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On the chromatic number of the power graph of a finite group
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ScienceDirect Indagationes Mathematicae 26 (2015) 626–633 www.elsevier.com/locate/indag
On the chromatic number of the power graph of a finite group Xuanlong Ma ∗ , Min Feng Sch. Math. Sci. & Lab. Math. Com. Sys., Beijing Normal University, Beijing, 100875, China Received 19 August 2014; received in revised form 7 April 2015; accepted 15 April 2015 Communicated by R. Tijdeman
Abstract The power graph ΓG of a finite group G is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. We investigate the chromatic number χ(ΓG ) of ΓG . A characterization of χ (ΓG ) is presented, and a conjecture in Mirzargar et al. (2012) is disproved. Moreover, we classify all finite groups whose power graphs are uniquely colorable, split or unicyclic. c 2015 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. ⃝
Keywords: Finite group; Power graph; Graph coloring; Split graph; Unicyclic graph
1. Introduction All groups considered in this paper are finite. Kelarev and Quinn [14] introduced the directed power graph of a group G, which is the digraph with vertex set G and there is an arc from x to y if and only if x ̸= y and y = x m for some positive integer m. The directed power graphs were also considered in [15,17,16]. Motivated by this, Chakrabarty, Ghosh and Sen [4] introduced the (undirected) power graphs. The power graph ΓG of a group G is the graph whose vertex set is G with two distinct vertices adjacent if one is a power of the other. Recently, many interesting results on the power graphs have been obtained, see [6,2,3,5,8,11,10,20,21,18]. In [1], Abawajy, Kelarev and Chowdhury gave a detailed list of results and open questions. ∗ Corresponding author.
E-mail addresses: [email protected] (X. Ma), fgmn [email protected] (M. Feng). http://dx.doi.org/10.1016/j.indag.2015.04.003 c 2015 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. 0019-3577/⃝
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For a graph Γ , by V (Γ ) and E(Γ ) we denote its vertex set and its edge set, respectively. The chromatic number of Γ , denoted by χ (Γ ), is the smallest number of colors needed to color the vertices of Γ so that no two adjacent vertices share the same color. Denote by Zn the cyclic group of order n. Mirzargar, Ashrafi and Nadjafi-Arani [20] computed the chromatic number of Zn and proposed the following conjecture. Conjecture 1.1 ([20, Conjecture 1]). Let G be a group. Then χ (ΓG ) = χ (ΓZn ), where n is the maximum order of an element in G. Motivated by the conjecture, we shall explore the coloring of the power graph of a non-cyclic group. A partition of the vertex set of a graph Γ is called a coloring if each set of the partition is an independent set of Γ . If there is a unique partition of V (Γ ) into χ (Γ ) independent sets, then Γ is said to be uniquely colorable. For more information on the uniquely colorable graph, see [13, Chapter 6, p. 113]. The clique number ω(Γ ) of a graph Γ is the maximum size of a clique in a graph. If χ (∆) = ω(∆) for each induced subgraph ∆ of Γ , then Γ is called a perfect graph. It was noted in [9, Theorem 1] and [11, Corollary 2.5] that all power graphs are perfect. A graph Γ is said to be split if V (Γ ) is the disjoint union of a clique and an independent set. Split graphs form a very useful class of perfect graphs. More information on the split graphs, can be found in [12,19]. A graph is called unicyclic if it is connected and has a unique cycle. In this paper, we disprove Conjecture 1.1 and characterize the chromatic number of the power graph of a finite group. Furthermore, we classify all finite groups whose power graphs are uniquely colorable, split or unicyclic. 2. Coloring In the following proposition the chromatic number of the power graph of a cyclic group is determined. Proposition 2.1 ([20, Theorem 2]). Let n = p1α1 p2α2 · · · prαr , where p1 < p2 < · · · < pr are prime numbers. Then χ (ΓZn ) = prαr +
j r −2 α j−1 αr −i ( pr −r −j−1 − 1) ) , φ( pr −i j=0
i=0
where φ is Euler’s totient function. Denote by |x| the order of element x in the group. Now we disprove Conjecture 1.1 by the following example. Example 2.2. Given a group G and an element g ∈ G of order n, Γ⟨g⟩ is an induced subgraph of ΓG which is isomorphic to ΓZn . It follows that χ (ΓG ) ≥ χ (Γ⟨g⟩ ) = χ (ΓZn ). Since power graphs are perfect, Conjecture 1.1 is equivalent to the statement that Γ⟨g⟩ contains a maximal clique of ΓG whenever g has maximum order. In fact, this is not always the case. Our counterexample to Conjecture 1.1 is the general linear group GL(2, p) of invertible 2 × 2 matrices over GF( p) for certain prime numbers p. By [22, Theorem 2], the maximum order
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of any element of GL(2, p) is p 2 − 1. This result also implies that there exists an element of GL(2, p) of order p( p − 1). Some such elements can be constructed as follows. Let α be a primitive element of GF( p 2 ). So |α| = p 2 − 1 in the multiplicative group GF∗ ( p 2 ). Then there are b, c ∈ GF( p) such that α 2 + bα + c = 0. Let 0 1 A= . −c −b Then we have 1 1 A =α . α α It follows that the order of A is p 2 − 1. Let β be a primitive element of GF( p). Hence |β| = p −1 in the multiplicative group GF∗ ( p). Let 0 1 B = β I2 + . 0 0 Then B = β I2 + nβ n
n
n−1
0 0
1 . 0
This implies that the order of B is p( p − 1). In general there is no way to determine the primitive element of GF( p) or GF( p 2 ), but we know that there are respectively φ( p − 1) and φ( p 2 − 1) many of each. Using Proposition 2.1, we compute χ (Γ⟨A⟩ ) and χ (Γ⟨B⟩ ). Take p = 11, so p 2 − 1 = 120 and p( p − 1) = 110. Now χ (Γ⟨A⟩ ) = 69 and χ (Γ⟨B⟩ ) = 91. We have verified that this construction also gives counterexamples to Conjecture 1.1 for all prime numbers 11 ≤ p < 100. The general problem is a subtle problem of number theory. Next, we study the chromatic number of the power graph of a non-cyclic group. Lemma 2.3 ([20, Lemma 1]). A subset of a group forms a clique in the power graph of the group if and only if the elements of the subset can be ordered so that each element is a power of the next. We say that a proper cyclic subgroup M of group G is maximal cyclic if M ≤ K ≤ G implies M = K or K = G, where K is cyclic. Denote by MG the set of maximal cyclic subgroups of G. Theorem 2.4. For any non-cyclic group G, χ (ΓG ) = max{χ (Γ M ) : M ∈ MG }. Proof. Note that ΓG is perfect. It suffices to prove that ω(ΓG ) = max{ω(Γ M ) : M ∈ MG }. Note that for any subgroup K of G, the subgraph Γ K of ΓG is an induced subgraph. Therefore max{ω(Γ M ) : M ∈ MG } ≤ ω(ΓG ). On the other hand, for any clique C of ΓG , there exists an element x in C such that C ⊆ ⟨x⟩ and ⟨x⟩ ∈ MG by Lemma 2.3. This implies that ω(ΓG ) ≤ max{ω(Γ M ) : M ∈ MG }. The proof is now complete. By Theorem 2.4, the following corollary is immediate. Corollary 2.5. Let G be a group with max{|x| : x ∈ G} = n. If n is a prime power, then χ(ΓG ) = n.
