Characterizing Minimum Gossip Graphs on 16 vertices - CiteSeerX

Characterizing Minimum Gossip Graphs on 16 vertices  Roger Labahn y Universitat Rostock, FB Mathematik 18051 Rostock, Germany Christian Pietsch z Forschungsinstitut fur Diskrete Mathematik Nassestr.2, 53113 Bonn, Germany October 12, 1993

submitted to Networks Abstract

We introduce a general method for constructing minimum gossip graphs on 2k vertices, and show that this construction is sucient to get all minimum gossip graphs on 16 vertices. A complete list of all bipartite or all of diameter 4 among them is given.

 Research supported by SFB 303 (DFG), Institute of Discrete Mathematics / Operations Research, University of Bonn. y Research supported by a special grant of the Alexander-von-Humboldt foundation. z Current address: Universit at Greifswald, FB Mathematik, Fr.-L.-Jahn-Str. 15a, 2200 Greifswald, Germany.

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1. Introduction

Let each vertex of a connected simple graph G = (V; E) with V = f1; . . . ; ng know an item of information unknown to any other vertex. An exchange of information is arranged in rounds such that at its end, every vertex knows all these items. Each round consists of some time-parallel calls on independent edges of G, that is, a call is made between the two vertices belonging to an edge, but every vertex can take part in at most one call per round. During such a call, both participants exchange all information they already know. Such a communication process is called "gossiping", and we refer to [1] for a survey of known results and more references. T(G) denotes the minimum number of rounds a complete information ow on G must have. Moreover, let T := minT(G) where the minimumis taken over all simple connected graphs on n vertices. If for G, T(G) = T , then introducing new edges in G will not destroy this property. For that reason we are always interested only in graphs G with a minimal number of edges. Here moreover, let us de ne E := min jE j where the minimum is taken over all simple connected n-vertex graphs G with T(G) = T , and call a graph G minimum gossip graph (mgg) if T(G) = T and jE j = E . Clearly, the edges of G used in a xed round i form a matching m . Any sequence M = (m1 ; m2; . . .) corresponding to a minimum time gossip scheme on G is called a matching sequence for G. Throughout this paper we consider mgg's G with 2 vertices (2 -mgg's). In [5] some results are presented for graphs having other numbers of vertices. For the case n = 2 , we know from [2] that T = k. Moreover, it is well-known and easy to see that in order to achieve this bound, in every round for every vertex, the number of items it knows must double. Consequently, any matching sequence consists of k pairwise disjoint perfect matchings, i.e. E = k
2 ?1 and every 2 -mgg is k-regular and k-edge-colorable. In Section 2, we list some more properties of 2 -mgg's and introduce a construction to get new from known 2 -mgg's. The next sections contain the proof that this construction suces to get all 2 -mgg's for k  4. For k = 4, we cannot give a complete list of all 16-mgg's but in Section 6, lists of all bipartite 16-mgg's and of all 16-mgg's with diameter 4 are presented. Finally, attempts to nd a nice graph characterization of 16-mgg are mentioned in Section 7. In the following we do not distinguish between a vertex and that item of information it knew before gossiping started. So, if after a certain round x 2 V knows what originally was known to y 2 V only then we shortly say "x knows y". For i = 0; . . . ; k and any vertex x 2 V , let  (x) denote the set of vertices x knows after round i whereby 0(x) := fxg. We know fxg = 0(x)  1(x)  . . .   (x) = V , and j (x)j = 2 . Throughout the following, let G = (V; E) be any 2 -mgg and M = (m1 ; . . . ; m ) be any matching sequence for G. For any subset of matchings, I = fm 1 ; . . . ; m )  M, let G(I) = G(i1 ; . . . ; i ) denote the corresponding n

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subgraph V; =1 m of G. For two vertices x; y 2 V , a path from x to y consisting of edges e1 2 m 1 ; e2 2 m 2 ; . . . ; e 2 m is said to be monotone i i1 < i2 < . . . < i . In our mgg, any pair (x; y) 2 V 2 of vertices is connected by exactly one monotone (x; y)-path. Finally we should mention that the facts listed above are known from the analysis of minimum broadcast graphs (see [1]) where broadcasting is the process of disseminating one item from a single generator to all other vertices in the graph. ij

