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Discussiones Mathematicae
Graph Theory 35 (2015)733–754
doi:10.7151/dmgt.1834
RAINBOW TETRAHEDRA IN CAYLEY GRAPHS
Italo J. Dejter
University of Puerto Rico
Rio Piedras, PR 00936-8377
e-mail: italo.dejter@gmail.com
Abstract
Let Γnbe the complete undirected Cayley graph of the odd cyclic group
Zn. Connected graphs whose vertices are rainbow tetrahedra in Γnare
studied, with any two such vertices adjacent if and only if they share (as
tetrahedra) precisely two distinct triangles. This yields graphs Gof largest
degree 6, asymptotic diameter |V(G)|1/3and almost all vertices with degree:
(a) 6 in G; (b) 4 in exactly six connected subgraphs of the (3,6,3,6)-semi-
regular tessellation; and (c) 3 in exactly four connected subgraphs of the
{6,3}-regular hexagonal tessellation. These vertices have as closed neigh-
borhoods the union (in a fixed way) of closed neighborhoods in the ten
respective resulting tessellations.
Keywords: rainbow triangles, rainbow tetrahedra, Cayley graphs.
2010 Mathematics Subject Classification: 05C15, 05C75, 05C62.
1. Introduction
Cayley graphs are very important because they have many useful applications
(cf. [11]) and are related to automata theory (cf. [12, 13]). In the present work,
we deal with Cayley graphs of a finite abelian group Gwith its identity denoted
0. Let Sbe a subset of Gsuch that 0 /∈Sand S=−S(that is: s∈Sif and
only if −s∈S). The Cayley graph Γ(G, S) on Gwith connection set Sis a graph
that has as its vertices the elements of Gand is such that it has an edge ejoining
vertices gand hif and only if h=g+s, for some s∈S. In this case, we say that
the edge ehas color s. A concept of “rainbow” has been used in various fashions
in a graph theory context, in [1, 2, 3, 8, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21] and
related papers. Ours is in relation to edge colors in Cayley graphs of finite cyclic
groups. Below, the complete graph Kn=K2k+1 will be viewed as the Cayley
graph Γn= Γ(Zn,[k]) of the cyclic group Znof integers modulo nwith connecting
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734 I.J. Dejter
set [k] = {1,2, . . . , k}. Relations among rainbow triangles and tetrahedra in Γn
(rainbow meaning here edges with pairwise different colors) will be shown to yield
a family G1of connected graphs G=Gn,4of largest degree ∆(G) = 6, asymptotic
diameter |V(G)|1/3and such that almost all its vertices vhave degree: (a) 6 in G;
(b) 4 in exactly six connected subgraphs of the (3,6,3,6)-semi-regular tessellation
([7], page 43); and (c) 3 in exactly four connected subgraphs of the {6,3}-regular
hexagonal tessellation ([7], page 43). We refer to each of these ten subgraphs
of Gas a D- or as an H-modeled subgraph of Gif it is as in (b) or as in (c)
above, respectively. On the other hand, based on rainbow triangles a family
G0of connected graphs G=Gn,3of largest degree ∆(G) = 3 and asymptotic
diameter |V(G)|1/2was introduced in [5]. See Section 3 below for a short survey
of [5] and for further developments ahead in this paper.
The mentioned asymptotic properties of the families G0and G1confirm the
following conjecture, further discussed in [6].
Conjecture 1. The asymptotic diameter of a family of graphs Gwith a common
∆(G)is a given (radical, logarithmic, ...)function of the vertex number of G.
2. Main Results
The present paper is devoted to the following results, containing the claimed
properties of G1. (For related properties, see [4] and its references.) The tessel-
lated neighborhood of a vertex vin a D- or H-modeled subgraph Gis formed by
vand its incident edges and faces as well as by the other edges adjacent to those
faces and the endvertices of these edges.
Figure 1. Tessellated neighborhoods of a vertex of Gn,4in the subgraphs of Theorem 2.
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Theorem 2. There exists an infinite family G1of finite connected graphs G=
Gn,4with asymptotic diameter |V(G)|1/3such that that the subset V6of vertices
v∈V(G)with deg(v) = ∆(G) = 6 has asymptotic order |V(G)|. In that case,
almost every v∈V6
1. is incident to three triangles T0, T1, T2in Gwith pairwise intersection {v}
determining exactly six planar D-modeled subgraphs Dk
i,j (i, j = 0,1,2;k=
0,1) such that Ti∪Tj=D0
i,j ∩D1
i,j for each pair {i, j} ⊂ {0,1,2}with i6=j;
2. is the intersection of the six D-modeled subgraphs of Gabove, in which
deg(v) = 4, and exactly four H-modeled subgraphs in G, in which deg(v) = 3,
and such that the closed neighborhood of vin Gis contained in a fixed way
in the union of the tessellated neighborhoods of vin the ten cited subgraphs,
comprising 43 vertices.
To give an idea of what is going on locally at almost every vertex in the
context of Theorem 2, Figure 1 shows on its left (respectively, right) side the
closed (respectively, tessellated) neighborhoods of a particular vertex v—given
by the edge-colored copy of K4(in Gn,4or G∞,4) depicted at the figure center,
see Section 5—in each of the ten subgraphs mentioned in the two items of the
statement, namely, in the six D-(respectively, four H-) modeled subgraphs of Gn,4
claimed above, for a value of nsufficiently large, with edges colored via a= 7,
b= 9, c= 2, d= 3, e= 1 and f= 6.