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Remark 2.6. By Proposition 2.1 and Theorem 2.4, we can see that for any group G, if the order of every maximal cyclic subgroup of G is known, then we may calculate χ (ΓG ). A group G is said to be an EPO-group if all non-trivial element orders of G are prime (see [7,24]). Denote by π(n) the set of the prime divisors of a positive integer n (and not the number of prime numbers ≤ n). Given a group G, we shall write π(G) instead of π(|G|). It is straightforward that χ (ΓG ) = 2 if and only if G is an elementary abelian 2-group. We now classify all groups whose power graphs have chromatic number 3. Proposition 2.7. Let G be a group. Then χ (ΓG ) = 3 if and only if G is one of the following groups: • The symmetric group S3 on three letters. • A 3-group of exponent 3. • An EPO-group of order 2m · 3 or 2 · 3m for some positive integer m ≥ 2. Proof. First assume that χ (ΓG ) = 3. Then ω(ΓG ) = 3. Hence, π(G) ⊆ {2, 3}. If G is a 2group then G has exponent 2 and so G is an elementary abelian 2-group, namely ω(ΓG ) = 2, a contradiction. If G is a 3-group, then every non-identity element of G is of order 3, that is, G is a 3-group of exponent 3, as desired. Thus, now we may assume that π(G) = {2, 3}. In this case the order of any non-identity element of G is 2 or 3. It follows that G is an EPO-group. Now the desired result follows from [7]. For the converse, the result follows trivially from Corollary 2.5. Now we classify all groups whose power graphs are uniquely colorable. Theorem 2.8. For a group G, ΓG is uniquely colorable if and only if G is an elementary abelian 2-group or a cyclic group of prime power order. Proof. Pick a cyclic subgroup of G and consider its set U of generators. Observe that the elements of U are mutually adjacent and have exactly the same set of neighbors in ΓG since each is a power of the others. In particular, in any partition of ΓG into independent sets, we may freely permute the elements of U in the cells and once again obtain a partition of ΓG into independent sets. Note that any cyclic subgroup of order at least three has at least two generators. Assume ΓG is uniquely colorable. Then the above implies that every element of order at least three is a maximal independent set in any partition of ΓG into as few independent sets a possible. Since we cannot enlarge this singleton independent set by moving a vertex from another cell of the partition, we conclude that every element of order at least three is adjacent to every vertex of the power graph of G. If G is an elementary abelian 2-group, we may partition the set of vertices of ΓG into two independent sets: one consisting of the identity, and one consisting of every other vertex. This is its unique partition into χ (ΓG ) = 2 independent sets. Suppose G is not an elementary abelian 2-group. If G does not have prime power order, then there are elements of distinct prime orders, one of which is at least three. By construction, they are non-adjacent in the power graph (neither is a power of the other). This contradicts the above, so G must have prime power order. Now let C be a maximal proper cyclic subgroup with generator x and take an element g not in C. Clearly, g is not a power of x. Thus since g and x are adjacent, it must be the case that g is a power of x, so C is contained in the subgroup generated by x. By maximality of C, it must
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Fig. 1. C4 .
