Structure Graphs of Complete Cayley Graphs Italo J. Dejter ∗ University of Puerto Rico Rio Piedras, PR 00931-3355
[email protected]
Abel A. Delgado University of Puerto Rico Rio Piedras, PR 00931-3355
[email protected]
Abstract A recent approach on the structure of complete undirected Cayley graphs, via graphs whose vertices are labelled by the triangular and tetrahedral equivalence classes of their equally-multicolored triangles and tetrahedra, respectively, taken as complete edge-connected systems, is presented, as well as some new advancements, yielding an infinite family of connected labelled graphs of maximum degree ∆ = 6 with diameter asymptotically of the order of the cubic roots of the number of vertices.
1
Introduction
As mentioned for example in [1], the design of large interconnection networks with bounded degree and diameter is a subject of intense activity in the case of symmetric networks, (see for example [2, 3, 4, 10, 12, 13, 14, 16]). In the present work, we concentrate on a somewhat different question: How do the diameters of connected graphs with a fixed maximum degree compare with their vertex-set cardinalities, that is their numbers of vertices? In this direction, [9] characterized a family of graphs of maximum degree 3 with diameter asymptotically of the order of the square root of the number of vertices, inspired on the way in which totally edge-multicolored triangles, (that is, with its edges colored differently), are organized inside the complete undirected Cayley graphs of the odd cyclic groups ZZ2k+1 . Following one step further in the same direction but inspired on tetrahedra, ([7, 8]), instead than just on triangles, in Section 9 we characterize a family of graphs of maximum degree 6 with diameter asymptotically of the order of the cubic root of the number of vertices, where k runs on the positive integers. Each of these graphs presents, away from an asymptotically small subset of vertices, an intertwining family of planar subgraphs modelled out of regular honeycomb hexagonal tessellations of Schl¨afi symbol {6, 3} and semiregular hinged star-of-David tessellations of ∗
This work was partially supported by FIPI, University of Puerto Rico.
1
symbol {3, 6, 3, 6}, ([11, 15, 5]), as depicted in our figures below, (four tessellations of the first type and six of the second type at each vertex away from the mentioned small subset). These tessellate planar subgraphs are fitted inside large triangles whose angles are 30, 60 and 90 deg and whose sides act in a mirror-like, or specular, manner. We do not know of any other developments from this asymptotic point of view for the relations between vertex numbers and diameters of graphs with given maximum degree. We do not know either how difficult the general problem for maximum degree ≥ 5 may result. We would like to know whether, for each 2 ≤ m ∈ ZZ, a family of graphs with fixed maximum degree ∆ = m(m + 1)/2 whose diameters are of the order of the mth root of their vertex-set cardinalities exists. For example, does there exists a family of graphs with fixed ∆ = 10 whose diameters are of the order the fourth root of their vertex numbers? In case this family exists, does it possess also interesting geometric and symmetric properties, as is our case for m = 2, 3?
2
K3-Types and K3-Type Graphs
Let 0 < n = 2k + 1 ∈ ZZ. Let Cayn = Cay(ZZn , In ) be the Cayley graph of ZZn with generating set In = {1, 2, ..., k} ⊂ ZZn , that is the complete undirected Cayley graph of ZZn . The elements x of In , referred to as the colors of Cayn , are in one-to-one correspondence with the pairs {x, −x} ⊂ ZZn \ {0}. This insures Cayn as an edge-colored version of Kn with degree of each color at each vertex equal to 2: just drop the orientations of the edges of Cayn while preserving their labelling colors. Indeed, this makes of Cayn into one such an undirected edge-colored graph Kn . A triangle in Cayn has K3 -type {a, b, c} if its edges have colors a, b, c ∈ In . If no confusion arises we suppress commas and parentheses, as in abc for {a, b, c}. More generally, a K3 -type abc = acb = bac = bca = cab = cba of ZZn is a 3-multiset {a, b, c} of In ∪ {0} such that a + b ∈ {c, −c} ∈ In , where the sum a + b is performed mod n. (This 3-multiset can be viewed as a class of at most six 3-tuples of colors of In ∪ {0}, one of which is abc). Example. The K3 -types {a, b, c} of ZZ7 with gcd(a, b, c) = 1 are {0, 1, 1}, {1, 1, 2}, {1, 2, 3}, {1, 3, −(1 + 3) = 3} and {2, 3, −(2 + 3) = 2}. 2 A complete subgraph of Cayn is totally multicolored, (tmc), if its edges have different colors. In the example above, only 123 = {1, 2, 3} is tmc. In Cayn each tmc triangle t and each edge of t determine exactly one tmc triangle t0 6= t with the same colors of t and sharing with t. Consider IN = {m ∈ ZZ : m ≥ 0} as a color set. A K3 -type abc of ZZ, simply called a K3 -type, is a 3-multiset {a, b, c} of IN such that the sum of the two least colors equals the greatest one. Given m, m0 , n ∈ IN with m0 ∈ In , we say that m0 ≡ m MOD n if and only if: for m00 ≡ m mod n with 0 ≤ m00 < n (1) if m00 > n/2 then m0 = n − m00 ; (2) otherwise m0 = m00 . Such m0 is said to be the reduction of m MOD n. In [9], a graph Gn was defined, for 0 < n = 2k + 1 ∈ ZZ, whose vertices are the K3 -types of ZZn such that any two such vertices, say v and v 0 , are adjacent by means of an edge if 2
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Figure 1: Representing a generic K4 -type abcdef and its cases MOD 13 and only if v and v 0 share exactly two objects (or possibly one repeated twice), say a and b, in which case is determined by v, v 0 , a and b. We have taken {a, b} (or ab) as a label for , so that Gn became an edge-labelled graph. In addition, it can be assumed that Gn does not have multiple edges, so we had incorporated this assumption into the definition of Gn .