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2. General Properties of 2k -mgg's Let G be a 2 -mgg. G is standard-constructible i there is a matching sequence M for G such that G(1; . . . ; k ? 1) is a graph with 2 components each being a 2 ?1-mgg. k

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Proposition 2.1 Let G be a 2 -mgg. If there is a matching sequence M for G such that G(1; 2) or G(k ? 1; k) is a Hamiltonian cycle then G is standard-constructible. k

Proof: Suppose, G(k ? 1; k) is Hamiltonian cycle C in G. Let any x 2 V be xed, and let y; z 2 V be its neighbors where xy 2 m ? and xz 2 m . Then  ? (y) =  ? (x) and  ? (z) = V n  ? (x) because after x and z have exchanged information in round k both must know all items. Application of this idea for all other vertices shows that after round k ? 1, every vertex knows either  ? (x) or V n ? (x). Hence, G(1; . . . ; k ?1) consists of two components on fv 2 V :  ? (v) =  ? (x)g resp. fv 2 V :  ? (v) = V n  ? (x)g, and k

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both are 2 ?1-mgg's. Clearly, the inverse sequence (m ; m ?1; . . . ; m1 ) is matching sequence for G, too. Therefore, if G(1; 2) is Hamiltonian cycle the proof is analog.  k

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Because the matching sequence M gives a k-coloring of G any two matchings of M form a 2-factor of G.

Proposition 2.2 Let G be a 2 -mgg and M one of its matching sequences. The 2-factors G(1; 2) and G(k ? 1; k) consist of disjoint cycles each of length divisible by 4. k

Proof: Assume, the vertices x1; . . . ; x4 +2 2 V form a cycle in G(k ? 1; k) and x1x2 2 m ?1 . The idea of the above proof yields  ?1(x1 ) =  ?1(x2 ) =  ?1(x5) =  ?1(x6 ) = . . . =  ?1(x4 +1 ) =  ?1(x4 +2 ). Because x1 x4 +2 2 m ,  (x1 ) =  ?1(x1 ) [  ?1(x4 +2 ) =  ?1(x1 ) 6= V in contradiction to the completeness of the considered gossiping scheme.  p

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Proposition 2.3 Let G be a 2 -mgg and M one of its matching sequences. If I is any subset of k ? 1 matchings such that one component G0 of G(I) is 2 ? -mgg, then G is k

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standard-constructible.

Proof: Let m be the matching not belonging to I and let xy 2 m be any

edge. If both x and y would belong to V (G0) information would ow from x to y along dierent monotone paths which is impossible. Hence, every edge of m connects a vertex of V (G0) with one of V n V (G0 ) because m is perfect, and there are no more edges between both these sets. Therefore, G(I) n G0 is also 2 ?1-mgg. Now, build another matching sequence M 0 by taking m as the last matching and not changing the sequence of the other matchings. Then M 0 is a matching sequence for G which shows that G is standard-constructible.  k

The following statement is easy to check.

Proposition 2.4

Any mgg does not contain neither K3 nor K2 3 as a subgraph. ;

Proof: For any assignment of the edges of those subgraphs to pairwise disjoint matchings, one nds a pair of vertices connected by two dierent monotone paths.  Now we describe a method to derive new mgg's from a known one.

Replacing Procedure: Let G be a 2 -mgg and M = (m ; . . . ; m ) one k

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of its matching sequences. Further, let I = (m ; . . . ; m + ?1 ) be a subsequence of l consecutive matchings of M such that G(I) contains a 2 -mgg G0 = (V 0; E 0) as a component with matching sequence M 0 = (m01 ; . . . ; m0 ) where m0 = m + ?1 \ E 0 (j = 1; . . . ; l). Replacing G0 and M 0 by another 2 -mgg H 0 = (V 0 ; E 00 ) with matching sequence N 0 = (n01 ; . . . ; n0 ) yields the  where E = E nE 0 [ E 00 and a matching sequence N = graph H = (V; E) (m1 ; . . . ; m ?1 ; n~1; . . . ; n~ ; m + ; . . . ; m ), where n~ = m + ?1nm0 [ n0 (j = 1; . . . ; l). i