Corollary 3. There is a subfamily G′
1of G1such that any Dk
i,j in a member G
of G′
1is a D-modeled subgraph restricted to a 30◦-60◦-90◦triangular region of the
Euclidean plane. Moreover, there are n−1pairwise distinct such subgraphs Dk
i,h
distributed, for y≥1, into two subsets of size n−1
2composed each by isomor-
phic subgraphs. By denoting these n−1
2-subsets by V−
yand V+
y, if k= 5 + 2y;
respectively, U−
yand U+
yif k= 4 + 2y, with |V−
y|<|V+
y|and |U−
y|<|U+
y|, then
|V−
y|=y2+y−1and |V+
y|= 3y2+ 3y−3−ǫ(k), where ǫ(k) = 1 if k≡1(mod 3)
and ǫ(k) = 0 if k6≡ 1(mod 3); respectively, |U−
y|=|V−
y|−yand |U−
y|=|V+
y|−3y.
Figure 9 of [4] illustrates the 30◦-60◦-90◦-triangular regions in Theorem 2;
alternatively, see Figures 6 and 7. The proofs of Theorem 2 and Corollary 3 in
Section 9 are composed by the arguments presented in Sections 3–9 and, for the
H-modeled subgraphs in item 2 of Theorem 2, by Theorem 2 of [4].
3. K3-Types and K3-Type Graphs
A triangle in Γnhas K3-type (a, b, c) if its edges have colors a, b, c ∈[k]. If no
confusion arises, we suppress commas and parentheses, so we write (a, b, c) = abc.
More generally, a K3-type abc =acb =bac =bca =cab =cba of Znis a 3-multiset
{a, b, c}of [k]∪ {0}such that a+b∈ {c, −c} ∈ [k], where a+bis taken modulo
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736 I.J. Dejter
n. (This 3-multiset can be viewed as a class of at most six 3-tuples of colors of
[k]∪ {0}, one of which is abc.)
Example 4. The K3-types {a, b, c}of Z7with 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}, where the greatest
common divisor gcd(J) of a finite multiset Jof nonnegative integers is the largest
common divisor of the nonzero integers of J.
Let Gnbe the graph whose vertices are the K3-types of Znand such that any
two of them, say vand v′, are adjacent via an edge ǫif and only if vand v′share
either two different colors of Γnor one color of Γnrepeated twice, say aand a′;
in either case we can consider ǫas determined by {v, v′}or by {a, a′}. We take
{a, a′}(= aa′, for short) as the color of ǫ, so that Gnbecomes an edge-colored
graph. In addition, we assume that Gndoes not have multiple edges. In the
example above, only 123 is rainbow. Each rainbow triangle tin Γnand edge ǫ
of tdetermine exactly one rainbow triangle t′6=twith the same colors of tand
sharing ǫwith t. For n= 2k+ 1 ≥7, let G′
n⊆Gnbe the subgraph of Gninduced
by the rainbow K3-types of Zn. Let Gn,3be the component of G′
ncontaining the
K3-type 123. Then all the remaining components of G′
nare isomorphic to graphs
Gm,3with 1 < m < n and m|n. Notice that the vertices of Gm,3are 3-sets. Now,
consider N={m∈Z:m≥0}as an infinite color set. A K3-type abc of Z,
simply called a K3-type, is a 3-multiset {a, b, c}of Nsuch that the sum of the
two least colors equals the greatest one. Let G∞,3be the graph whose vertices
are the K3-types abc with gcd(a, b, c) = 1 and whose edges are as defined above
for Gn. Given m, m′, n ∈Nwith m′∈[k], we say that m′≡m(mod n) whenever
if for m′′ ≡m(mod n) with 0 ≤m′′ < n:
(1) if m′′ > n/2, then m′=n−m′′;
(2) if not, then m′=m′′.
Here, m′is said to be the reduction of m(mod n). It was shown in [5],
Proposition 2.16, that for odd n≥7, Gn,3can be obtained, from a connected
subgraph Fof G∞,3containing 011, 112, 123 and the remaining K3-types with
colors ≤n, by reducing modulo nall the colors of K3-types of F. Let φ(n) be
the value of Euler’s totient function at the positive integer n. It was shown in
Theorem 2.17 of [5] that |V(Gn,3)|=O(nφ(n)) and subsequently, in Theorems
2.20 and 2.21, that the diameter of Gn,3is both Ω(n) and O(|V(Gn,3)|1/2). The
family G0in the introductory section above is formed by these graphs Gn,3.
4. K4-Types and K4-Type Graphs
AK4-type of Zn(respectively, Z) is a maximal class of 6-tuples abcdef of col-
ors of [k] (respectively, N) such that abc,cde,aef and bdf are K3-types of Zn
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Rainbow Tetrahedra in Cayley Graphs 737
(respectively, Z). Such a class has at most twenty-four 6-tuples. A 6-tuple in a
K4-type tis called a card of t. If no confusion arises, we represent a K4-type by
one of its cards. A card abcdef will be represented
(i) either as a tetrahedron each of whose edges bears a color, as in Figure
2(a);
(ii) or by keeping only the locations of the colors in (i) in an enclosure, as
shown in Figure 2(b). The colors in Figure 2(a) split into three different pairs of
opposite colors: {a, d},{b, e},{c,f }, (opposite in the sense that each pair is held
by a corresponding pair of edges of K4with no vertices in common, the remaining
edges forming a 4-cycle).
Figure 2. Representing a generic K4-type abcdef and its cases modulo 13.
Any 6-multiset of Ndetermines at most one K4-type of Z. This is not true
for (Zn,[k]) in place of (Z,N). For example, the two rainbow K4-types 123645
and 246153 of Z13 represented in Figures 2(c1) and 2(c2), respectively, are distinct
but have the same underlying multiset.
Arainbow K4-type is one with six different colors. Given n= 2k+ 1 ≥13,
let G′
n,4be the graph whose vertices are the rainbow K4-types abcdef of Znwith
gcd(a, b, c, d, e,f, n) = 1 and such that any two such vertices, say tand t′, are
adjacent via an edge ǫif and only if tand t′looked upon as K4-types share
precisely two K3-types vand v′. In this case, vand v′share exactly one color a
of [k]. We take aas the (weak)color of ǫand this makes G′
n,4into an edge-colored
graph.