3
K4-types and More on K3-Type Graphs
In this section and in the following one we review and continue to develop the ideas of [6, 7, 8]. A K4 -type of ZZn , (ZZ), is a class of (at most 24) 6-tuples abcdef of colors of In , (IN), such that abc, cde, aef and bdf are K3 -types of ZZn , (ZZ). A 6-tuple in a K4 -type t is called a card of t. If no confusion arises, we represent a K4 -type by any of its cards. A K4 -type is tmc if its six colors are pairwise different. We represent the card abcdef in two possible ways in Figure 1: either (a) as a copy of K4 each of whose edges bears a color; or more succinctly (b) keeping only the locations of the color denominations in (a) inside a prototypical rectangular frame. The colors in Figure 1(a) split into three different pairs of opposite colors: {a, d}, {b, e}, {c, f } (opposite in the sense that each of them is held by a corresponding pair of edges in K4 with no vertices in common, the remaining edges forming a 4-cycle). Any 6-multiset of IN determines at most one K4 -type (of ZZ). This is not true for ZZn in place of ZZ: the two tmc K4 -types 123645 and 246153 of ZZ13 , represented in Figure 1(a1 -a2 ) respectively, are different but have the same underlying (multi)set. If 3 ≤ k ∈ ZZ and 0 < s ∈ ZZ then a finite collection {K 1 , K 2 , . . . , K s } of complete subgraphs Kk of order k in a graph G is said to be a complete edge-connected system of Kk ’s if for each i (1 ≤ i ≤ s) and for each edge of K i , there exists a unique j (1 ≤ j ≤ s, j 6= i) such that K i ∩ K j contains solely and its endvertices. Each tmc K3 -type abc, (tmc K4 -type abcdef ), of Zn represents exactly 2n K3 ’s, (2n K4 ’s), of Cayn . These 2n tmc K3 ’s or K4 ’s constitute a complete edge-connected system of equally-multicolored K3 ’s or K4 ’s, which for K3 ’s is just a subgraph Cayn (abc) of Cayn induced by triangles of K3 -type abc and embedded into the 2-dimensional torus. The graph Gn of Section 2 sheds light on the structure of Cayn because the traversal of each of its edges in any of its two directions represents a transformation between complete edge-connected systems of triangles. In fact, the traversal of an edge = uv of Gn , where u and v represent respectively subgraphs Cayn (abc) and Cayn (abd), can be interpreted as the removal of the c-colored edges of Cayn (abc) followed by the replacement of the d-colored edges of Cayn (abd). This allows to associate to the corresponding intersecting graph Cayn (ab) = 3
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Figure 2: A neighborhood of 123745 in G00∞,4 Cayn (abc) ∩ Cayn (abd), clearly induced by equally-multicolored 4-cycles embedded into the 2-dimensional torus. The K4 -type graphs considered from the next section on admit a generalized interpretation of this that we omit.
