The isomorphism problem for Cayley graphs

The isomorphism problem for Cayley graphs over a group H will be considered and solved completely for cyclic groups H of prime power order for every prime p including p = 2. The method we use to obtain our solution is based on Schur rings. This method can be applied as well to colored Cayley graphs, and its scope admits further generalizations. 1. How to solve the isomorphism problem? In this paper (which is a totally revised version of the technical report [MuzP99]) the isomorphism problem for Cayley graphs over a group H will be considered and solved completely for cyclic groups H of prime power order. (Cayley graphs over cyclic groups are known as circulant graphs.) We believe that the method we use for our solution can be applied to a much wider class of combinatorial objects. Let H be a nite group (in multiplicative notation) and let S H. The Cayley (di)graph over H determined by S is the directed graph with vertex set H and edge set Cay(H;S) := f(x; y) 2 H j xy 1 2 Sg : (1.1) For simplicity this binary relation Cay(H;S) is itself called a Cayley graph, and we refer to S = fx 2 H j (x; e) 2 Cay(H;S)g as the connection set of Cay(H;S). Let = (D1; :::; Dr) be a partition of H which, for convenience, we represent as a tuple of subsets so as to establish an ordering. Sometimes we refer to as an ordered partition. Then Cay(H; ) := (Cay(H;D1); :::;Cay(H;Dr)) (1.2) is called the colored Cayley graph over H determined by (see [AlsN99]). Each Cayley graph Cay(H;S) may be considered as a colored one by setting = (S;H n S). Two colored Cayley graphs Cay(H; ) and Cay(H; ) are isomorphic if there exists a permutation f 2 Sym(H) such that Cay(H; ) = Cay(H; ). (For more details see 2.2.) The isomorphism problem is the problem of determining whether or not two speci ed colored Cayley graphs are isomorphic. However, it is not clear a priori what constitutes a \good" solution to the isomorphism problem. Indeed, one always has the trivial solution of checking all permutations in Sym(H) to decide if one is an isomorphism, but this method is both esthetically displeasing and prohibitively time-consuming for large H. So one might ask for a decision algorithm of low complexity. 1991 Mathematics Subject Classi cation. Primary 05C60, 05E99; Secondary 05C25, 20B25. The rst author was partially supported by grant A/00/24054 of the DAAD. 3 4 MIKHAIL MUZYCHUK AND REINHARD P OSCHEL In order to attack the isomorphism problem it is convenient to introduce the notion of a solving set ([Muz99]). This is just a list of permutations which it is su cient to check to determine whether or not two Cayley graphs are isomorphic. More explicitly, if D is an arbitrary subclass of the class C(H) of all Cayley graphs over a group H, then a set P Sym(H) of permutations is a solving set for D in C(H) if two arbitrarily chosen colored Cayley graphs Cay(H; ) 2 D and Cay(H; ) 2 C(H) are isomorphic if and only if there is an isomorphism in P . Now it becomes clear what might constitute a solution to the isomorphism problem for a class D with respect to C: indicate an explicit solving set. A \good" solution corresponds to a small solving set and a nal (optimal) solution of the isomorphism problem is reached if a solving set of minimal size (a so-calledminimal solving set) is provided. Minimality of P ensures that all the permutations in P are really needed in order to check if some graph in D is isomorphic to some graph in C. We are now faced with a dichotomous situation: The larger the class D of graphs the larger becomes the size of a minimal solving set. Thus, in attacking the isomorphism problem one must decide whether to look for a more general solution covering many graphs with one (predictably large) solving set, or, alternatively, consider a small class of graphs (perhaps even a single graph) yielding a hopefully small solving set. In the case of cyclic groups there is a clear and natural resolution (which may work also for other classes). As a rst step we divide the class of all colored Cayley graphs into subclasses such that it is easy to decide to which subclass a given graph belongs, and graphs from di erent subclasses are not isomorphic. For this purpose, the notion of key vector k( ) of a colored Cayley graph Cay(H; ) is introduced. Two graphs will belong to the same subclass if and only if they have the same key vector; thus we may refer to a given