be the case that G is cyclic with generator x. Thus G is either an elementary abelian 2-group or a cyclic group of prime power order. For the converse, observe that ΓG is complete bipartite or complete. The desired result follows trivially. 3. Split In this section we shall characterize the groups whose power graphs are split. It was shown in [12], that a graph is split if and only if it does not have an induced subgraph isomorphic to one of the three forbidden graphs, C4 , C5 or 2K 2 . Note that the power graph of a group is perfect. Then ΓG has no induced subgraph isomorphic to C5 . Theorem 3.1. For any group G, ΓG is split if and only if ΓG has no induced subgraph isomorphic to the forbidden graph 2K 2 . Proof. Necessity is clear so we just need to prove sufficiency. Suppose that ΓG has no induced subgraph isomorphic to 2K 2 . In order to prove that ΓG is split, it suffices to show that ΓG has no induced subgraph isomorphic to C4 . Assume, to the contrary, that ΓG has an induced subgraph isomorphic to C4 (see Fig. 1). Without loss of generality, let ⟨x⟩ ⊆ ⟨y⟩. It follows that ⟨v⟩ ⊆ ⟨y⟩, ⟨v⟩ ⊆ ⟨u⟩ and ⟨x⟩ ⊆ ⟨u⟩. Hence |y| > 2 and |u| > 2. Note that y and u are not adjacent in ΓG . So the subgraph of ΓG induced by the set {y, y −1 , u, u −1 } is isomorphic to 2K 2 , a contradiction. The following result is obvious by Theorem 3.1. Corollary 3.2. Let G be a group and let ΓG be split. If there exists an element x in G such that |x| > 2, then G has precisely one cyclic subgroup of order |x|. It follows from Theorem 3.1 that if ΓG is split, then |G| is a divisor of 2n p m for some odd prime p and some nonnegative integers n and m. In the following we shall discuss the groups whose powers are split. Lemma 3.3. Let G be a non-abelian 2-group, and M be a maximal cyclic subgroup of G. Then ΓG is split if and only if G satisfies the following conditions: Every element y ∈ G\M is an involution and ⟨y⟩ is not normal in G M is normal in G and G/M is elementary abelian (1) Suppose M = ⟨x⟩. Then for all y ∈ G\M, yx y = x −1 Proof. We first assume that ΓG is split. Let M = ⟨x⟩. Since G is non-abelian, one has |x| > 2. By Corollary 3.2, M is the unique cyclic subgroup of order |x|. Hence M is normal in G. Take y ∈ G\M. Then x and y are non-adjacent in ΓG . If |y| > 2, then the subgraph induced by
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{y, y −1 , x, x −1 } is isomorphic to 2K 2 , contrary to Theorem 3.1. It follows that any element in G\M is an involution. Now we verify that G/M is elementary abelian. Since y ̸∈ M, one has yx ̸∈ M, and so |yx| = 2. It means that yx y = x −1 . Suppose that ⟨y⟩ is normal in G. Then G contains an abelian subgroup isomorphic to Z2 ×Z2m where 2m = |M|. Note that there exist two elements (0, 1), (1, 1) ∈ Z2 × Z2m such that |(0, 1)| = |(1, 1)| = 2m and ⟨(0, 1)⟩ ̸= ⟨(1, 1)⟩. This contradicts Corollary 3.2. Thus, ⟨y⟩ is not normal in G. Now the proof of the necessity is complete. For the converse, observe that M is a clique and G\M is an independent set. It means that ΓG is split. Example 3.4. Denote by D2·2m−1 the dihedral group of order 2m , where m ≥ 3. Then D2·2m−1 satisfies the conditions in (1). Lemma 3.5. Let G be an abelian 2-group. Then ΓG is split if and only if G is cyclic or elementary abelian. Proof. Suppose that G is a non-cyclic group such that ΓG is split. Let M be a maximal cyclic subgroup of G. Since G is non-cyclic, G has at least two distinct involutions. This implies that there exists an involution in G that does not belong to M. If |M| = n > 2, then G has a subgroup isomorphic to Z2 × Zn , and so ⟨(0, 1)⟩ ̸= ⟨(1, 1)⟩ and |(0, 1)| = |(1, 1)| = n, contrary to Corollary 3.2. Thus, we have |M| = 2 and so G is elementary abelian. The converse is straightforward. Lemma 3.6. Let G be a p-group, where p is an odd prime number. Then ΓG is split if and only if G is cyclic. Proof. Suppose that ΓG is split. By Corollary 3.2, G has just one subgroup of order p, it follows that G is cyclic. The converse is clear. n Lemma 3.7. Suppose that G ∼ = Zm 2 × Z p , where p is an odd prime number and m and n are two positive integers. Then ΓG is split if and only if G ∼ = Z2 p .
Proof. If G ∼ = Z2 p , then by Proposition 2.1 ω(ΓG ) = 2 p − 1 and hence ΓG is split. Conversely, we first suppose that n ≥ 2. Note that G has a subgroup of order 2 p n . It follows that there exist two elements x, y ∈ G such that |x| = 2 p and |y| = p 2 . This implies that the subgraph of ΓG induced by {x, x −1 , y, y −1 } is isomorphic to 2K 2 . In view of Theorem 3.1, one has a contradiction. Thus n = 1. Suppose that G has a subgroup isomorphic to Z22 × Z p ∼ = Z2 × Z2 p . Then G has two distinct elements u, v such that |u| = |v| = 2 p and ⟨u⟩ ̸= ⟨v⟩, contradicting Corollary 3.2. Lemma 3.8. Let G be a non-abelian group of order 2m p n , and let M be a Sylow p-subgroup of G, where p is an odd prime number and m, n ≥ 1. Then ΓG is split if and only if G satisfies the following conditions: G∼ = Zm 2 n Z pn Every element y ∈ G\M is an involution and ⟨y⟩ is not normal in G (2) Suppose M = ⟨x⟩. Then for all y ∈ G\M, yx y = x −1 Proof. It is similar to the proof of Lemma 3.3.
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Example 3.9. Let p be an odd prime number and m be a positive integer. Then D2· pm satisfies the conditions in (2). We now state our main theorem of this section as follows. Theorem 3.10. Let G be a group. Then ΓG is split if and only if G is isomorphic to one of the groups: • Z pm for some prime p and positive number m. • Z2 p for some odd prime number p. • Zm 2 for some positive number m. • A non-abelian 2-group satisfying the conditions in (1). • A non-abelian group of order 2m p n satisfying the conditions in (2). Proof. Note that if there exist two distinct odd prime numbers p, q ∈ π(G), then G has elements x and y such that |x| = p and |y| = q, and so the subgraph induced by {x, x −1 , y, y −1 } is 2K 2 . Consequently, if ΓG is split, then by Theorem 3.1 one has π(G) ⊆ {2, p} for some odd prime number p. Now our result follows from Lemmas 3.3, 3.5, 3.6, 3.7 and 3.8. 4. Unicyclic power graphs The proof of the following proposition is straightforward. Proposition 4.1. For group G, ΓG has no cycles if and only if G is an elementary abelian 2group. Note that in this case the power graph is a star graph. A unicyclic graph is either a cycle or a cycle with trees attached. In this section we shall classify all groups whose power graphs are unicyclic. We first quote the “N /C” lemma in group theory. Lemma 4.2 ([23, Theorem 7.1(1)]). Let G be a group and H a subgroup of G. Then the quotient group N G (H )/C G (H ) is isomorphic to a subgroup of the automorphism group Aut(H ) of H . Lemma 4.3. There exist no EPO-groups G of order 2n · 3 for some positive integer n ≥ 2 such that ΓG is unicyclic. Proof. Suppose that G is an EPO-group of order 2n · 3 such that its power graph is unicyclic, where n ≥ 2. Let P and Q be a Sylow 3-subgroup and a Sylow 2-subgroup of G, respectively. Clearly, P ∼ = Z3 , and Q is an elementary abelian 2-group of order 2n . Set P = ⟨x⟩. Since ΓG is unicyclic, G has precisely two distinct elements of order greater than 2. It follows that P is normal in G. By Lemma 4.2, G/C G (P) can be imbedded in Aut(P). Suppose there exists an element y in G\P such that x y = yx. Then |y| = 2. Since |x| and |y| are coprime, one has |x y| = 6, which is impossible. It means that C G (P) = P. It is well known that Aut(P) ∼ = Z2 . Hence, G/C G (P) ∼ = 1 or Z2 , in the former case one has G = P, in contradiction to the order of G. In the latter case one has |G| = 6, also a contradiction. In [5, Theorem 11], it is shown that for an abelian group G, ΓG is unicyclic if and only if G is the cyclic group of order 3. Now we determine all groups whose power graphs are unicyclic. Theorem 4.4. Let G be a group. Then ΓG is unicyclic if and only if G is isomorphic to Z3 or S3 . Proof. We first assume that ΓG is unicyclic. If G has an element y of order greater than or equal to 4, then the subgraph of ΓG induced by ⟨y⟩ has at least two distinct cycles, which is impossible.