4
K4-Type Graphs
Given n = 2k + 1 ≥ 13, let G0n,4 be the graph whose vertices are the tmc K4 -types of the form abcdef of ZZn with gcd(a, b, c, d, e, f, n) = 1 and such that any two such vertices, say t and t0 , are adjacent by means of an edge if and only if t and t0 , looked upon as K4 -types, share exactly two K3 -types v and v 0 . In this case, v and v 0 share exactly one color a of In . We take a as a label for , so that G0n,4 becomes an edge-labelled graph. Figure 2 may be interpreted as a neighborhood N of the K4 -type 123745 in a supergraph Gn,4 of G0n,4 (defined below in Theorem 9.1), where n > 15 is odd; (notice that the two lowermost-rightmost K4 -types in Figure 2 are not tmc). Any edge with endvertices t, t0 in such an N is determined by the two K3 -types v, v 0 that t and t0 have in common, (t, t0 looked upon as K4 -types), where v, v 0 share a unique color a. Motivated by this, we define in the next paragraph a graph whose vertices are K4 -types (of ZZ) with adjacency between any two vertices if and only if they share exactly two K3 -types. Let G00∞,4 be a graph whose vertices are the K4 -types abcdef satisfying gcd(a, b, c, d, e, f ) = 1 but not of the form abcdef = abcabc unless abcdef = 011011 and such that any two such vertices are endvertices of an edge if and only if they share exactly two K3 -types (which therefore determine ). Such a graph G00∞,4 is uniquely determined whenever its edges are restricted to: (a) possess different endvertices (i.e. no loops) and (b) have multiplicity µ = 1 (by replacing µ ≥ 1 by µ = 1). We assume restrictions (a)-(b) in order to consider complete the definition of G00∞,4 . Figure 2 can be used to illustrate Theorem 4.1. An edge between two vertices t and 0 t of G00∞,4 with respective cards r and r0 determines a K3 -type s common to t and t0 and 4
equally located in r and r0 in the sense that the component colors of s are equally located in r and r0 , just as the K3 -type s = 123 is not only common but, moreover, equally located in both the central card in Figure 2 and the one horizontally located at its right, where s occupies the three uppermost-leftmost locations in r and r0 . The locations gr in the cards r0 of Theorem 4.1 obtained from the central card r at the center of Figure 2 are shown inside small circles, to ease comprehension. Theorem 4.1 Let t ∈ V (G00∞,4 ) possess a card r with color g at location gr and color g 0 at the location gr0 opposite to gr . Then t has a neighbor t0 with card r0 differing from r just on: (a) the color at gr and (b) the permutation of the colors at the two locations 6= gr0 of just one of the two K3 -types common to r and r0 that contain (the color at) gr . Proof. t0 is determined from t as follows. Let s, s0 be the two K3 -types not containing gr in r. Then s, s0 contain gr0 . We can assume s0 has its colors equally located in r and r0 . Let i, j be the colors of r at the two locations ir , jr 6= gr0 of s. Thus s = ijg 0 . The other two K3 -types in t apart from s and s0 are of the form gij 0 and gji0 , with s0 = i0 j 0 k. We want r0 to have the colors i, j interchanged with respect to r. Thus we take (ir0 , jr0 ) = (jr , ir ). Let ν(a, b) = {| a − b |} ∪ {a + b}, for each pair of integers a, b ≥ 0. There is at least a color h ∈ H = ν(i, j) ∩ ν(i0 , j 0 ) 6= ∅ that yields r0 when located at gr (which should be called hr0 in r0 ) so that r0 is formed by the K3 -types s = ijg 0 , s0 = i0 j 0 g 0 , hii0 and hjj 0 . Moreover, r0 does not depend on the card r of t selected. In fact h = h(r, gr ) depends only on r and gr . If r = 011011 and g = 0 then h equals either 0, yielding t0 = t, not a distinct neighbor of t in G00∞,4 so we discard it, or 2, yielding a neighbor t0 of t. Otherwise, since no other vertex of G00∞,4 is of the form abcabc 6= 011011, then |H| = 1, even if (r, g) = (011011, 1). Thus, if either r 6= 011011 or (r, g) = (011011, 1), then h is unique. 2 Examples. In the examples (1)-(3) that follow g assumes subsequently colors f , a and d of the K4 -type t possessing card r = abcdef , for the particular values of a, b, c, d, e, f , in each case. (1) Applying Theorem 4.1 to (r, g) = (112354, 4) yields t0 = t, where gr = fr . (2-3) Applying Theorem 4.1 to (r, g) = (011011, 0) yields, for h = 2, neighbors t0 , t00 with respective cards r0 , r00 , where gr = ar , dr respectively, so r0 = 211011 and r00 = 011211, yielding t0 = t00 . 2
5
Canonical Triangles and Connectedness