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Lemma 2.1

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[Replacement Lemma]

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 is 2 -mgg and N is In the above Replacing Procedure, the graph H = (V; E) matching sequence for H . k

Proof: Take two vertices x; y 2 V . In G there is a monotone path P from x

to y. If P does not use any edge from G0 then information is conveyed from x to y in H, too. Suppose now that P uses edges from G0. These are consecutive on P. There are uniquely determined vertices x0; y0 2 V 0 in which the path P is entering or leaving G, respectively. Because H 0 is a 2 -mgg on V 0 there is a l

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unique monotone path P 0 from x0 to y0 inside H 0. Hence, (P n E 0 ) [ P 0 is the unique monotone path connecting x and y in H. 

3. Constructing 2k -mgg's We know from the Replacement Lemma that replacing a sub-mgg in a 2 -mgg yields another 2 -mgg. Let G and H be two 2 -mgg's. We say that H is directly R-constructible from G (G  H), if a matching sequence for H can be obtained from a suitably chosen matching sequence for G by replacing a j-submgg. Furthermore, H is R-constructible from G (G  H) if there is a sequence G1; G2; . . . ; G of 2 -mgg's such that G = G1,H = G , and for i = 1; . . . ; s ? 1, it holds G +1  ( ) G where j(i) = 2 for suitable integers l , 2  l  k ? 1. It is easy to see that " " is an equivalence relation in the set of all 2 -mgg's. A certain characterization of 2 -mgg's is to list all equivalence classes for xed k. In the following sections we show that for k = 1; 2; 3; 4, there is only one such class. Accordingly, we should introduce a standard gossiping on 2 vertices which is a well-known scheme on the k-dimensional hypercube Q (see e.g. [2]): As m1 ; . . . ; m , take the k pairwise disjoint sets each of 2 ?1 pairwise parallel edges in one xed direction in Q . This shows that indeed, Q is 2 -mgg. k

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Remark: By symmetry, any permutation of these perfect matchings yields a matching sequence for gossiping on Q . All of them are considered to be standard. k

Now, our main result is:

Theorem 3.1 Let some k=1,2,3, or 4 be xed. Then a graph G is 2 -mgg if and only if G Q . k

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Note that this is trivially true for k = 1; 2 because the only 2- or 4-mgg's are Q1 or Q2, respectively. From the Theorem it follows by induction that it is sucient to consider consecutive replacing of 4-cycles only.

Corollary 3.1

Let a graph G be 2 -mgg for some k=1,2,3, or 4. Then there is a sequence G1; G2; . . . ; G of 2 -mgg's such that G = G1,G = Q , and for i = 1; . . . ; s ? 1, it holds G +1 4 G . k

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4. Minimum Gossip Graphs on eight vertices We know that every 8-mgg has to be connected, 3-regular, and triangle-free. In [7] it was shown that only two non-isomorphic graphs satisfy both these conditions: the 3-dimensional hypercube Q3 and the twisted hypercube shown in Fig. 1 with a gossiping on it. Figure 1: The twisted 3-dimensional hypercube

Pu PPP 1 u 2 PPP 2 u 1 PPu 3

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Consequently, we have

Lemma 4.1

Every 8-mgg is isomorphic to either the 3-dimensional hypercube or the twisted 3-dimensional hypercube.

Corollary 4.1

If G is an 8-mgg then G  Q3 . R

Proof: We consider the matching sequence shown in Fig. 1. It contains a

4-submgg on the vertices a; b; c; d. This can be replaced by another 4-mgg by changing ac and bd into ad and bc in m2 . The arising graph is isomorphic to Q3. 

