ∈ C such that u and z are adjacent, however, u−1 and z are adjacent and u−1 ∈ C, a contradiction. It follows that V(ΓG*) = C, and so every nontrivial element of G has order 3, hence (ii) follows.

Now if G is a finite group of exponent 3, then G* is a total perfect code of ΓG*. It follows that (i) and (ii) are equivalent. Similarly, we can conclude that (ii) and (iii) are equivalent.□

Acknowledgement

We are grateful to the referees for many useful suggestions and comments.

Ma’s research was supported by National Natural Science Foundation of China (61472471) and Innovation Talent Promotion Plan of Shaanxi Province for Young Sci-Tech New Star (No. 2017KJXX-60). Lu’s research was supported by National Natural Science Foundation of China (51609201). Zhao’s research was supported by National Natural Science Foundation of China (11626187).

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Received: 2017-6-11
Accepted: 2017-9-21
Published Online: 2017-12-9

© 2017 Ma et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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https://doi.org/10.1515/math-2017-0123
Keywords for this article
Power graph; Enhanced power graph; Finite group; Perfect code; Total perfect code
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