Let G∞,4 be the supergraph of G00∞,4 obtained by adding to G00∞,4 \{011011} all the loops obtained by means of Theorem 4.1 taken each with multiplicity 1. Then any edge or loop between vertices t and t0 in G∞,4 is labelled by the associated pair (s, s0 ), as in the notation of the proof of Theorem 4.1, and is alternatively labelled by the only color g 0 of s and s0 that remains at its location gr0 = gr0 0 both in r and r0 . Now let G0∞,4 be obtained from G∞,4 by restriction to the tmc K4 -types. Corollary 5.1 The graphs G0∞,4 and G0n,4 are edge-disjoint unions of triangles, at most three such triangles incident to each vertex. 5
Proof. Applying Theorem 4.1 to the colors g, g 0 of a pair of opposite edges of a vertex t of G∞,4 looked upon as a K4 -type with card r yields h(r, g) = h(r, g 0 ). This determines two respective neighboring cards r0 and r00 of r, i.e. cards representing neighbors t0 and t00 of t, respectively. The two K3 -types that r0 and r00 share and those two that r and r0 , (r and r00 ), share constitute the four K3 -types of r0 , (r00 ). Finally, each G0n,4 can be obtained from G0∞,4 via reduction MOD n. 2 The triangles in Corollary 5.1 are called canonical triangles (ct’s). When two or three K4 -types in a ct T = {t, t0 , t00 } obtained as in Theorem 4.1 coincide, e.g. t = t0 6= t00 or t = t00 6= t0 or t 6= t0 = t00 or t = t0 = t00 , we say that T is a degenerate ct. Examples. (1) There are cases in Corollary 5.1 for which t0 = t00 . For example, if t has r = abcdef with a, b > 0, c = a + b, d = a, e = b, f = |a − b| and (gr , gr0 ) ∈ {(ar , dr ), (br , er )}, then t0 = t00 . This example yields two degenerate ct’s, the vertices of each one of which are of the form t, t0 , t00 = t0 , where edge tt0 must be considered = tt00 and edge t0 t00 is a loop of G∞,4 . (2) Theorem 4.1 applied to t = 000111 yields three degenerate ct’s, each representable by just two vertices, namely t (twice) and t0 = 011011, a link tt0 and a loop at t; these three ct’s coincide, so that by the edge-multiplicity assumption, we consider them as only one ct. (3) Theorem 4.1 applied to t = 132112 yields three ct’s incident to t, one of which, obtained by making value changes in both cases of color g = 2 at opposite locations in t, has its three vertices equal to t, so this ct reduces to a looped vertex in G∞,4 . The other two ct’s incident to t are {t, 202111, 132201} and {t, 431122, 132421}. 2 Theorem 5.2 G∞,4 is connected. Proof. There exists a path in G∞,4 whose ends have cards of the form abcdef and abcydx. This path has length 2 and middle vertex card abcfxd. The first edge of this 2-path can be labelled by {abc, bdf } and the second edge by {abc, adx}. Now, let cde and cxy be K3 -types of the integers with gcd(c, d, e) = gcd(c, x, y). Then there exists a path in G∞,4 whose ends have cards of the form abcdef and abcxyz. This uses the fact that if gcd(c, d, e) = gcd(c, x, y), then there is a path in G∞,3 from cde to cxy, ([9]), where G∞,3 is the graph whose vertices are the K3 -types abc with gcd(a, b, c) = 1 and whose edges are as in Gn , Section 2. Using this, we have that if abcdef ∈ V (G∞,4 ), then there exist: (a) a path in G∞,4 from 110110 to 110aa(a + 1); (b) a path in G∞,4 from 110aa(a + 1) to aa0bbc; (c) a path in G∞,4 from aa0bbc to abcdef . Thus, every vertex of G∞,4 can be connected to 110110. 2
6
A Planar-Subgraph Generating Algorithm
Theorem 6.1 The ct’s of G∞,4 are in 1-1 correspondence with the family of multisets {a, b, c, d} = abcd of colors in IN such that: (a) ν(a, b) ∩ ν(c, d) 6= ∅ (or ν(a, c) ∩ ν(b, d) 6= ∅ or ν(a, d) ∩ ν(b, c) 6= ∅). (b) gcd(a, b, c, d) = 1, so at least one of a, b, c, d is nonzero. Proof. From Theorem 4.1 and Corollary 5.1, each ct of G∞,4 has its vertices as K4 -types sharing exactly four colors as in the statement. 2 6
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Figure 3: Unfolded coverings of subgraphs of G∞,4 Example. In Figure 2, the upper, (lower-left, lower-right), ct has its vertices sharing 1357, (1247, 2345). 