In order to distinguish the D- and H-modeled subgraphs that we claim G′
n,4
contains, we introduce the graph G′′
∞,4as the simple graph (i.e., graph without
loops or multiple edges) whose vertices are the K4-types abcdef with a6=d,b6=e
and c6=funless abcdef = 011011 and satisfying gcd(a, b, c, d, e,f ) = 1, with
two vertices uand vdetermining an edge if and only if they share precisely two
K3-types in differing locations of the representation of the K4-types that stand
for uand vas in Figure 1.
Figure 3 illustrates G′′
∞,4as well as Theorem 5 below. The figure represents a
neighborhood Nof the K4-type 123745 in G′′
∞,4. Notice that the two right-lower
K4-types in Figure 3 (joined by the edge colored with 6) are not rainbow. An edge
ǫjoining two vertices tand t′of G′′
∞,4with respective cards rand r′determines a
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738 I.J. Dejter
Figure 3. A neighborhood of 123745 in G′′
∞,4.
K3-type scommon to tand t′and equally located in rand r′in the sense that the
component colors of soccupy the same positions in rand r′, just as the K3-type
s= 123 is not only common to but also equally located in the central card in
Figure 3 and the card horizontally located at its right, with soccupying the three
upper-left locations in rand r′. The locations grof the colors in the cards r′of the
statement of Theorem 5 obtained from the central card rat the center of Figure
3 are shown encircled. Also, the K3-type sis highlighted in a sub-enclosure of
its own. Observe that in each of the six enclosures representing the neighbors of
the central vertex in Figure 3 the two colors outside the sub-enclosure and the
encircled color are permuted in their positions.
Theorem 5. Let t∈V(G′′
∞,4). Let rbe a card of twith color gat location gr
and color g′at the location g′
ropposite to gr. Then thas a neighbor t′with card
r′differing from rjust in
(a) the color at grand
(b) a permutation of the colors at the two locations 6=g′
rin just one of the two
K3-types common to rand r′that contain the color at gr.
Proof. t′is determined from tas follows. Let s, s′be the two K3-types not
containing grin r. Then sand s′contain g′
r. We can assume that s′has its
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Rainbow Tetrahedra in Cayley Graphs 739
colors equally located in rand r′. Let i, j be the colors of rat the two locations
ir6=g′
rand jr6=g′
rof s. Thus s=ijg′. The two other K3-types in tapart
from sand s′are of the form gij′and gji′with s′=i′j′k. We take r′as
having the colors i, j exchanged with respect to r. So (ir′, jr′) = (jr, ir). Let
ν(a, b) = {| a−b|} ∪ {a+b}for each pair of integers a, b ≥0. There is at least
one color h∈ν(i, j)∩ν(i′, j ′)6=∅that yields r′when located at gr(which should
be called hr′in r′) so that r′is formed by the K3-types s=ijg′,s′=i′j′g′,
hii′and hjj′. Moreover, r′does not depend on the selected card rof t. In fact
h=h(r, gr) depends only on rand gr. If r= 011011 and g= 0 then hequals
either 0, yielding t′=t, not a distinct neighbor of tin G′′
∞,4so we discard it, or
2, yielding a neighbor t′of t. Otherwise, since no remaining vertex of G′′
∞,4is of
the form abcabc 6= 011011, then |ν(i, j)∩ν(i′, j′)|= 1, even if (r, g) = (011011,1).
Thus, if either r6= 011011 or (r, g) = (011011,1), then his unique.
Example 6. In the following special cases, gassumes subsequently colors f,a
and din a K4-type tof card r=abcdef :
(A) applying Theorem 5 to (r, g) = (112354,4) (so g=f) yields t′=twhere
gr=fr= 4rbecause exchanging dr= 1rand er= 1rdoes not produce
changes from r;
(B) applying Theorem 5 to (r, g) = (011011,0) (so g=a, d) yields, for h= 2,
neighbors t′, t′′ with respective cards r′= 211011 and r′′ = 011211 where
gr=ar, drrespectively, but observe that t′=t′′ .
5. Canonical Triangles
Let G∞,4be the supergraph of G′′
∞,4obtained by adding to the vertices of
G′′
∞,4\{011011}the loops offered by the method of vertex adjacency in Theo-
rem 5 and Figure 3, taking each maximal set of loops incident to a common
vertex and with a common color to have multiplicity 1. Then, a link or loop
joining vertices tand t′in G∞,4has the pair (s, s′) in the proof of Theorem 5
as its strong color and the only color g′in sand s′that remains at the location
g′
r=g′
r′both in rand r′as its weak color. Let G′
∞,4be the graph obtained from
G∞,4by restriction to the vertices that are rainbow K4-types.
Applying Theorem 5 to the colors g, g′of a pair of opposite edges of a vertex
tof G∞,4looked upon as a K4-type with card ryields h(r, g) = h(r, g′). This
determines in rtwo corresponding neighboring cards r′and r′′ representing re-
spective neighbors t′and t′′ of t. The two K3-types that r′and r′′ share and those
two that rand r′(respectively, rand r′′) share constitute the four K3-types of r′
(respectively, r′′ ). The resulting triangle, whose vertices t, t′, t′′ have respective
cards r, r′, r′′ , is said to be a canonical triangle, or CT. Since there are three pairs
of opposite vertices in the card rassociated to the vertex tof G∞,4, then there
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740 I.J. Dejter
are at most three CTs incident to t. Since each G′
n,4can be obtained from G′
∞,4
via reduction modulo n, we have completed the proof of the following corollary.
Corollary 7. The graphs G′
∞,4and G′
n,4are edge-disjoint unions of CTs, at
most three such CTs incident to each vertex.