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Thus, χ (ΓG ) = 2 or 3 by Corollary 2.5. By Propositions 4.1 and 2.7 we have that G ∼ = Z3 , S3 or an EPO-group of order 2n · 3 for some positive integer n ≥ 2. Now Lemma 4.3 completes the proof of the necessity. The sufficiency is straightforward. Acknowledgments The authors are indebted to Prof. R. Tijdeman and the anonymous reviewers for some corrections and suggestions, the construction of Example 2.2, and the proof of Theorem 2.8. This research is supported by National Natural Science Foundation of China (11271047, 11371204), the Scientific Research Foundation of Yunnan Educational Committee (2014Y500). References [1] J. Abawajy, A. Kelarev, M. Chowdhury, Power graphs: A survey, Electron. J. Graph Theory Appl. 1 (2) (2013) 125–147. [2] P.J. Cameron, The power graph of a finite group, II, J. Group Theory 13 (6) (2010) 779–783. [3] P.J. Cameron, S. Ghosh, The power graph of a finite group, Discrete Math. 311 (13) (2011) 1220–1222. [4] I. Chakrabarty, S. Ghosh, M.K. Sen, Undirected power graphs of semigroups, Semigroup Forum 78 (3) (2009) 410–426. [5] T.T. Chelvam, M. Sattanathan, Power graph of finite abelian groups, Algebra Discrete Math. 16 (1) (2013) 33–41. [6] B. Curtin, G.R. Pourgholi, Edge-maximality of power graphs of finite cyclic groups, J. Algebraic Combin. 40 (2) (2014) 313–330. [7] M. Deaconescu, Classification of finite groups with all elements of prime order, Proc. Amer. Math. Soc. 106 (3) (1989) 625–629. [8] A. Doostabadi, A. Erfanian, M. Farrokhi D.G., On power graphs of finite groups with forbidden induced subgraphs, Indag. Math. (NS) 25 (3) (2014) 525–533. [9] A. Doostabadi, A. Erfanian, A. Jafarzadeh, Some results on the power graph of groups, in: The Extended Abstracts of the 44th Annual Iranian Mathematics Conference 27–30 August 2013, Ferdowsi University of Mashhad, Iran. http://profdoc.um.ac.ir/articles/a/1036567.pdf. [10] M. Feng, X. Ma, K. Wang, The full automorphism group of the power (di)graph of a finite group, 2014. Preprint arXiv:1406.2788v1 [math.GR]. [11] M. Feng, X. Ma, K. Wang, The structure and metric dimension of the power graph of a finite group, European J. Combin. 43 (2015) 82–97. [12] S. Foldes, P.L. Hammer, Split graphs, in: Proceedings of the 8th South-Eastern Conference on Combinatorics, Graph Theory and Computing, 1977, pp. 311–315. [13] C. Godsil, G. Royle, Algebraic Graph Theory, Springer, New York, 2001. [14] A.V. Kelarev, S.J. Quinn, A combinatorial property and power graphs of groups, in: D. Dorninger, G. Eigenthaler, M. Goldstern, H.K. Kaiser, W. More, W.B. Mueller (Eds.), Contrib. General Algebra 12, Springer-Verlag, 2000, pp. 229–235. 58. Arbeitstagung Allgemeine Algebra (Vienna University of Technology, June 3–6, 1999). [15] A.V. Kelarev, S.J. Quinn, Directed graphs and combinatorial properties of semigroups, J. Algebra 251 (1) (2002) 16–26. [16] A.V. Kelarev, S.J. Quinn, A combinatorial property and power graphs of semigroups, Comment. Math. Univ. Carolin. 45 (1) (2004) 1–7. [17] A.V. Kelarev, S.J. Quinn, R. Smolikova, Power graphs and semigroups of matrices, Bull. Aust. Math. Soc. 63 (2001) 341–344. [18] X. Ma, M. Feng, K. Wang, The rainbow connection number of the power graph of a finite group, 2014. Preprint arXiv:1412.5849v1 [math.GR]. [19] R. Merris, Split graphs, European J. Combin. 24 (4) (2003) 413–430. [20] M. Mirzargar, A.R. Ashrafi, M.J. Nadjafi-Arani, On the power graph of a finite group, Filomat 26 (6) (2012) 1201–1208. [21] A.R. Moghaddamfar, S. Rahbariyan, W.J. Shi, Certain properties of the power graph associated with a finite group, J. Algebra Appl. 13 (7) (2014) 1450040. 18 pp. [22] I. Niven, Fermat theorem for matrices, Duke Math. J. 15 (3) (1948) 823–826. [23] J.J. Rotman, An Introduction to the Theory of Groups, Springer-Verlag, New York, 1995. [24] W.J. Shi, W.Z. Yang, A new characterization of A5 and the finite groups in which every non-identity element has prime order, J. Southwest-China Teach. College 9 (1) (1984) 36–40. (in Chinese).