2 From now on, each ct is denoted by its associated multiset, according to Theorem 6.1. Given a tmc K4 -type t = abcdef , the ct’s incident to t are obtained via deleting from t each one of the three pairs ad, be and cf , yielding bcef , acdf and abde. Consider the union C ∪ D of any two ct’s of G∞,4 having a vertex abcdef in common. To fix ideas let C = acdf and D = abde. We set such a union as a plane graph B(t, a, d) as follows. C and D are set as congruent equilateral triangles C and D in the plane with central labels a and with the other colors of C and D labelling internally the vertices of C and D, respectively, where d is the label of t in both C and D. The sides of C and D incident to t are drawn on two straight lines at external angles of 120 deg. We label each edge of C (or D) with the alternative label of the first paragraph in Section 5: the labelling color of the corresponding edge of C (or D). The resulting label of each edge of C forms: (a) a K3 -type s() together with the labels of the endvertices of in C; (b) another K3 -type s0 (), together with the central label of C and the label of the vertex opposite to in C. Notice that s() and s0 () are the pair of K3 -types set originally as a label of the image of in G∞,4 in the first paragraph of Section 5. Let C and D be respective edges of C and D at an angle of 120 deg at t, that is: not in the same straight line. Then the label d of t in both C and D forms with the labels of C and D the K3 -type s(C ) = s(D ). Instances (a) and (b) of Figure 3 can be used to illustrate Theorem 6.2 in order to grow a planar subgraph of G∞,4 starting out from B(t, a, d). Theorem 6.2 (1) Given a ct C = afgh, let a be the central label of C and f be the color labelling vertex u of C inside C. Then there exists a color i such that (a) ν(a, h) ∩ ν(f, g) = 7
{i}; (b) the edge = uu0 in C with label g of u0 in C has label `() = i. (2) Let L be the line passing through u parallel to the edge of C\u. Then each pair (u, C) determines at most one other ct D 6= C sharing u with C such that D = ρL (C), where ρL is reflection of the plane with respect to L, and having (a) a as central label; (b) the label f of u in C as label of u in D; (c) for each one of the two edges = uu0 of C (i) `() as the label of ρL (u0 ) in D and (ii) the label of u0 in C as the color of ρL (). (3) u is the K4 -type formed by the following K3 -types determined by each one of the two edges of D incident to u: (a) a and the colors labelling and the vertex opposite to in D; (b) the colors labelling and the endvertices of in D. Proof. The statement follows from a series of adequate applications of the previous ideas via Theorem 4.1, as illustrated in Figure 3 (a- b). 2 The union of two ct’s C and D sharing exactly one vertex v is said to be a butterfly and denoted CvD. In that case, v is said to be the central vertex of CvD. Note that the labels of v in C and D equal a fixed color d, which we call the butterfly label of CvD. For example, B(t, a, d) above is a butterfly CtD with central label a and butterfly label d, say with C = acdf and D = abde. Corollary 6.3 Let t = abcdef be a tmc K4 -type. Any sequence of applications of Theorem 6.2 starting at the two ct’s in B(t, a, d) and then at each new ct obtained first from those two initial applications and then from all the posterior induction-step applications of Theorem 6.2 in all permissible direction instances constitutes an algorithm yielding a planar graph H 0 = H 0 (t, a) = H 0 (t, a, d) which is a covering of an edge-disjoint union H = H(t, a) = H(t, a, d) of butterflies in G∞,4 with central label a. A representation of H 0 by means of equilateral triangles as in Theorem 6.2 covers the whole plane. Proof. The continuation of the procedure in Theorem 6.2 via a sequence of its applications out of a butterfly B(t, a, d) and in all feasible directions constitutes an algorithm covering the plane while producing H 0 . By identifying all the possible objects in H 0 having the same labels, a quotient H of H 0 is obtained so that H 0 becomes a covering of H. 2 Examples. Figure 3(a) shows part of an H 0 as in Corollary 6.3, a star-of-David-tessellated subgraph, ([11] pg.42-43). We will see that if such an H 0 is not a subgraph of G∞,4 then it can be folded along at most two symmetry axes (sa’s) to yield H. The dotted line in Figure 3(a) is such an sa, (in particular, labels coincide by reflection). Such an H will be seen to be a subgraph of H 0 spanning a connected region of the plane limited by such sa’s. Edges crossing an sa at 90 deg will yield loops of H and each ct in H 0 will be incident to three hexagons. 2 Proposition 6.4 Given a vertex t of H 0 (t, a, d), the three ct’s incident to t according to Theorem 4.1 are: (a) the two ct’s incident to t in H 0 (t, a, d) and (b) the ct formed by the labels of the four edges of the two ct’s in item (a) which are incident to t. Proof. The proposition follows from Theorem 4.1 and Corollary 6.3. 8
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7
Canonical Hexagons