5. Minimum Gossip Graphs on sixteen vertices In the present section we show the following 6

Lemma 5.1

If G is a 16-mgg then G  Q4 . R

Together with the Replacement Lemma which yields the converse direction this is the nal step for proving Theorem 3.1. In order to show the Lemma we investigate the structure of certain 2-factors of 16-mgg's generated by our perfect matchings. Throughout this section let G = (V; E) be a 16-mgg and M = (m1 ; m2 ; m3; m4 ) one of its matching sequences. We already know that G(1; 2) and G(3; 4) are collections of cycles of length divisible by 4. We say that e 2 E is an (i)-edge if e 2 m . Furthermore, two (3)-edges e = xy 2 m3 and e = xy 2 m3 are said to be opposite if 3(x) = V n 3(x). In this situation it is possible to make a call between x or y and x or y in round 4 because all information would be available. For any xed (3)-edge e = xy, let xu 2 m4 and uv 2 m3 . Then uv and xy are opposite, i.e. there is at least one opposite edge for e. i

Proposition 5.1

If for the 3-edge e, there is exactly one opposite edge e, then e and e lie in one 4-cycle of G(3; 4).

Proof: In round 4, two calls have to connect pairwisely the vertices of e and 

e.

The main tool for proving Lemma 5.1 is replacement of 4-cycles. Any 4-cyclecomponent of G(i; i+1) (1  i  3) is a 4-sub-mgg and can be substituted by any other 4-mgg. All ways of doing that are described in the following proposition.

Proposition 5.2

dierent 4-mgg's: (1) a i b i+1 i+1 c i d (4)

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On 4 xed vertices a; b; c; d there are exactly 6 pairwise (2)

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Proof: Consider all non-isomorphic assignments of the labels a; b; c; d to the vertices of the standard 4-mgg (1).  To describe how 4-mgg's are replaced one by another, we call the transformation changing (1) into (2) i-switching, (3) i + 1-switching, (4) turning, (5) i-changing, and (6) i+1-changing. Obviously, these transformations are dependent, in particular, they form the Symmetric Group which could be generated by two of them. Next we de ne two more procedures basing on replacement of a 4-mgg. Suppose, C  = C4 is a component of G(i; i + 1) where 0  i  2. Then either there is exactly one cycle D  G(i + 1; i + 2) containing both (i + 1)-edges, or there are two cycles E; F  G(i + 1; i + 2) each containing one such edge. Turning C moves one situation into the other. We call this dividing D or joining E and F, respectively. The following proposition shows that we are done if the graph turns out to be standard-constructible.

Proposition 5.3

If G is a standard-constructible 2 -mgg then G  Q . k

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Proof: By de nition, G consists of two 2 ? -mgg's G ; G forming the components of G(1; . . . ; k ? 1), and the perfect matching m joining those subgraphs. k

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In the rest of the proof of Lemma 5.1 we use the notation C + C for the graph which consists of two disjoint components one a cycle of length p and the other one a cycle of length q. Similarly, pC is used for p components each being a C . Now we show how the arbitrarily chosen 16-mgg G can be changed to Q4 by replacement of smaller mgg's, in particular 4-cycles. The proof is divided into parts according to the lengths of the cycles of G(1; 2). p

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Lemma 5.2

If G(1; 2)  = 4C4 then G  Q4. R

Proof: Let A ; A ; A ; A be the four components of G(1; 2). Because G is 1

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trianglefree, for i = 1; 2; 3; 4, there are four (3)-edges leaving A . Hence, we nd a component, w.l.o.g. A2 , connected to A1 by at least two (3)-edges. Case 1: A1 and A2 are connected by four (3)-edges. Then G(1; 2; 3) consists of two 8-mgg's, i.e. G is standard-constructible. Case 2: A1 and A2 are connected by precisely three (3)-edges. i

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Let e 2 m3 be the remaining (3)-edge incident to A1 and, w.l.o.g. to A3 . For e, there is a unique opposite edge e. By Prop. 5.1 we nd a 4-cycle in G(3; 4) containing both e and e. Turning and if necessary 3-switching of the latter cycle leads to the situation of Case 1. Case 3: A1 and A2 are connected by precisely two (3)-edges. If the remaining (3)-edges connect A1 with A3 as well as A4 double application of the above method yields a standard-constructible graph. So let two (3)-edges connect A1 and A3 . The other (3)-edges connect A4 with A2 and A3 in pairs. Suitable transformations of A1 ; A2; A3; A4 always lead to the situation shown in Fig. 2 where G(1; 3)  = C16 and G(2; 3)  = 4C4. By turning all Figure 2: Illustration for the proof of Lemma 5.2

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components of G(2; 3) we get a graph with G(1; 2)  = C16 which is standardconstructible by Prop. 2.1. 