ScienceDirect Indagationes Mathematicae 26 (2015) 626–633 www.elsevier.com/locate/indag
On the chromatic number of the power graph of a finite group Xuanlong Ma ∗ , Min Feng Sch. Math. Sci. & Lab. Math. Com. Sys., Beijing Normal University, Beijing, 100875, China Received 19 August 2014; received in revised form 7 April 2015; accepted 15 April 2015 Communicated by R. Tijdeman
Abstract The power graph ΓG of a finite group G is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. We investigate the chromatic number χ(ΓG ) of ΓG . A characterization of χ (ΓG ) is presented, and a conjecture in Mirzargar et al. (2012) is disproved. Moreover, we classify all finite groups whose power graphs are uniquely colorable, split or unicyclic. c 2015 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. ⃝
Keywords: Finite group; Power graph; Graph coloring; Split graph; Unicyclic graph
1. Introduction All groups considered in this paper are finite. Kelarev and Quinn [14] introduced the directed power graph of a group G, which is the digraph with vertex set G and there is an arc from x to y if and only if x ̸= y and y = x m for some positive integer m. The directed power graphs were also considered in [15,17,16]. Motivated by this, Chakrabarty, Ghosh and Sen [4] introduced the (undirected) power graphs. The power graph ΓG of a group G is the graph whose vertex set is G with two distinct vertices adjacent if one is a power of the other. Recently, many interesting results on the power graphs have been obtained, see [6,2,3,5,8,11,10,20,21,18]. In [1], Abawajy, Kelarev and Chowdhury gave a detailed list of results and open questions. ∗ Corresponding author.
E-mail addresses: [email protected] (X. Ma), fgmn [email protected] (M. Feng). http://dx.doi.org/10.1016/j.indag.2015.04.003 c 2015 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. 0019-3577/⃝
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For a graph Γ , by V (Γ ) and E(Γ ) we denote its vertex set and its edge set, respectively. The chromatic number of Γ , denoted by χ (Γ ), is the smallest number of colors needed to color the vertices of Γ so that no two adjacent vertices share the same color. Denote by Zn the cyclic group of order n. Mirzargar, Ashrafi and Nadjafi-Arani [20] computed the chromatic number of Zn and proposed the following conjecture. Conjecture 1.1 ([20, Conjecture 1]). Let G be a group. Then χ (ΓG ) = χ (ΓZn ), where n is the maximum order of an element in G. Motivated by the conjecture, we shall explore the coloring of the power graph of a non-cyclic group. A partition of the vertex set of a graph Γ is called a coloring if each set of the partition is an independent set of Γ . If there is a unique partition of V (Γ ) into χ (Γ ) independent sets, then Γ is said to be uniquely colorable. For more information on the uniquely colorable graph, see [13, Chapter 6, p. 113]. The clique number ω(Γ ) of a graph Γ is the maximum size of a clique in a graph. If χ (∆) = ω(∆) for each induced subgraph ∆ of Γ , then Γ is called a perfect graph. It was noted in [9, Theorem 1] and [11, Corollary 2.5] that all power graphs are perfect. A graph Γ is said to be split if V (Γ ) is the disjoint union of a clique and an independent set. Split graphs form a very useful class of perfect graphs. More information on the split graphs, can be found in [12,19]. A graph is called unicyclic if it is connected and has a unique cycle. In this paper, we disprove Conjecture 1.1 and characterize the chromatic number of the power graph of a finite group. Furthermore, we classify all finite groups whose power graphs are uniquely colorable, split or unicyclic. 2. Coloring In the following proposition the chromatic number of the power graph of a cyclic group is determined. Proposition 2.1 ([20, Theorem 2]). Let n = p1α1 p2α2 · · · prαr , where p1 < p2 < · · · < pr are prime numbers. Then χ (ΓZn ) = prαr +
j r −2 α j−1 αr −i ( pr −r −j−1 − 1) ) , φ( pr −i j=0
i=0
where φ is Euler’s totient function. Denote by |x| the order of element x in the group. Now we disprove Conjecture 1.1 by the following example. Example 2.2. Given a group G and an element g ∈ G of order n, Γ⟨g⟩ is an induced subgraph of ΓG which is isomorphic to ΓZn . It follows that χ (ΓG ) ≥ χ (Γ⟨g⟩ ) = χ (ΓZn ). Since power graphs are perfect, Conjecture 1.1 is equivalent to the statement that Γ⟨g⟩ contains a maximal clique of ΓG whenever g has maximum order. In fact, this is not always the case. Our counterexample to Conjecture 1.1 is the general linear group GL(2, p) of invertible 2 × 2 matrices over GF( p) for certain prime numbers p. By [22, Theorem 2], the maximum order
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of any element of GL(2, p) is p 2 − 1. This result also implies that there exists an element of GL(2, p) of order p( p − 1). Some such elements can be constructed as follows. Let α be a primitive element of GF( p 2 ). So |α| = p 2 − 1 in the multiplicative group GF∗ ( p 2 ). Then there are b, c ∈ GF( p) such that α 2 + bα + c = 0. Let 0 1 A= . −c −b Then we have 1 1 A =α . α α It follows that the order of A is p 2 − 1. Let β be a primitive element of GF( p). Hence |β| = p −1 in the multiplicative group GF∗ ( p). Let 0 1 B = β I2 + . 0 0 Then B = β I2 + nβ n
n
n−1
0 0
1 . 0
This implies that the order of B is p( p − 1). In general there is no way to determine the primitive element of GF( p) or GF( p 2 ), but we know that there are respectively φ( p − 1) and φ( p 2 − 1) many of each. Using Proposition 2.1, we compute χ (Γ⟨A⟩ ) and χ (Γ⟨B⟩ ). Take p = 11, so p 2 − 1 = 120 and p( p − 1) = 110. Now χ (Γ⟨A⟩ ) = 69 and χ (Γ⟨B⟩ ) = 91. We have verified that this construction also gives counterexamples to Conjecture 1.1 for all prime numbers 11 ≤ p < 100. The general problem is a subtle problem of number theory. Next, we study the chromatic number of the power graph of a non-cyclic group. Lemma 2.3 ([20, Lemma 1]). A subset of a group forms a clique in the power graph of the group if and only if the elements of the subset can be ordered so that each element is a power of the next. We say that a proper cyclic subgroup M of group G is maximal cyclic if M ≤ K ≤ G implies M = K or K = G, where K is cyclic. Denote by MG the set of maximal cyclic subgroups of G. Theorem 2.4. For any non-cyclic group G, χ (ΓG ) = max{χ (Γ M ) : M ∈ MG }. Proof. Note that ΓG is perfect. It suffices to prove that ω(ΓG ) = max{ω(Γ M ) : M ∈ MG }. Note that for any subgroup K of G, the subgraph Γ K of ΓG is an induced subgraph. Therefore max{ω(Γ M ) : M ∈ MG } ≤ ω(ΓG ). On the other hand, for any clique C of ΓG , there exists an element x in C such that C ⊆ ⟨x⟩ and ⟨x⟩ ∈ MG by Lemma 2.3. This implies that ω(ΓG ) ≤ max{ω(Γ M ) : M ∈ MG }. The proof is now complete. By Theorem 2.4, the following corollary is immediate. Corollary 2.5. Let G be a group with max{|x| : x ∈ G} = n. If n is a prime power, then χ(ΓG ) = n.