H 0 = H 0 (t, a, d) has two 6-cycles incident at each vertex. These 6-cycles can be represented as regular hexagons in the plane whenever the ct’s of H 0 are represented by equilateral triangles and their edges are represented along lines parallel to any fixed such equilateral triangle, as in Section 6. Proposition 7.1 presents this specifically. Proposition 7.1 Let bdf = s and cde = s0 be K3 -types. Then t = abcdef is contained in 6-cycles q = a.s and q 0 = a.s0 of H 0 (t, a, d). The edge-label sets of q and q 0 are respectively {b, d, f } and {c, d, e}, each of their elements labelling opposite edges. Moreover, the color that labels t in its incident ct’s in H 0 and those that label the two edges in q, (q 0 ), incident to t conform s, (s0 ). Furthermore, d is the color that labels t in its incident ct’s in H 0 and the parallel edges of a.bdf and a.cde incident neither to t nor to its corresponding opposite vertices. Proof. Let t, a and s be as in the statement. We will see that there exists a 6-cycle (t0 , t1 , t2 , t3 , t4 , t5 ) of vertices of H 0 passing through t = t0 and determined by means of the following algorithm that yields ti when ti−1 is given, for i = 1, 2, 3, 4, 5, (and returns to t0 = ti from ti−1 = t5 if i = 6 ≡ 0 with indices taken mod 6): (a) declare the card ri of the K4 -type ti (as in Figure 1(b)) to have color a fixed in the respective location ari = ar0 through the entire algorithm; (b) denote locations bri = br0 , cri = cr0 and eri = er0 , even though this algorithm could change their color values from their original values, namely b, c and e, respectively; (c) define color hi = b, (hi = f ), if i is even, (odd); (d) establish an exchange of colors, i indicated via a new denomination of locations at the i-th level: dri = hi−1 r i−1 and hr i = dr i−1 ; (e) the color eri , (cri ), if i is even, (odd), takes the unique value in ν(ari , fri ) ∩ ν(cri , dri ), (ν(ari , bri ) ∩ ν(dri , eri )). This determines a well-defined ri . Of course, we just gave one of the location instances for the determination of a 6-cycle as claimed, the other ones being essentially equivalent to this one. The rest of the statement follows. 2 Examples. A 6-cycle generated textually as in the proof of Proposition 7.1 and starting at t0 = 123745 is a.s = 1.257 = (123745, 123587, 156287, 156712, 176512, 176245). Its accompanying coplanar 6-cycle a.s0 is 1.347 = (123745, 187345, 187434, 134734, 134376, 123476). An essentially equivalent 6-cycle to these and sharing its first two vertices with a.s as just given, is 7.145 = (123745, 583741, 48c751, 1bc754, 5b6714, 426715), where lowercase hexadecimal notation is used, and its accompanying coplanar 6-cycle is 7.123 = (123745, 321785, 23178a, 13279a, 312796, 213746). 2 Given K3 -types bcd and bc0 d0 with b < c < d and b < c0 < d0 , define bcd < bc0 d0 if c + d < c0 + d0 . A graph H 0 = H 0 (t, a, d) as in Corollary 6.3 is said to be a star-of-Davidtessellated subgraph (T-subgraph), ([11] pg.42-43), and denoted a(s), where s is the smallest K3 -type 6= 000 labelling a 6-cycle of H 0 under ’ 0. There is a labelled supergraph Gn,4 of G0n,4 and a well-defined transformation Φn from G∞,4 onto Gn,4 that operates by replacing all colors of IN intervening in labels of vertices, edges, ct’s and ch’s of G∞,4 by their image colors under reduction MOD n in the sense that all vertices, respectively edges, having a common image label can be identified. Proof. Let AF be the subset of vertices of G∞,4 whose labels have component color labels ≤ k and let BF be the set of neighbors of vertices of AF in G∞,4 . Let F be the graph induced by AF ∪ BF in G∞,4 . By reducing MOD n all the colors which are constituents of labels of F , the resulting label identifications in F yield Gn,4 . Of course, the reductions MOD n happen solely for the vertices of BF . Once these vertices are reduced MOD n, they have the same labels as some vertices of AF , so they must be identified correspondingly, and the edges from AF to BF are then transformed into edges between vertices of AF which were not originally induced by AF in G∞,4 . Now, Φn is defined by replacing the labels of the objects in G∞,4 , (vertices, edges, ct’s and ch’s), by their reductions MOD n, which yields the corresponding objects in Gn,4 . 2 Corollary 9.2 The graph Gn,4 is an edge-disjoint union of ct’s, possibly degenerate, at most three incident to each vertex. Proof. Immediate from Corollary 5.1.
2
Theorem 9.3 Gn,4 is connected, for any odd positive integer n. Proof. From applying Theorems 5.2 and 9.1 to the continuous map Φn : G∞,4 → Gn,4 .