When two or three K4-types in a CT T={t, t′, t′′}obtained as in Theorem 5
coincide (e.g., either t=t′6=t′′ or t=t′′ 6=t′or t6=t′=t′′ or t=t′=t′′), then
we say that Tis a degenerate CT.
Example 8. (A) If thas r=abcdef with a, b > 0, c=a+b,d=a,e=b,
f=|a−b|and (gr, g′
r)∈ {(ar, dr),(br, er)}, then t′=t′′. This yields two
degenerate CTs with vertices of the form t,t′and t′′ =t′, where tt′=tt′′
and t′t′′ is a loop of G∞,4.
(B) Theorem 5 applied to t= 000111 yields three degenerate CTs, each rep-
resentable by: two vertices, namely t(twice) and t′= 011011, a link tt′
and a loop at t; these three CTs coincide, since edges are assumed to have
multiplicity 1.
(C) Theorem 5 applied to t= 132112 yields three CTs 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 two remaining CTs incident to tare {t, 202111,132201}
and {t, 431122,132421}.
Corollary 9. G∞,4is connected.
Proof. Given t=abcdef and t′=abcydx in G∞,4there exists a 2-path in G∞,4
from tto t′with middle vertex card abcfxd and edge strong colors {abc, bdf }and
{abc, adx}. Let cde and cxy be K3-types of Zwith gcd(c, d, e) = gcd(c, x, y).
Then there exists a path in G∞,4whose 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∞,3from cde to cxy [5]. Thus, if abcdef ∈V(G∞,4), then there exist: (a) a
path in G∞,4from 110110 to 110aa(a+ 1); (b) a path in G∞,4from 110aa(a+ 1)
to aa0bbc;(c) a path in G∞,4from aa0bbc to abcdef . Hence, every vertex of G∞,4
can be connected to 110110.
6. Generation of D-Modeled Subgraphs
Corollary 10. The set of CTs of G∞,4is in 1-1correspondence with the family
of 4-multisets or quadruples abcd of colors of Nsuch 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.
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Rainbow Tetrahedra in Cayley Graphs 741
Proof. From Theorem 5 and Corollary 7, each CT of G∞,4has its vertices as
K4-types sharing precisely four colors as in the statement.
Example 11. In Figure 3, the upper (respectively, lower-left, lower-right) CT
has its vertices sharing the quadruple 1357 (respectively, 1247, 2345).
From now on, each CT will be denoted by its associated multiset in Corol-
lary 10. Given a rainbow K4-type t=abcdef , the CTs incident to tare obtained
by deleting from teach one of the three pairs ad,be and cf, which yields respec-
tively bcef,acdf and abde.
Let abcdef be a vertex of G∞,4and let C=acdf and D=abde be two CTs in
G∞,4sharing just abcdef . Then C∪Dis represented as a colored 5-vertex plane
graph B(t, a, d) where C and D participate as respective equilateral triangles C
and D, respectively, that share solely a vertex t(i.e., C∩D={t}) that stands
for abcdef and is center of a point symmetry that takes Conto Dand viceversa.
Thus, pairs of sides of Cand Dincident to tare set collinearly as in Figure 4.
We require ato tag the centers of both Cand D, and the remaining colors of
Cand Dto tag respectively the vertices of Cand Dinternally. Then, dis the
color tagging tinternally in both Cand D. We tag each edge of C(respectively,
D) with the weak color of the corresponding edge of C(respectively, D), such
that the weak color of each edge ǫof Cforms: (a) aK3-type s(ǫ) with the colors
tagging the endvertices of ǫin C;(b) another K3-type s′(ǫ), with the central
tagging color of Cand the color tagging the vertex opposite to ǫin C. Notice
that {s(ǫ), s′(ǫ)}is the strong color of the image of ǫin G∞,4. Let ǫCand ǫDbe
edges of Cand D, respectively, meeting at an angle of 120◦at vertex t. Then
the color dtagging tin both Cand Dforms with the colors tagging ǫCand ǫD
the K3-type s(ǫC) = s(ǫD).
6.1. Growth of a D-modeled subgraph
The growth of a D-modeled subgraph of G∞,4sprouting from B(t, a, d) =
C∪Dvia Theorem 5 can be performed via the following properties deducible via
Theorem 5 and enjoyed by the objects conceived in the previous paragraph with
their tagging notation around r=abcdef as shown in Figure 4(c) and illustrated
in Figure 4(a)–(b).
(1) Given a CT C=afgh, let abe the central tag of Cand let color ftag a
vertex uin C. Then there is a color iso that (a) ν(a, h)∩ν(f, g) = {i};(b) the
edges ǫ=uu′in Cwith u′having tag gor hin Chave color i, denoted γ(ǫ) = i.
(2) Let ℓbe the line containing uand parallel to the unique edge of C\u.
Then each pair (u, C) determines at most one remaining CT D6=Csharing u
with C, so that D=ρℓ(C), where ρℓis reflection of the plane on ℓ, and having
(a) aas central tag;
(b) the tag fof uin Cas tag of uin D;
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742 I.J. Dejter
Figure 4. Unfoldings of subgraphs of G∞,4.
(c) for each edge ǫ=uu′of C:
(i) γ(ǫ) as the tag of ρℓ(u′) in Dand (ii) the tag of u′in Cas the tag of ρℓ(ǫ).
(3) The vertex uis the K4-type formed by the K3-types determined by each
edge ǫof Dincident to uand formed by:
(a) aand the tags of ǫand the vertex opposite to ǫin D;
(b) the tags of ǫand the endvertices of ǫin D.