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Remark 2.6. By Proposition 2.1 and Theorem 2.4, we can see that for any group G, if the order of every maximal cyclic subgroup of G is known, then we may calculate χ (ΓG ). A group G is said to be an EPO-group if all non-trivial element orders of G are prime (see [7,24]). Denote by π(n) the set of the prime divisors of a positive integer n (and not the number of prime numbers ≤ n). Given a group G, we shall write π(G) instead of π(|G|). It is straightforward that χ (ΓG ) = 2 if and only if G is an elementary abelian 2-group. We now classify all groups whose power graphs have chromatic number 3. Proposition 2.7. Let G be a group. Then χ (ΓG ) = 3 if and only if G is one of the following groups: • The symmetric group S3 on three letters. • A 3-group of exponent 3. • An EPO-group of order 2m · 3 or 2 · 3m for some positive integer m ≥ 2. Proof. First assume that χ (ΓG ) = 3. Then ω(ΓG ) = 3. Hence, π(G) ⊆ {2, 3}. If G is a 2group then G has exponent 2 and so G is an elementary abelian 2-group, namely ω(ΓG ) = 2, a contradiction. If G is a 3-group, then every non-identity element of G is of order 3, that is, G is a 3-group of exponent 3, as desired. Thus, now we may assume that π(G) = {2, 3}. In this case the order of any non-identity element of G is 2 or 3. It follows that G is an EPO-group. Now the desired result follows from [7]. For the converse, the result follows trivially from Corollary 2.5. Now we classify all groups whose power graphs are uniquely colorable. Theorem 2.8. For a group G, ΓG is uniquely colorable if and only if G is an elementary abelian 2-group or a cyclic group of prime power order. Proof. Pick a cyclic subgroup of G and consider its set U of generators. Observe that the elements of U are mutually adjacent and have exactly the same set of neighbors in ΓG since each is a power of the others. In particular, in any partition of ΓG into independent sets, we may freely permute the elements of U in the cells and once again obtain a partition of ΓG into independent sets. Note that any cyclic subgroup of order at least three has at least two generators. Assume ΓG is uniquely colorable. Then the above implies that every element of order at least three is a maximal independent set in any partition of ΓG into as few independent sets a possible. Since we cannot enlarge this singleton independent set by moving a vertex from another cell of the partition, we conclude that every element of order at least three is adjacent to every vertex of the power graph of G. If G is an elementary abelian 2-group, we may partition the set of vertices of ΓG into two independent sets: one consisting of the identity, and one consisting of every other vertex. This is its unique partition into χ (ΓG ) = 2 independent sets. Suppose G is not an elementary abelian 2-group. If G does not have prime power order, then there are elements of distinct prime orders, one of which is at least three. By construction, they are non-adjacent in the power graph (neither is a power of the other). This contradicts the above, so G must have prime power order. Now let C be a maximal proper cyclic subgroup with generator x and take an element g not in C. Clearly, g is not a power of x. Thus since g and x are adjacent, it must be the case that g is a power of x, so C is contained in the subgroup generated by x. By maximality of C, it must
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Fig. 1. C4 .