subclass as the \class with key vector k." As a second step we solve the isomorphism problem for each such class by indicating a minimal solving set P (k) for the class with key vector k. For the cyclic group H = Zp we obtain the following theorem (cf. [MuzP99, Thm. 6.6]). Main Theorem. Let n = p be a prime power for an arbitrary prime number p and let Cay(Zn; ) and Cay(Zn; ) be two colored Cayley graphs over Zn. Then we have: (i) If k( ) 6= k( ) then Cay(Zn; ) and Cay(Zn; ) are not isomorphic. (ii) If k( ) = k( ) =: k then the following are equivalent: (ii1) Cay(Zn; ) and Cay(Zn; ) are isomorphic, (ii2) there exists a permutation f 2 P (k) such that Cay(Zn; ) f = Cay(Zn; ) ; (ii3) there exists a permutation f 2 P (k) such that f = 0 . As mentioned above, the set P (k) is a minimal solving set for the class of all colored Cayley graphs with key vector k. This of course implies that P (k) is also a solving for each individual colored Cayley graph with key vector k though it need not be minimal for that particular graph. As an example, a minimal solving set for the complete graph Cay(Zn;Zn n f0g) consists of just a single permutation, namely the identity permutation, while the minimal solving set P (k((Zn n f0g; f0g))) of the class to which Cay(Zn;Zn n f0g) belongs consists of (p 1) permutations. (Interestingly, however, there always exist colored graphs with key vector k for which P (k) is minimal). Nevertheless P (k) is of reasonable size which never exceeds '(n) = jZ nj (here Z n denotes the set of all x 2 Zn relatively prime to n), cf. 5.6. This is a remarkable fact since THE ISOMORPHISM PROBLEM FOR CAYLEY GRAPHS 5 '(n) is the size of a solving set for circulant graphs which satisfy the so-called Ad am's conjecture. This conjecture claims that two circulant graphs, i.e. Cayley graphs Cay(Zn; S) and Cay(Zn; T ) over Zn, are isomorphic if and only if there exists a multiplier m 2 Z n such that mS = T (or equivalently that fm : Zn ! Zn : x 7! mx is an isomorphism). It is well known that Ad am's conjecture holds for every prime number n (and certain additional numbers) but fails for proper prime powers. We note that we can also represent the permutations in our solving sets P (k) in terms of such multipliers which satisfy certain compatibility conditions (see (*) in the proof of 6.3). This result is not new and appeared for the rst time in [KliP80] for n = p, p an odd prime. However, here we use another approach which includes p = 2 and independently reproves the isomorphism theorem in [KliP80]. Once the isomorphism problem is solved for circulant graphs of prime power order there is some hope for a complete solution for arbitrary circulants. We formulate the following: Conjecture([MuzP99, 4.7]): Let n = p 1 1 p m m . If Pi is a solving set for C(Zp i i ) (i 2 f1; : : : ;mg) then P1 Pm is a solving set for C(Zn). (Here the action of (f1; : : : ; fm) 2 P1 Pm on Zn is given by (x1; : : : ; xm) (f1;:::;fm) := (x1 1 ; : : : ; x fm m ) via the natural isomorphism Zn = Zp 1 1 Zp m m .) In this paper we show how the isomorphism problem for Cayley graphs can be solved. Our main tool toward this end is the theory of S-rings. Isomorphisms of Cayley graphs can be described by automorphisms of S-rings (5.12). We shall give a brief desciption of the remaining sections in an attempt to clarify the basic ideas. In Section 2 we characterize isomorphisms and solving sets by group theoretical conditions (Prop. 2.6, 2.8). As a crucial step toward solving the isomorphism problem we introduce a quasi-order relation i v which we de ne in 2.10: whenever i v then each solving set for Cay(H; ) is also a solving set for Cay(H; ). Thus we have only to consider maximal elements with respect to this quasi-order. It turns out that such maximal elements can be conveniently described in terms of S-rings over H (Prop. 3.14). General properties of S-rings are given in Section 3. In particular, it is shown how they can be exploited for the purpose of solving the isomorphism problem and constructing solving sets (KP-scheme 3.17, 3.23). The S-rings which correspond to the aforementioned maximal elements are p-S-rings in the case of p-groups H (Prop. 3.19). This motivates our search for p-S-rings in Section 4. We deduce some structural properties of p-S-rings (Prop. 4.4, 4.6) and introduce their key vectors (4.5). Isomorphisms turn out to be also automorphisms (Prop. 4.11) and can be characterized by \multipliers" (Thm. 4.12). Further we introduce special S-rings, called unipotent, which are de ned by so-called Catalan vectors (4.7). We later see (Thm. 7.1) that in the case of cyclic groups H of prime power order p , these unipotent p-S-rings characterize the aforementioned maximal elements and therefore play a crucial role in the solution of the isomorphism problem. (For p > 2, unipotence of p-S-rings is guaranteed, cf. Thm. 4.10.) Unipotent p-S-rings are investigated in more detail in Section 5. Their automorphisms can be characterized uniquely by multipliers (5.7, 5.9, 5.10) which leads to a minimal solving set for the corresponding Cayley graph (5.12). To each colored Cayley graph Cay(H; ) there is a least unipotent S-ring which contains (de nition cf. 3.1) the key vector of which serves as the key vector k( ) of Cay(H; ). An algorithm for computing k( ) is provided in 5.3. 6 MIKHAIL MUZYCHUK AND REINHARD P OSCHEL In Section 6 the case p = 2 is considered. It is shown that for the isomorphism problem it is su cient to consider unipotent S-rings, and the manner in which this result is exploited is by correcting each key vector k to its associated Catalan vector k+, cf. 6.3. Combining the results of previous sections, the isomorphism problem is treated for Cayley graphs over Zp in Section 7. A solving set for a Cayley graph can be obtained as a solving set for a unipotent S-ring (7.1) which in turn can be directly constructed from the (corrected) key vector of the given Cayley graph (7.3). The Main Theorem above will be proved in this section. Condition (ii) of the Main Theorem leads directly to an isomorphism criterion for circulant graphs in terms of multipliers (7.4), already characterized in Section 5. Finally, in Section 8 we provide an example which may be consulted by the reader for a better understanding of the notions de ned and the methods used. Acknowledgements. Our thanks are due to M. Klin for valuable hints and for his continued encouragement throughout the preparation of the manuscript. Further we thank A. Woldar for careful critical reading of a previous version of the manuscript and many remarks which considerably improved the paper. We also thank the German Academic Exchange Service (DAAD Grant A/00/24054) and the Israeli Ministry of Absorption who partially supported the rst author. 2. Isomorphisms and Solving sets 2.1. Notation. In what follows let H be a group with identity e. We denote by oPart(H) the set of all ordered partitions of H and by Part(H) the set of all unordered partitions of H. Given = (D1; : : : ; Dr) 2 oPart(H), we shall often denote the corresponding unordered partition fD1; : : : ; Drg by e . Let and be two ordered/unordered partitions of H. We write v if is a re nement of , i.e., if each block in is the union of certain blocks of . The relation v is both re exive and transitive. It is antisymmetric on Part(H), but not antisymmetric on oPart(H), however, for ; 2 oPart(H) we have ( v ) ^ ( v ) () e = e (that is, i and coincide up to some permutation of components). For f 2 Sym(H) and x 2 H let x denote the image of x under f . For any D H and for any binary relation on H let D := fx j x 2 Dg and f := f(x ; y ) 2 H H j (x; y) 2 g. Now, for = (D1; :::; Dr) 2 oPart(H), let f := (D 1 ; : : : ; D f r ) and let Cay(H; ) denote the colored graph (Cay(H;D1) f ; :::;Cay(H;Dr) f ). (We caution the reader that Cay(H; ) = Cay(H; f ) does not hold in general, but see 2.4.) For unordered partitions e we put (e ) := fD 1 ; : : : ; D r g. 2.2. Definitions. Two colored Cayley graphs Cay(H; ) and Cay(H; ) ( ; 2 oPart(H)) are said to be (combinatorially) isomorphic if there exists a permutation f 2 Sym(H) such that Cay(H; ) = Cay(H; ). The permutation f is here called a (combinatorial) isomorphism. As mentioned earlier, each Cayley graph Cay(H;S) may be interpreted as the colored Cayley graph Cay(H; (S;H n S)), in which case isomorphisms correspond to the usual ones. An isomorphism f is called normalized if e = e. We mention that it is an easy excercise to see that an isomorphism between two Cayley graphs exists if and only if there exists a normalized one. THE ISOMORPHISM PROBLEM FOR CAYLEY GRAPHS 7 For two Cayley graphs Cay(H;S) and Cay(H;T ), let Iso(Cay(H;S);Cay(H;T )) denote the set of all isomorphisms from Cay(H;S) onto Cay(H;T ). The automorphism group of Cay(H;S) is then de ned as Aut(S) := Aut(Cay(H;S)) := Iso(Cay(H;S);Cay(H;S)):