2
Φn applied to the charts of G∞,4 yields charts of Gn,4 . The collection of charts of G∞,4 , (Gn,4 ), whose ct centers are labelled i, for each i ∈ {1, . . . , n/2}, is called an i-atlas. Theorem 9.4 Let ρn : In → {atlases of Gn,4 } be such that ρn (i) = i-atlas of Gn,4 , for each i ∈ In . Then ρn (i), where gcd(n, i) = 1 < i < n/2, is obtained from ρn (1) by replacing each color a labelling a vertex, edge, ct or ch of ρn (1) by a color a0 defined as follows: let a00 = a.i; let a0 be the reduction MOD n of a00 . If n is prime, applying Φn to the i-atlases of G∞,4 yields bn/2c i-atlases of Gn,4 , which are graph isomorphic. Proof. The way presented to get ρn (i) from ρn (1) yields a labelled-graph isomorphism whenever gcd(n, i) = 1. The given modular reduction identifies oppositely signed colors mod n. 2 Chart ρ13 (1) depicted in Figure 9 of [8] is an example of the following theorem. Theorem 9.5 ρn (1) is representable inside a plane triangle T (n, 1) whose sides are sa’s of 1[011] ⊂ G∞,4 , namely two sa’s of type (2) and one of type (1). The angle between the sa’s 13
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Figure 5: Superposition of drawings for σn (1) and τn (1) of type (2) is 60 deg. The angles between each of these and the sa of type (1) are 30 and 90 deg. This 30 deg angle has its vertex at the center v of 1.000 and ρn (1) is represented in a twelfth part of the total 360 deg angle of v. The 90 deg angle has its vertex at 0jj1jj, where j = (n − 1)/2. There is only one maximal path Ln,1 of ρn (1) passing through 0jj1jj with its edges in color j and cutting the opposite side of T (n, 1) at 90 deg on a double-trace edge. The 60 deg angle has its vertex at the center of the ct 1hhh, where h = (n − 5)/2. Proof. The statement follows by combining 1[011], 2[011], 3[112] under the isomorphisms ρn (1) → ρn (i), Theorem 9.4. 2 q
Theorem 9.6 The diameter of Gn,4 is Ω(n) and O( 3 |V (Gn,4 )| ). Thus, its diameter is asymptotically of the order of the cubic root of its vertex-set cardinality. Proof. In [9], a component Gn,3 of the subgraph of Gn induced by the tmc K3 -types of ZZn was defined, for n = 2k + 1 ∈ ZZ odd > 0, which is the one containing the K3 -type 123. Let us see that |V (Gn,3 )| = O(n.φ(n)), where φ(n) is the Euler characteristic of n. Every aa0, where gcd(a, n) = 1, belongs to Gn,3 . Thus, there are bφ(n)/2 − 1c paths whose ends are 011 and 0aa, with 0 < a ≤ bn/2c and gcd(a, n) = 1. But the distance from 0aa to 011 in Gn,3 is ≤ a, yielding our claim. Second, let us see that |V (Gn,4 )| = O(n2 .φ(n)). If we fix a K3 -type of abcdef ∈ Gn,4 , say abc, then for each color d MOD n there are at most two different values for e, yielding a unique value for f . This way, there are at most n.φ(n).(2bn/2c) different K4 -types MOD n, yielding our claim. Let us see now that the diameter of Gn,4 is Ω(n). A path of length n+1 between 110110, (1, 1, 2,qn−1, n, n) happens along the image of L(1, 2, 2). 2 Thus, the diameter of Gn,4 is Ω(n) and O( 3 |V (Gn,4 )| ). A representation of the charts of G0n,4 is presented now , leading to the asymptotic property claimed in the Introduction, as well as to a conjecture about the connectedness of G0n,4 for n ≥ 17. Let σn (1) be the restriction of ρn (1) induced by the tmc vertices. We superpose the Tsubgraph representation of σn (1) with a honeycomb tessellation τn (1) formed by congruent regular hexagons such that: (a) each edge of σn (1) is traversed by an edge 0 of τn (1) at 90 deg at the common midpoint of and 0 ; (b) each ch of σn (1) contains in its interior a regular hexagon of τn (1). Figure 5 contains, without labels, a superposition of partial representations of σn (1) and τn (1). In Figure 6, representing τn (1) for odd n = 13, . . . , 25, each tmc vertex of σn (1) is given by an hexagon of τn (1) in which a positive integer is written. Each integer-filled hexagon 14
τ13 (1)
65 34
66 • 56 • • 6 23 • 2 45 34
τ15 (1)
76
• 5 34 67 • • 56 • 4 7 • 7 23 2 2 •
34
77
45
56
τ17 (1)
98 87 • 45 4 8 6 34
88 78 • • 67 3 • • 56 • 4 7 5 8 • 8 23 2 2 2 • 34 45 56 67
a9 56 • 98 99 • 6 8 45 89 • 4 78 • 5 9 • 9 7 34 67 3 3 • • 56 • 4 7 5 8 6 9 • 9 23 2 2 2 2 • 34 45 56 67 78
τ19 (1)