be the case that G is cyclic with generator x. Thus G is either an elementary abelian 2-group or a cyclic group of prime power order. For the converse, observe that ΓG is complete bipartite or complete. The desired result follows trivially. 3. Split In this section we shall characterize the groups whose power graphs are split. It was shown in [12], that a graph is split if and only if it does not have an induced subgraph isomorphic to one of the three forbidden graphs, C4 , C5 or 2K 2 . Note that the power graph of a group is perfect. Then ΓG has no induced subgraph isomorphic to C5 . Theorem 3.1. For any group G, ΓG is split if and only if ΓG has no induced subgraph isomorphic to the forbidden graph 2K 2 . Proof. Necessity is clear so we just need to prove sufficiency. Suppose that ΓG has no induced subgraph isomorphic to 2K 2 . In order to prove that ΓG is split, it suffices to show that ΓG has no induced subgraph isomorphic to C4 . Assume, to the contrary, that ΓG has an induced subgraph isomorphic to C4 (see Fig. 1). Without loss of generality, let ⟨x⟩ ⊆ ⟨y⟩. It follows that ⟨v⟩ ⊆ ⟨y⟩, ⟨v⟩ ⊆ ⟨u⟩ and ⟨x⟩ ⊆ ⟨u⟩. Hence |y| > 2 and |u| > 2. Note that y and u are not adjacent in ΓG . So the subgraph of ΓG induced by the set {y, y −1 , u, u −1 } is isomorphic to 2K 2 , a contradiction. The following result is obvious by Theorem 3.1. Corollary 3.2. Let G be a group and let ΓG be split. If there exists an element x in G such that |x| > 2, then G has precisely one cyclic subgroup of order |x|. It follows from Theorem 3.1 that if ΓG is split, then |G| is a divisor of 2n p m for some odd prime p and some nonnegative integers n and m. In the following we shall discuss the groups whose powers are split. Lemma 3.3. Let G be a non-abelian 2-group, and M be a maximal cyclic subgroup of G. Then ΓG is split if and only if G satisfies the following conditions: Every element y ∈ G\M is an involution and ⟨y⟩ is not normal in G M is normal in G and G/M is elementary abelian (1) Suppose M = ⟨x⟩. Then for all y ∈ G\M, yx y = x −1 Proof. We first assume that ΓG is split. Let M = ⟨x⟩. Since G is non-abelian, one has |x| > 2. By Corollary 3.2, M is the unique cyclic subgroup of order |x|. Hence M is normal in G. Take y ∈ G\M. Then x and y are non-adjacent in ΓG . If |y| > 2, then the subgraph induced by
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{y, y −1 , x, x −1 } is isomorphic to 2K 2 , contrary to Theorem 3.1. It follows that any element in G\M is an involution. Now we verify that G/M is elementary abelian. Since y ̸∈ M, one has yx ̸∈ M, and so |yx| = 2. It means that yx y = x −1 . Suppose that ⟨y⟩ is normal in G. Then G contains an abelian subgroup isomorphic to Z2 ×Z2m where 2m = |M|. Note that there exist two elements (0, 1), (1, 1) ∈ Z2 × Z2m such that |(0, 1)| = |(1, 1)| = 2m and ⟨(0, 1)⟩ ̸= ⟨(1, 1)⟩. This contradicts Corollary 3.2. Thus, ⟨y⟩ is not normal in G. Now the proof of the necessity is complete. For the converse, observe that M is a clique and G\M is an independent set. It means that ΓG is split. Example 3.4. Denote by D2·2m−1 the dihedral group of order 2m , where m ≥ 3. Then D2·2m−1 satisfies the conditions in (1). Lemma 3.5. Let G be an abelian 2-group. Then ΓG is split if and only if G is cyclic or elementary abelian. Proof. Suppose that G is a non-cyclic group such that ΓG is split. Let M be a maximal cyclic subgroup of G. Since G is non-cyclic, G has at least two distinct involutions. This implies that there exists an involution in G that does not belong to M. If |M| = n > 2, then G has a subgroup isomorphic to Z2 × Zn , and so ⟨(0, 1)⟩ ̸= ⟨(1, 1)⟩ and |(0, 1)| = |(1, 1)| = n, contrary to Corollary 3.2. Thus, we have |M| = 2 and so G is elementary abelian. The converse is straightforward. Lemma 3.6. Let G be a p-group, where p is an odd prime number. Then ΓG is split if and only if G is cyclic. Proof. Suppose that ΓG is split. By Corollary 3.2, G has just one subgroup of order p, it follows that G is cyclic. The converse is clear. n Lemma 3.7. Suppose that G ∼ = Zm 2 × Z p , where p is an odd prime number and m and n are two positive integers. Then ΓG is split if and only if G ∼ = Z2 p .
Proof. If G ∼ = Z2 p , then by Proposition 2.1 ω(ΓG ) = 2 p − 1 and hence ΓG is split. Conversely, we first suppose that n ≥ 2. Note that G has a subgroup of order 2 p n . It follows that there exist two elements x, y ∈ G such that |x| = 2 p and |y| = p 2 . This implies that the subgraph of ΓG induced by {x, x −1 , y, y −1 } is isomorphic to 2K 2 . In view of Theorem 3.1, one has a contradiction. Thus n = 1. Suppose that G has a subgroup isomorphic to Z22 × Z p ∼ = Z2 × Z2 p . Then G has two distinct elements u, v such that |u| = |v| = 2 p and ⟨u⟩ ̸= ⟨v⟩, contradicting Corollary 3.2. Lemma 3.8. Let G be a non-abelian group of order 2m p n , and let M be a Sylow p-subgroup of G, where p is an odd prime number and m, n ≥ 1. Then ΓG is split if and only if G satisfies the following conditions: G∼ = Zm 2 n Z pn Every element y ∈ G\M is an involution and ⟨y⟩ is not normal in G (2) Suppose M = ⟨x⟩. Then for all y ∈ G\M, yx y = x −1 Proof. It is similar to the proof of Lemma 3.3.
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Example 3.9. Let p be an odd prime number and m be a positive integer. Then D2· pm satisfies the conditions in (2). We now state our main theorem of this section as follows. Theorem 3.10. Let G be a group. Then ΓG is split if and only if G is isomorphic to one of the groups: • Z pm for some prime p and positive number m. • Z2 p for some odd prime number p. • Zm 2 for some positive number m. • A non-abelian 2-group satisfying the conditions in (1). • A non-abelian group of order 2m p n satisfying the conditions in (2). Proof. Note that if there exist two distinct odd prime numbers p, q ∈ π(G), then G has elements x and y such that |x| = p and |y| = q, and so the subgraph induced by {x, x −1 , y, y −1 } is 2K 2 . Consequently, if ΓG is split, then by Theorem 3.1 one has π(G) ⊆ {2, p} for some odd prime number p. Now our result follows from Lemmas 3.3, 3.5, 3.6, 3.7 and 3.8. 4. Unicyclic power graphs The proof of the following proposition is straightforward. Proposition 4.1. For group G, ΓG has no cycles if and only if G is an elementary abelian 2group. Note that in this case the power graph is a star graph. A unicyclic graph is either a cycle or a cycle with trees attached. In this section we shall classify all groups whose power graphs are unicyclic. We first quote the “N /C” lemma in group theory. Lemma 4.2 ([23, Theorem 7.1(1)]). Let G be a group and H a subgroup of G. Then the quotient group N G (H )/C G (H ) is isomorphic to a subgroup of the automorphism group Aut(H ) of H . Lemma 4.3. There exist no EPO-groups G of order 2n · 3 for some positive integer n ≥ 2 such that ΓG is unicyclic. Proof. Suppose that G is an EPO-group of order 2n · 3 such that its power graph is unicyclic, where n ≥ 2. Let P and Q be a Sylow 3-subgroup and a Sylow 2-subgroup of G, respectively. Clearly, P ∼ = Z3 , and Q is an elementary abelian 2-group of order 2n . Set P = ⟨x⟩. Since ΓG is unicyclic, G has precisely two distinct elements of order greater than 2. It follows that P is normal in G. By Lemma 4.2, G/C G (P) can be imbedded in Aut(P). Suppose there exists an element y in G\P such that x y = yx. Then |y| = 2. Since |x| and |y| are coprime, one has |x y| = 6, which is impossible. It means that C G (P) = P. It is well known that Aut(P) ∼ = Z2 . Hence, G/C G (P) ∼ = 1 or Z2 , in the former case one has G = P, in contradiction to the order of G. In the latter case one has |G| = 6, also a contradiction. In [5, Theorem 11], it is shown that for an abelian group G, ΓG is unicyclic if and only if G is the cyclic group of order 3. Now we determine all groups whose power graphs are unicyclic. Theorem 4.4. Let G be a group. Then ΓG is unicyclic if and only if G is isomorphic to Z3 or S3 . Proof. We first assume that ΓG is unicyclic. If G has an element y of order greater than or equal to 4, then the subgraph of ΓG induced by ⟨y⟩ has at least two distinct cycles, which is impossible.