Lemma 5.3

If G(1; 2)  = 2C4 + C8 then G  Q4. R

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Proof: Let A ; A be the two 4-cycles, and B be the 8-cycle. We distinguish 1

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cases according to the number of (3)-edges between vertices of B. Possible values are 0,1,2,3, and 4. Note that such edges can only connect vertices which belong to (2)-edges diametral on B because otherwise there would be a pair of vertices connected by at least two monotone paths. Case 1: There is no (3)-edge connecting two vertices of B. We consider the four (3)-edges between A1 and B. Suppose rstly, we nd two of them adjacent to one (2)-edge e1 of B. Then, by turning or changing A1 it can be organized that these edges belong to a 4-cycle D1 in G(2; 3). Moreover, there is another (2)-edge e2 of B both adjacent (3)-edges of which lead to A2 , and A2 can be transformed such that these belong to another 4-cycle D2 in G(2; 3). Fig. 3 shows this situation whereby only one of the two possible locations of e1 and e2 on B is illustrated. Turning D1 and D2 leads to a Figure 3: Illustration for the proof of Lemma 5.3

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Hamiltonian cycle consisting of all (1)- and (2)-edges. Therefore by Prop. 2.1, G is standard-constructible. Now let the (3)-edges be such that every (2)-edge of B is connected to A1 as well as to A2. Then for every (3)-edge, there is exactly one opposite edge, and by Prop. 5.1, G(3; 4)  = 4C4. Using the inverse information ow we may apply Lemma 5.2, i.e. G is R-constructible from Q4. Case 2: Precisely one (3)-edge connects vertices of B. This edge has an uniquely determined opposite edge. By Prop. 5.1, both together belong to a 4-cycle of G(3; 4). Turning this leads to the situation of Case 1. Case 3: Precisely two (3)-edges connect vertices of B. Let e be one of these edges. Clearly, four (3)-edges leave B and the remaining two (3)-edges connect A1 and A2 . By suitable transformations of A1 and A2 , the latter edges may be arranged in one 4-cycle D of G(2; 3). Two (4)-edges 10

f1; f2 must connect e with D. If e; f1 ; f2 do not belong to one 4-cycle of G(3; 4), we can reach that by 3-switching in D. Turning the constructed 4-cycle leads to the situation of Case 2. Case 4: At least three (3)-edges connect vertices of B. We nd two of them belonging to one 4-cycle D of G(2; 3). This can be used to divide B into two 4-cycles, whereby, if necessary, a 3-switching is made in D before that. Then, G(1; 2)   = 4C4, and Lemma 5.2 can be applied.

Lemma 5.4

If G(1; 2)  = C4 + C12 then G  Q4. R

Proof: Let A and B denote the 4-cycle and the 12-cycle, respectively. We

have four (3)-edges between A and B. If two of them are adjacent to the same (2)-edge of B then after a suitable transformation of A, both belong to a 4-cycle of G(2; 3). Turning this cycle joins A and B to a C16, i.e. by Prop. 2.1 G is standard-constructible. Suppose now that the four (3)-edges from A to B end in pairwise dierent (2)-edges of B. Then, for every (3)-edge, we nd exactly one opposite edge, i.e. G(3; 4)  = 4C4 by Prop. 5.1. Hence, for the inverse matching sequence we may apply Lemma 5.2. 

Lemma 5.5

If G(1; 2)  = 2C8 then G  Q4. R

Proof: Let A and B be the 8-cycles in G(1; 2). If G(1; 2; 3) still has two

components, then these are 8-mgg and consequently G is standard-constructible. So we may assume that there is an (3)-edge e between A and B. Opposite edges to e must connect a1 or a2 with b1 or b2 (see Fig. 4). If there are two of them then they lie in a 4-cycle of G(2; 3) turning of which joins A and B to a C16, i.e. G is standard-constructible by Prop. 2.1. If otherwise there is a unique opposite edge e, then e and e belong to a 4-cycle of G(3; 4) by Prop. 5.1. Hence, for the inverse matching sequence we nd a 4-cycle in G(1; 2) what allows to apply Lemma 5.2, Lemma 5.3, or Lemma 5.4.  Finally, if G(1; 2)  = C16 then Prop. 2.1 yields G  Q4 . Altogether, Lemma 5.1 is proved. R