The union of two CTs Cand Dthat share precisely one vertex vis said to
be a butterfly and denoted CvD. In this case, vis called the central vertex of
CvD. Note that the colors of vin Cand Dequal a fixed color dwhich we call
the butterfly color of CvD. For example, B(t, a, d) above is a butterfly CtD with
central color aand butterfly color d, say with C=acdf and D=abde. Given
a simple graph Gand a pseudograph H(i.e., His a non-simple graph in which
each vertex may be incident to one or more loops), then Gis an unfolding of H
if there exists a surjective map f:V(G)→V(H) such that for each v∈V(G)
there exists a 1-1 correspondence induced by ffrom the links incident to vin G
to the edges incident to f(v) in H.
6.2. Maximal D-modeled graphs
Let t=abcdef be a rainbow K4-type. A maximal D-modeled graph H′=
H′(t, a) = H′(t, a, d)⊃B(t, a, d) that is an unfolding of an edge-disjoint union
H=H(t, a) = H(t, a, d) of butterflies in G∞,4with common central color ais
generated by repeated application of item (2), Subsection 6.1, at gradients 0◦,
60◦, 120◦, 180◦, 240◦, 300◦of the line ℓin the item.
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Example 12. Both Figure 4(a) and 4(b) show parts of an H′as above.
We will see that if such an H′is not a subgraph of G∞,4, then it can be folded
along at most two symmetry axes, or SAs, to yield H. The dotted line in Figure
4(a) represents such an SA. In particular, edge colors will coincide by reflection
in an SA. The graph obtained from Hby removing the resulting loops will be
seen to be a subgraph of H′spanning a connected region of the plane delimited
by SAs. Edges crossing an SA at 90◦will yield loops of Hand each CT in H′
will be incident to three hexagons.
Observation 13. Given a vertex tof H′(t, a, d), the three CTs incident to t
according to Theorem 5are:
(a) the two CTs incident to tin H′(t, a, d)and
(b) the CT formed by the colors of the four edges of the two CTs in item (a)
which are incident to t.
7. Presence and Properties of 6-Cycles
The graph H′(t, a, d) in Subsection 6.2 has two edge-disjoint 6-cycles with just
the vertex tin common which are given by regular hexagons in the plane when
the CTs of H′(t, a, d) are represented as equilateral triangles as in the discussion
after Example 11. This is the specific case in Subsection 7.2 below. If qis any of
these 6-cycles, then its edges are colored with the component colors of a K3-type
s. In that case, we denote q=a.s, where ais the central color of the six CTs
adjacent to q.
7.1. A procedure to determine 6-cycles
Let bdf =sand cde =s′be K3-types, where t=abcdef is a vertex of H′(t, a, d).
We will see that there exists a 6-cycle (t0, t1, t2, t3, t4, t5) in H′(t, a, d) containing
t=t0. It will be determined by the following procedure that yields tiwhen ti−1
is given, successively for i= 1,2,3,4,5, (and returns to t0=tifrom t5=ti−1, if
i= 6 ≡0 with indices taken modulo 6.
(a) Declare the card riof the K4-type tito have color a(as in Figure 2(b))
fixed in the location ar0(so that ari=ar0) during the entire procedure;
(b) denote locations bri=br0,cri=cr0and eri=er0regardless of changes
in their color values from the initial ones, namely b,cand erespectively along
the running of the procedure;
(c) define color hi=b(respectively, hi=f) if iis even (respectively, odd);
(d) establish a color exchange via a redesignation of locations at the i-th
level: dri=hi−1
ri−1and hi
ri=dri−1;
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744 I.J. Dejter
(e) the color eri(respectively, cri) if iis even (respectively, odd) takes the
only value from ν(ari,fri)∩ν(cri, dri) (respectively, ν(ari, bri)∩ν(dri, eri)).
This determines a well-defined card riand yields a location instance for the
determination of a 6-cycle as claimed.
Example 14. A 6-cycle generated by the procedure in the previous paragraph
and starting at t0= 123745 is
a.s = 1.257 = (123745,123587,156287,156712,176512,176245).
Its accompanying coplanar 6-cycle a.s′is
1.347 = (123745,187345,187434,134734,134376,123476).
An essentially equivalent 6-cycle to this and sharing its first two vertices with
a.s′as just given is 7.145 = (123745,583741,48C751,1BC754,5B6714,426715),
where capital hexadecimal notation is used, and its accompanying coplanar 6-
cycle is 7.123 = (123745,321785,23178A, 13279A, 312796,213746),
sharing its first two vertices with a.s.
7.2. On 6-cycles containing specific K4-types
Each tas above is contained in precisely two 6-cycles q=a.s and q′=a.s′of
H′(t, a, d). The edge-color sets of qand q′are respectively {b, d,f }and {c, d, e},
each color tagging opposite edges. Moreover, the color tagging tin its incident
CTs in H′(t, a, d) and those tagging the two edges in q(respectively, q′) that are
incident to tconform s(respectively, s′). Furthermore, dis the color tagging t
in its incident CTs in H′(t, a, d) as well as tagging the two parallel edges of a.bdf
(respectively, a.cde) incident neither to tnor to its corresponding opposite vertex.
Given K3-types bcd and bc′d′with b < c < d and b < c′< d′, define bcd < bc′d′
if and only if c+d < c′+d′. A graph H′=H′(t, a, d) as in Subsection 6.2 is
said to be a T-subgraph and denoted a(s), where sis the smallest K3-type 6= 000
coloring a 6-cycle of H′under ’<’, while H=H(t, a, d) is denoted a[s]. Hexagons
a.s of an H′(t, s, d) and their images in H(t, a, d) are called canonical hexagons
or CHs.
Proposition 15. Let H′=H′(t, a, d), where t=abcdef is common to C=acdf
and D=abde, with C∪D⊂H′(t, a, d)and dtagging tin both Cand D. Then,
the T-subgraph H′′ =H′(t, d, a)has tcommon to a flipped copy Dof Dand a
direct copy Cof C. As a result, d.caf and d.bae contain the colors of the CTs
incident to tin H′′. Moreover, H′′ =H′if and only if f=cand e=b.