98 a9 • 7 56 aa • 5 98 21 9a • • a 7 9 45 a9 • 67 23 4 • 4 89 6 ba 78 • 5 9 6 a • a 8 34 bb • 7 a 8 56 a9 56 67 25 3 3 3 • • ab • 5 ba • 9a • 6 b • b 8 a 45 56 • 4 7 5 8 6 9 7 a • a 23 cb • 8 a 67 • 2 2 2 2 2 4 4 • 4 89 cc 6 6 45 56 67 78 89 34 78 • 5 9 6 a 7 b • b 9 34 bc • • c 8 b 9 56 67 3 3 3 3 • • ab 5 • 5 56 • 4 7 5 8 6 9 7 a 8 b • b 23 9a • 6 b 7 c • c 9 b 45 • 2 2 2 2 2 2 4 4 4 • 4 89 34 45 56 67 78 89 9a 78 • 5 9 6 a 7 b 8 c • c a 34 67 3 3 3 3 3 • • 56 • 4 7 5 8 6 9 7 a 8 b 9 c • c 23 • 2 2 2 2 2 2 2 45 56 67 78 89 9a ab 34
τ (1)
τ (1)
τ (1)
Figure 6: The representations τn (1), for n = 13, . . . , 25 representing a vertex of σn (1) is the intersection of two hexagon towers of τn (1). There are three directions of parallelism for appearing hexagon towers: one of them is horizontal and the other two are at angles of ±60 deg. Each such a tower is headed on the boundary of τn (1) by a partially-drawn double-trace hexagon containing a pair of integers. Assume the integer in an hexagon ζ of τn (1) is i and the integer pair heading its two hexagon towers are (p, q) and (r, s). Then the K3 -types composing ζ are: 1pq, 1rs and either ipr and iqs or ips and iqr. An hexagon contains a bullet • instead of an integer if it represents a non-tmc K4 -type. Each empty hexagon stands for a corresponding ch. It follows that each σn (1) has at least two isolated vertices, represented in τn (1) by: (1) the hexagon containing 2, at the lower-left corner of τn (1), (K4 -type 134265); (2) the hexagon containing bn/2c, at the lower-right corner of τn (1), (K4 -type 123k(k − 2)(k − 1), where n = 2k + 1). If n 6= 0 mod 3 then these are the only two isolated vertices of σn (1). Otherwise, there is exactly one more isolated vertex in σn (1), determined by the hexagon containing n/3, at the upper-right corner of τn (1), (K4 -type 1(k − 2)(k − 1)k(k + 1)(k + 2)), as shown in Figure 6 for n = 15, 21. Theorem 9.7 For each unit i MOD n, the set of vertices of Gn,4 − G0n,4 in ρn (i) has cardinality convergent to zero as k = b n−1 c tends to infinity. 2 Proof. The set of vertices of Gn,4 − G0n,4 in ρn (i) are of one or more of the following four types: (1) those which are contained in an sa; (2) those adjacent to a loop, that is those which are images of vertices adjacent to a half-edge (orthogonal to an sa) in a chart H 0 , in particular those as in item (2) of Proposition 8.3; (3) those represented by bullets in τn (i), which are as in item (1) of Proposition 8.3, and (4) those not represented in τn (i). Now, the sa’s and the straight-line paths formed by bullets or by looped vertices in τn (i) 15
are straight lines, and therefore 1-dimensional. Thus, the set of vertices of Gn,4 − G0n,4 in ρn (i) has cardinality O(n). On the other hand, the set of the other interior vertices of ρn (i), namely those of σn (i), which cover its star-of-David tessellation in some 2-dimensional region of the plane, has cardinality O(n2 ). 2 For n ≥ 17, the isolated vertices of σn (1) are nonisolated in the other charts σn (i), where i 6= 1 ranges over the units MOD n from 2 to bn/2c. This yields our final conjecture. Conjecture 9.8 G0n,4 is a connected graph, for n ≥ 17. On the other hand, the six charts τ13 (i), for i = 1, . . . , 6, represent the same pair of isolated vertices shown in Figure 1(c) and (d), which are thus the only components of G013,4 . In addition, the four charts τ15 (i), for i = 1, 2, 4, 7, represent only a ct and 4 isolated vertices.
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[11] L. Fejes Toth, Regular Figures, Pergamon Press, Oxford, 1964. [12] P.R. Hafner, Large Cayley graphs and digraphs with small degree and diameter, in W.Bosma and Alf van der Poorten, eds., Computational Algebra and Number Theory Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht/Boston/London (1995), 291–302. [13] B.D. McKay, M. Miller, J. Siran, A note on large graphs of diameter two and given maximum degree, J. Combin. Theory Ser. B 74 (1998) 110-118. [14] M. Sampels and S. Sch¨of, Massively parallel architectures for parallel discrete event simulation, Proc. 8th European Simulation Symposium (ESS’96), vol. 2, (1996), 374378. [15] E. W. Weisstein, Mathworld, mathworld.wolfram.com/Tessellation.html. [16] O. Wohlmuth, A New Dense Group Graph Discovered by an Evolutionary Approach, Paralleles und Verteiltes Rechnen Beitraege zum 4, Workshop ueber Wissenschaftlichen Rechnen, Shaker Verlag, 1996, 21-27.
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