6. Subclasses of 16-mgg's We can present better results for 16-mgg's which are bipartite or have diameter 4. Our rst observation even holds in general. Note that a 2 -mgg may have diameter at most k. k

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Figure 4: Illustration for the proof of Lemma 5.5

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Proposition 6.1

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Let G be any 2 -mgg, and let u and ?
v be vertices of G with distance k. Then for p = 0; 1; . . . ; k, there are exactly vertices having distance p from u. k

k p

Proof: Fix any complete gossip scheme on G = (V; E), and consider the subgraph containing all edges along which u's item is transmitted. This is the well-known minimum broadcast tree on 2 vertices whichPis even ?
uniquely determined up to isomorphism (see [6]). As shown in [3] it has =0 vertices at distance  p from the root u. Now, let U (x) denotePthe set ?
of vertices at distance not exceeding i from x 2 V . Hence, jU (u)j  =0 . The analog holds for v. In the following, let p be xed arbitrarily. Since u and v have distance k, U (u) \ U ? ?1(v) = ;. Therefore, 2 = jV j  jU (u) [ U ? ?1(v)j = jU (u)j + jU ? ?1(v)j  k

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k + k = k + j =2 : i i i = +1 =0 =0 =0 ?
P Consequently, we must have equality, i.e. jU (u)j = =0 . Using this for p = 0; 1; 2; . .. successively yields the assertion.  In the remaining part of this section we consider k = 4. It is easy to check that it does not exist a connected, trianglefree, 4-regular 16-graph with diameter 2. Therefore, 16-mgg have diameter either 3 or 4, and the following theorem characterizes all of maximum diameter.



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Theorem 6.1

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Every 16-mgg of diameter 4 is isomorphic to Q4 or one of the two "twisted

Q4" shown in Fig. 5.

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Figure 5: "Twisted" Q4 of diameter 4

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Proof: We use all notation introduced in the previous proof. Moreover, let U 0(x) denote the set of vertices at distance precisely i from x 2 V . Because by Prop. 2.4, G is trianglefree, for every vertex x 2 U 0 (u), there are three edges i

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between x and V 0 := U20 (u) = U20 (v), and the analog holds for all vertices in U10 (v). Consequently, there is no edge incident to two vertices of V 0 . Moreover, for any vertex y 2 V 0, we have two of its incident edges leading into U10 (u) and both the other into U10 (v) because otherwise G would contain a K2 3 . It is easy to check that up to isomorphism there is only one possibility how to put the 12 edges connecting U10 (v) and V 0 without producing K2 3 as a subgraph. Finally, we must check all ways of assigning two vertices of U10 (u) =: fa; b; c; dg to each of the six vertices of V 0 =: f1; 2; 3; 4;5;6g. Assigning the same pair twice would again generate a K2 3, i.e. each of the six possible pairs ab; ac; ad; bc; bd; cd is used exactly once. Analogously, we nd a vertex in U10 (u), say a, which is adjacent neither to 1 and 6, nor to 2 and 5, nor to 3 and 4. But then the three neighbors of a in V 0 are such that any two of them have a common neighbor in U10 (v). Hence, w.l.o.g. we may assume that after a suitable rearrangement of U10 (v) and U10 (u) there are the edges 1a; 2a; 3a; 1b; 2c; 3d. Checking the remaining 6 bijections between f4; 5; 6g and fbc; bd; cdg gives Q4 and the two "twisted" Q4 shown in Fig. 5. These are pairwise non-isomorphic because the number of pairs of vertices having distance 4 is 8, or 4, or 2, respectively. Note that the isomorphy classes correspond to the cycle structure of the permutations in the symmetric group S3 .  In particular, the 16-mgg's of diameter 4 are bipartite. The next theorem completes the characterization of all bipartite 16-mgg's. ;

;

;

Theorem 6.2

Every bipartite 16-mgg of diameter 3 is isomorphic to one of the "twisted

Q4" shown in Fig. 6.