Proof. H′′ =H′(t, d, a) is established as follows:
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Rainbow Tetrahedra in Cayley Graphs 745
(1) represent H′′ as a temporarily uncolored T-subgraph and set tas one of
its vertices;
(2) represent Cand Din H′′ as the respective CTs Cand Dof H′with
common vertex tbut set the locations of aand din Cand D, instead, as those
of dand ain Cand D, respectively;
(3) the vertex colors cand fin Care exchanged with respect to their locations
in Cwhile the two vertex colors band ein Dare left as in D.
The remaining colors of H′′ can be set uniquely as in Subsection 6.1 above. If
H′′ 6=H′, then reflection with respect to the line perpendicular to the line ℓin
Subsection 6.1 through ttakes each edge color of Din H′′ to its location in D,
while the edge colors of Cremain as in C. The statement follows immediately,
as illustrated in Figure 4, where (b), at right, represents part of the T-subgraph
H′′ corresponding to the T-subgraph H′, partly represented itself in (a), with
t= 235142 at the center in both representations.
8. From D-Modeled Subgraphs to Charts
Local plane representations of some subgraphs a[s] = a[bcd] of G∞,4are provided
in Figure 5 with notation given before Proposition 15, a= 10, d = 13, g = 16
and thin (respectively, thick) edges for links (respectively, loops). In fact, the
subgraphs induced by the set of links of these a[s] yield subgraphs of the cor-
responding graphs a(s) = a(bcd). Concretely, Figure 5 upper-left (respectively,
upper-right) shows a plane region delimited by two dotted lines ℓand ℓ′that form
an internal angle of 30◦(respectively, 90◦) and determine a partial representation
of H′(s, 1) = 1(011) (respectively, H′(s, 2) = 2(011)), where s= 110001 (respec-
tively, s= 211011). This representation can be identified with H(s, 1) = 1[011]
(respectively, H(s, 2) = 2[011]) by interpreting as a loop each thick edge inter-
rupted perpendicularly by some dotted line ℓ. Moreover, H′(s, 1) (respectively,
H′(s, 2)) is obtained by unfolding H(s, 1) (respectively, H(s, 2)) along the SAs
formed by the lines in the finite sequence ℓ0=ℓ,ℓ1=ℓ′,...,ℓi= reflected line of
ℓi−2on the line ℓi−1, for i= 2,...,k−1, where additionally ℓk−1= reflected line
of ℓ1on the line ℓ0, with k= 360◦/30◦= 12 (respectively, k= 360◦/90◦= 4).
The extensions of these partial pictures to the plane will be referred to as
charts. Observe that the two charts in the previous paragraph are the only charts
of the form H′(t, a) with a= 1,2. However, no remaining value of aproduces
just one chart. For example, there are two charts H′(s, 3), one of which is 3(112),
with 3[112] partially shown in the bottom of Figure 5, where two straight lines
ℓ0and ℓ1at an angle of 60◦delimit its representation, and with finite sequence
ℓ0, ℓ1,..., as above, of length k= 360◦/60◦= 6. The remaining H′(s, 3) is 3(011),
with 3[011] having exactly one SA, delimiting a semi-plane representation. As a
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746 I.J. Dejter
increases its value, the first chart Hnot having an SA is H= 6(123) = 6[123].
Figure 5. Charts for 1[011], 2[011] and 3[112].
8.1. Unfolding charts
To see how the unfolding of a graph a(bcd) onto its corresponding a[bcd] takes
place, we observe that if H(t, a)6=H′(t, a), then H(t, a) is obtained by folds of
H′(t, a) along SAs of two types:
1. SAs dividing all CHs of the form a.0cc in symmetric halves through vertices
colored with 0 in CTs of the form a0cd, i.e., through all vertices of the forms
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Rainbow Tetrahedra in Cayley Graphs 747
0bbcca and 0ccdda;
2. SAs dividing all CHs of the form a.0cc in symmetric halves through vertices
colored with 0 in CTs of the form a0cd, i.e., through all vertices of the forms
0bbcca and 0ccdda; the form a.bbc in symmetric halves and passing at 90◦
through the midpoints of their edges colored with c(which are thick edges
that yield loops) and through the vertices opposite to them in corresponding
CTs.
In a chart H′, a thick edge halved perpendicularly in its middle point by
some SA yields a half-edge of H, and a CT that contains a half-edge yields a
half-CT of H. Degenerate CT 1113, shown in the lower-left corner of the chart
3[112] in Figure 5, has its center as the intersection of two SAs (and three SAs in
3(112)) and constitutes the only one-sixth-CT of any chart of G∞,4. See also the
example (C) before Corollary 9 in Section 5, where the CTs in their shown order
are 1113, 1122 and 1123, the first two present in 3[112]. The following properties
are observed:
1. A maximal connected region of an H′(t, a) delimited by SAs but with its
interior not intersecting any remaining SA yields a chart of H(t, a).
2. Charts a(bcd) and a[bcd] exist, for b≤c≤d, if and only if c+d≤a.
3. Every loop of G∞,4not in CTs 0011,1111,0112,1113 appears as a half-edge
in two different charts and as a thick edge in a different one. The CT that
contains such a loop: (a) is of the form aabc, where a, b, c are pairwise
different and (2a, b, c) is a K3-type; (b) appears as a half-CT obtained by
halving a degenerate CT as in the example (A) in Section 5 by means of
an SA in b[112] or c[112], and as a 3-cycle in a[011].
Two edges in a butterfly B(t, a, d) are said to be opposite if none has tas an
endvertex. Each butterfly has just one pair of opposite edges.
8.2. Color-alternating infinite paths
Any infinite path of H′=H′(t, a) = a(bcd) contained in a line has successive
edge tags in alternating colors fand geither differing in or adding up to a, the
latter occurring precisely if both f≤aand g≤a.