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Thus, χ (ΓG ) = 2 or 3 by Corollary 2.5. By Propositions 4.1 and 2.7 we have that G ∼ = Z3 , S3 or an EPO-group of order 2n · 3 for some positive integer n ≥ 2. Now Lemma 4.3 completes the proof of the necessity. The sufficiency is straightforward. Acknowledgments The authors are indebted to Prof. R. Tijdeman and the anonymous reviewers for some corrections and suggestions, the construction of Example 2.2, and the proof of Theorem 2.8. This research is supported by National Natural Science Foundation of China (11271047, 11371204), the Scientific Research Foundation of Yunnan Educational Committee (2014Y500). References [1] J. Abawajy, A. Kelarev, M. Chowdhury, Power graphs: A survey, Electron. J. Graph Theory Appl. 1 (2) (2013) 125–147. [2] P.J. Cameron, The power graph of a finite group, II, J. Group Theory 13 (6) (2010) 779–783. [3] P.J. Cameron, S. Ghosh, The power graph of a finite group, Discrete Math. 311 (13) (2011) 1220–1222. [4] I. Chakrabarty, S. Ghosh, M.K. Sen, Undirected power graphs of semigroups, Semigroup Forum 78 (3) (2009) 410–426. [5] T.T. Chelvam, M. Sattanathan, Power graph of finite abelian groups, Algebra Discrete Math. 16 (1) (2013) 33–41. [6] B. Curtin, G.R. Pourgholi, Edge-maximality of power graphs of finite cyclic groups, J. Algebraic Combin. 40 (2) (2014) 313–330. [7] M. Deaconescu, Classification of finite groups with all elements of prime order, Proc. Amer. Math. Soc. 106 (3) (1989) 625–629. [8] A. Doostabadi, A. Erfanian, M. Farrokhi D.G., On power graphs of finite groups with forbidden induced subgraphs, Indag. Math. (NS) 25 (3) (2014) 525–533. [9] A. Doostabadi, A. Erfanian, A. Jafarzadeh, Some results on the power graph of groups, in: The Extended Abstracts of the 44th Annual Iranian Mathematics Conference 27–30 August 2013, Ferdowsi University of Mashhad, Iran. http://profdoc.um.ac.ir/articles/a/1036567.pdf. [10] M. Feng, X. Ma, K. Wang, The full automorphism group of the power (di)graph of a finite group, 2014. Preprint arXiv:1406.2788v1 [math.GR]. [11] M. Feng, X. Ma, K. Wang, The structure and metric dimension of the power graph of a finite group, European J. Combin. 43 (2015) 82–97. [12] S. Foldes, P.L. Hammer, Split graphs, in: Proceedings of the 8th South-Eastern Conference on Combinatorics, Graph Theory and Computing, 1977, pp. 311–315. [13] C. Godsil, G. Royle, Algebraic Graph Theory, Springer, New York, 2001. [14] A.V. Kelarev, S.J. Quinn, A combinatorial property and power graphs of groups, in: D. Dorninger, G. Eigenthaler, M. Goldstern, H.K. Kaiser, W. More, W.B. Mueller (Eds.), Contrib. General Algebra 12, Springer-Verlag, 2000, pp. 229–235. 58. Arbeitstagung Allgemeine Algebra (Vienna University of Technology, June 3–6, 1999). [15] A.V. Kelarev, S.J. Quinn, Directed graphs and combinatorial properties of semigroups, J. Algebra 251 (1) (2002) 16–26. [16] A.V. Kelarev, S.J. Quinn, A combinatorial property and power graphs of semigroups, Comment. Math. Univ. Carolin. 45 (1) (2004) 1–7. [17] A.V. Kelarev, S.J. Quinn, R. Smolikova, Power graphs and semigroups of matrices, Bull. Aust. Math. Soc. 63 (2001) 341–344. [18] X. Ma, M. Feng, K. Wang, The rainbow connection number of the power graph of a finite group, 2014. Preprint arXiv:1412.5849v1 [math.GR]. [19] R. Merris, Split graphs, European J. Combin. 24 (4) (2003) 413–430. [20] M. Mirzargar, A.R. Ashrafi, M.J. Nadjafi-Arani, On the power graph of a finite group, Filomat 26 (6) (2012) 1201–1208. [21] A.R. Moghaddamfar, S. Rahbariyan, W.J. Shi, Certain properties of the power graph associated with a finite group, J. Algebra Appl. 13 (7) (2014) 1450040. 18 pp. [22] I. Niven, Fermat theorem for matrices, Duke Math. J. 15 (3) (1948) 823–826. [23] J.J. Rotman, An Introduction to the Theory of Groups, Springer-Verlag, New York, 1995. [24] W.J. Shi, W.Z. Yang, A new characterization of A5 and the finite groups in which every non-identity element has prime order, J. Southwest-China Teach. College 9 (1) (1984) 36–40. (in Chinese).