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Figure 6: "Twisted Q4 " with diameter 3

t t t t t t t t

t t t t t t t t t t %

t t t t t t t t %

t t t t t t

Proof: We apply similar techniques as in the proof of Theorem 6.1 and outline the steps here. For any vertex u 2 V , it follows jU 0 (u)j = jU 0 (u)j = 4 and jU 0 (u)j = 7 where for two vertices of U 0 (u), one incident edge leads to U 0 (u) 1

3

and three to U30 (u) while for the remaining 5 vertices of U20 (u), two incident edges lead to each of U10 (u) and U30 (u). Hence, w.l.o.g. G looks like drawn in Fig. 7. Note that 1b; 5b and 2c; 4c are to make sure that 1 and 5 resp. 2 and 4 2

2

1

Figure 7: Illustration for the proof of Theorem 6.2

u

u

u u u u u u u u u u u u u u u a

1

b

c

3

2

d

4

5

have distance 2. Moreover, 1a or 2a would generate a K2 3 , i.e. we may assume that 4 is one of the neighbors of a. If we have the edge a3, then necessarily ;

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we must also have c5, b3, d1, and d2 which gives one solution. If, otherwise, there is a5, then we might have d1; d2; b3; c3, or d2; d3; b3; c1, or d1; d3; b2;c3. The rst graph is another solution while in both the other cases we get a graph isomorphic to the one obtained rstly. To see that both graphs in Fig. 6 are nonisomorphic, consider all possible representations in the pattern used for Fig. 7. For each graph and xed vertex u, the corresponding sets fa; dg and fb; cg as well as the corresponding vertex 3 2 U20 (u) are uniquely determined. Now, for the left graph and any u, two edges join 3 with fb; cg while for the right graph and any u only one edge connects 3 with fb; cg and the other leads from 3 to fa; dg.  Note that in particular, all bipartite 16-mgg are standard-constructible.

7. Other Attempts to Characterize 16-mgg's When we started these investigations our intention was to nd a purely graphtheoretical characterization of all 16-mgg's. Unfortunately, all of these attempts failed. To support further work on that subject we include a list of counterexamples to earlier conjectures. Basically, these show that 16-mgg's may be "more dierent" from Q4 than we expected. 1. The standard constructibility de ned in Section 2 can be considered as a "product" of a collection of 2 ?1-mgg's with 2-mgg's (edges). We generalize this idea: A 2 -mgg G with matching sequence M is decomposable i for some xed p, 1  p  k ? 1, G is product of 2 -mgg's with 2 ? -mgg's, i.e. G(1; . . . ; p) has 2 ? components each being a 2 -mgg and G(p + 1; . . . ; k) has 2 components each a 2 ? -mgg. Note that then every component of G(1; . . . ; p) must intersect every component of G(p + 1; . . . ; k) in exactly one vertex. Clearly, by this operation plenty of 2 -mgg's can be produced using smaller mgg's, but unfortunately k

k

p

k

p

k

p

p

k

p

p

k

There is an indecomposable 16-mgg.

Proof: Let V = f0; . . . ; 15g be the vertex set of the graph G to be constructed. We give the four perfect matchings which together form G's edge-set. 



f2i; 2i + 1 (mod 16)g : i = 0; . . . ; 7  f4i; 4i + 3g; f4i + 1; 4i + 2g : i = 0; 1; 2; 3  f0; 4g; f8; 12g; f2; 6g; f10;14g; f7;15g; f3; 11g; f5;13g; f1;9g  f4; 8g; f2; 12g; f6; 14g; f10;0g; f1;7g; f11; 15g; f3;13g; f5;9g Note that G(1; 2)  = 4C4 and G(3; 4)  = 2C8. Then it can easily be checked that m1 m2 m3 m4

= = = =

this is a matching sequence for G.

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Now check that there is no perfect matching deleting of which leaves over two components of size 8. Hence, G is not standard-constructible or, equivalently, no product of 8-mgg's with 1-mgg's. G is no product of 4-mgg's with 4-mgg's (4cycles), because one can nally check that it does not contain two edge-disjoint 2-factors G1; G2 where G1  = 4C4 and every component of G1 intersects = G2  every component of G2. 