Denoting a path H′as above by L(f, g, a), we have:
1. f=gwhenever f=a/2∈Zor g=a/2∈Z; in this case, d=a/2 if
d≥b, c;
2. the edges colored 2ain L(a, 2a, a) are thick.
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748 I.J. Dejter
If two such paths are parallel and contiguous in H′then they are expressible as
L(f, g, a) and L(h,f, a), with |g−h|= 2aor g+h= 2a, the latter occurring
precisely if both g≤2aand h≤2a. Here, g , h are the edge colors opposite in the
butterflies taking place between L(f, g, a) and L(h,f, a). The edges of L(f, g, a)
and L(h,f, a) colored with fare divided into pairs of opposite edges of the CHs
lying between L(f, g, a) and L(h,f, a).
Observation 16. Given a vertex vof H(t, a), let f, g, h, i be the colors of the
edges incident to an unfolding vertex of vin H′(t, a). If ais odd or if vis not
in an L(a/2, a/2, a)then there is exactly one other vertex uof Hsuch that the
edges incident to any unfolding vertex of uin H′have colors f, g, h, i. In this
case uand vbelong to s=fghi and the edge uv has color a.
We may assume that vis shared in H(t, a) by a.fgj and by a.hij so that the
edge of shaving vas an endvertex but not having uas an endvertex is colored
with j, and jcolors vin s.
9. K4-Types of Zn
Proposition 17. Let 0< n = 2k+ 1 ∈Z. There is a colored supergraph Gn,4
of the graph G′
n,4introduced in Section 4 and a well-defined transformation Φn
from G∞,4onto Gn,4that operates by replacing all colors of Ntagging the objects,
e.g. vertices, edges, CTs and CHs of G∞,4, by their image colors under reduction
modulo nin the sense that all vertices (respectively, edges)with a common image
modulo ncolor disposition can be identified to a corresponding vertex (respectively,
edge).
Proof. Let Abe the subset of vertices of the graph G∞,4introduced in Section 5
whose colors have exclusively constituents ≤kand let Bbe the set of neighbors of
vertices of Ain G∞,4. Let Fbe the graph induced by A∪Bin G∞,4. By reducing
modulo nall the colors tagging objects of F, the resulting color identifications in
Fyield Gn,4. Note that the reduction modulo nfor vertices happens solely for
the vertices of B. Once these vertices are reduced modulo n, they have the same
colors as some vertices of A, so they must be identified correspondingly, and the
edges from Ato Bare then transformed into edges joining vertices of Awhich
were not originally induced by Ain G∞,4. Now, Φnis defined by replacing the
colors of the objects in G∞,4(vertices, edges, CTs and CHs) by their reductions
modulo n, which yields the corresponding objects in Gn,4.
Observation 18. The graph Gn,4is an edge-disjoint union of possibly degenerate
CTs, at most three incident to each vertex.
Corollary 19. Gn,4is connected, for any odd positive integer n.
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Rainbow Tetrahedra in Cayley Graphs 749
Proof. Apply Corollary 9 and Proposition 17 to the (continuous) map Φn:
G∞,4→Gn,4.
Application of Φnto the charts of G∞,4yields charts of Gn,4. The collection of
charts of Gn,4, (G∞,4), whose CT centers are colored i, for each i∈ {1,...,n/2},
is called an i-atlas.
Corollary 20. Let ρn: [k]→ {atlases of Gn,4}be the assignment given by
ρn(i) = i-atlas of Gn,4, for each i∈[k]. If gcd(n, i) = 1 < i < n/2, then
ρn(i)is obtained from ρn(1) by replacing each color ctagging a vertex, edge, CT
or CH of ρn(1) by the reduction modulo nof c.i. If nis prime, applying Φnto
the i-atlases of G∞,4yields ⌊n/2⌋i-atlases of Gn,4, which are graph isomorphic.
Proof. The given reduction modulo nidentifies oppositely signed colors modulo
n.
Chart ρ13(1), depicted in Figure 6 (where a superposition of part of the {6,3}-
regular hexagonal tessellation Hwith its edges intersecting at 90 deg some of the
edges of ρ13(1) is shown in relation to Figure 7 below) exemplify the following
properties, which follow by combining the images of the subgraphs 1[011], 2[011],
3[112] under the isomorphisms ρn(1) →ρn(i):
1. Chart ρn(1) is representable in a plane triangle T(n, 1) whose sides are SAs
of the subgraph 1[011] ⊂G∞,4, namely two SAs of type (2) and one of type
(1), as in Subsection 8.1.
2. The internal angle between the SAs of type (2) is 60◦. The internal angles
between each of these and the SA of type (1) are 30◦and 90◦. The angle
of 30◦has its vertex at the center vof the CH 1.000 so ρn(1) is represented
as a twelfth part of the total angle of 360◦at v. The angle of 90◦has its
vertex at 0jj1jj, where j= (n−1)/2.
3. There is only one maximal path Ln,1of ρn(1) passing through 0jj1jj with
its edges having color jand cutting the opposite side of T(n, 1) at 90◦on
a thick edge.
4. The angle of 60◦has its vertex at the center of the CT 1hhh, where h=
(n−5)/2.
Proposition 21. The diameter of Gn,4is both Ω(n)and O(|V(Gn,4)|1/3), so that
the asymptotic diameter of Gn,4is |V(Gn,4)|1/3.
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750 I.J. Dejter
Figure 6. Superposition of drawings for σn(1) and τn(1).