Remark: In the above construction, let G(3; 4) consist of the two 4-cycles

0,4,12,8 and 3,7,15,11, and the 8-cycle 2,6,13,9,1,5,14,10. The arising 16-mgg is standard-constructible, but no product of 4-mgg's with 4-mgg's. On the other hand, we get a 16-mggwhich is product of 4-mgg's with 4-mgg'sbut not standard-constructible when G(3; 4) consists of the four 4-cycles 0,4,8,12; 1,6,9,13; 2,11,5,14; 3,7,15,10. This shows a certain independence between both construction methods. 2. It seems to be unlikely that 2 -mgg's can be characterized by forbidden subgraphs. We found the graph drawn in Fig. 8 which shows There is a connected, 4-regular graph on 16 vertices containing neither K3 nor K2 3 as subgraphs which is not a 16-mgg. All required graph properties are easy to check in the drawing. We used a computer program to see that information exchange cannot be completed in 4 rounds. But note that this graph is a minimum broadcast graph, i.e. for every k

;

Figure 8: K3 or K2 3 are not contained, but it is no 16-mgg!

u

;

u u u u u u u u u u u u u u u 16

xed vertex v, one can convey v's item of information to every other vertex within 4 rounds. 3. 2 -mgg's are connected, trianglefree, k-regular graphs on 2 vertices. Do these properties suce to ensure that a graph has (edge) chromatic index k? This is another important property of 2 -mgg's which is more dicult to check than the properties listed above. For k = 3, the answer is yes. For k = 5, following a hint of D. Welsh we found a graph satisfying all of our conditions but with a cut-vertex. Hence, this graph cannot be 5-edge-colorable. For k = 4, we did not nd any counterexample, but the only known result in "positive" direction was reported to us by M.Laurent who proved the existence of two edge disjoint perfect matchings in any connected, trianglefree, 4-regular 16-graph. But do these graphs always have chromatic index 4? k

k

k

8. Concluding Remarks Obviously, we did not succeed to give a satisfying characterization of 16-graphs which allow gossiping in 4 rounds. The main reason for this is that we do not know enough graph properties for distinguishing between mgg's and non-mgg's, in particular, Theorem 3.1 does not provide tools for checking whether a given graph is 16-mgg or not. Moreover, the methods mentioned in our paper already produce "many" mgg's, but it is unknown which of these are isomorphic. We still believe the number of pairwise non-isomorphic 2 -mgg's grows too fast to hope for a complete list of all of such graphs. Even to generalize the type of result presented here one would probably need more insight in purely graph theoretical properties of 2 -mgg's rather than enough patience to check all possible con gurations of 2-factors exhaustively. The number of cases to be distinguished grows terribly, particularly because Prop. 2.2 is not valid for other 2-factors. Nevertheless, we conjecture that our Replacement Procedure is general enough to describe all possible minimumtime gossip schemes on 2 vertices, i.e. we believe that Theorem 3.1 holds in general. k

k

k

Acknowledgement The investigations presented here were initiated during some of the daily lunchtime discussions at the Institute of Discrete Mathematics, University of Bonn. We thank all colleagues who participated in these discussions during the Spring semester 1991 for their helpful hints and comments.

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References [1] S.M. Hedetniemi, S.T. Hedetniemi, and A.L. Liestman, A survey of gossiping and broadcasting in communication networks, Networks 18 (1988) 319-349. [2] W. Knodel, New gossips and telephones, Discrete Math. 13 (1975) 95. [3] R. Labahn, Extremal broadcasting problems, Discrete Applied Math. 23 (1989) 139-155. [4] R. Labahn, and Ch. Pietsch, Characterizing minimum gossip graphs on 16 vertices, Preprint 91707, Institute of Discrete Mathematics, University of Bonn. [5] R. Labahn, Some minimum gossip graphs, Networks 23 (1993) 333-341. [6] A. Proskurowski, Minimum broadcast trees, IEEE Trans. Comput. 30 (1981) 363-366. [7] D.B. West, A class of solutions to the gossip problem, Part I, Discrete Math. 39 (1982) 307-326.

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