Proof. First, we claim that |V(Gn,3)|=O(nφ(n)), where φ(n) = Euler char-
acteristic of n. Every aa0, where gcd(a, n) = 1, belongs to Gn,3. Thus, there
are ⌊φ(n)/2−1⌋paths whose ends are 011 and 0aa, with 0 < a ≤ ⌊n/2⌋and
gcd(a, n) = 1. But the distance from 0aa to 011 in Gn,3is no more than a, yield-
ing our claim. If we fix a K3-type of abcdef ∈Gn,4, say abc, then for each color
dmodulo nthere are at most two different values for ebut a unique value for f.
This way, there are at most nφ(n)(2⌊n/2⌋) different K4-types modulo n. Thus,
|V(Gn,4)|=O(n2φ(n)). Let us see now that the diameter of Gn,4is Ω(n). A
path of length n+ 1 between 110110 and 112(n−1)nn happens along the image
of L(1,2,2). Thus, the diameter of Gn,4is both Ω(n) and O(|V(Gn,4)|1/3).
A representation of the charts of G′
n,4leading to the connectedness of G′
n,4
for nlarge is introduced. Let σn(1) be the restriction of ρn(1) induced by the
rainbow K4-types. We superpose the T-subgraph representation of σn(1) with a
{6,3}-regular hexagonal tessellation H=τn(1) ([7], page 43) such that:
(a) each edge ǫof σn(1) is traversed by an edge ǫ′of τn(1) at 90◦at the
common midpoint of ǫand ǫ′;
(b) each CH of σn(1) contains in its interior a regular hexagon of τn(1). Figure
6 contains a superposition of a representation of σ13(1), with the two rainbow
K4-types indicated as bullets •and the part of τ13(1) used to represent σ13(1) in
Figure 7.
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Rainbow Tetrahedra in Cayley Graphs 751
Figure 7. The representations τn(1), for n= 13,...,25.
In Figure 7, representing τn(1) for odd n= 13,...,25, each rainbow K4-
type of σn(1) is given by an hexagon of τn(1) tagged by a positive integer, as
suggested in Figure 6 for n= 13 by the indicated superposition. Each tagged
hexagon representing a vertex of σn(1) is the intersection of two tagged-hexagon
sequences in τn(1). There are three directions of parallelism for existing tagged-
hexagon sequences: one horizontal and the other two at angles of ±60◦from
the horizontal. Each such sequence is headed on the boundary of τn(1) by a
partially-drawn thick-trace hexagon tagged by a pair of integers. Assume the
integer tagging an hexagon ζof τn(1) is iand the integer pairs heading its two
tagged-hexagon sequences are (p, q) and (r, s). Then the K3-types composing ζ
are: 1pq, 1rs and either ipr and iqs or ips and iqr. Here, an hexagon is tagged
with a bullet •instead of an integer if it represents a non-rainbow K4-type. Each
remaining (non-tagged) 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 tagging 2 at the lower-left corner of τn(1) (that is the K4-type
134265);
(2) the hexagon tagged by ⌊n/2⌋, at the lower-right corner of τn(1) (that is
the K4-type 123k(k−2)(k−1), where n= 2k+ 1).
If n6= 0(mod 3) then these are the only two isolated vertices of σn(1). Oth-
erwise, there is exactly one more isolated vertex in σn(1) and this is determined
by the hexagon tagged by n/3 at the upper-right corner of τn(1) (that is the
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752 I.J. Dejter
K4-type 1(k−2)(k−1)k(k+ 1)(k+ 2)).
For n≥17, the isolated vertices of σn(1) are nonisolated in the remaining
charts σn(i), where i6= 1 ranges over the units modulo nfrom 2 to ⌊n/2⌋. This
suggests the following conjecture.
Conjecture 22. G′
n,4is a connected graph, for n≥17.
The six charts τ13(i), for i= 1,...,6, represent the same pair of isolated
vertices shown in Figure 2(c1) and 2(c2), which are thus the only components of
G′
13,4. In addition, the four charts τ15(i), for i= 1,2,4,7, represent only a CT
and four isolated vertices.
10. Proofs of the Main Results
Proof of Theorem 2.By Proposition 21, the asymptotic diameter of Gn,4is
|V(Gn,4)|1/3. The vertices v∈V6in any member G=Gn,4of G1are the rainbow
K4-types in G. The four K3-types of each such rainbow K4-type form three
distinct pairs of K3-types, each corresponding to a respective triangle of G. This
yields three triangles T0, T1, T2almost always distinct as in the statement, so
that each pair {Ti, Tj}with i6=jdetermines two different butterflies at vand
respective charts D0
i,j and D1
i,j. Let S⊆V6be composed by these vertices v.
Clearly, |S|is asymptotically |V6|. Now, V(G)\V6has its vertices at distance
no more than 2 both from the boundary of charts τn(i) and from the diagonal
paths η(i) in them, with these paths departing from boundary vertices realizing
angles of 90◦as in the upper right representation in Figure 5 and as in Figure
6. This insures that |V(G)\V6|grows linearly as nincreases, while |V6|has a
quadratic growth with respect to n, so V6has asymptotic order |V(G)|. Each
of the four K3-types composing the K4-type associated with a vertex of Soffers
three positive integers that color the edges of a corresponding chart modeled on
Has in [4], Theorem 2. Each of these three integers colors the edges of a parallel
class of edges in that chart. These completes the proof of Theorem 2.
Proof of Theorem 3.Let G′
1⊂ G1be formed by the Gn,4with nan odd prime.
Then, the charts τn(i) are pairwise isomorphic. They are related with the graphs
Dk
i,j as follows, for i= 1,...,n
2. Each τn(i) has two components formed by
vertices representing rainbow K4-types. These components are: (a) contained
in a 30◦-60◦-90◦triangle R(formed by the three delimiting SAs); (b) separated
by the path η(i) in τn(i). The union of the two 30◦-60◦-90◦triangles delimited
by the SAs and η(i) yields τn(i). By Corollary 20, there are ⌊n/2⌋charts τn(i).
We consider stripping bands of the delimiting SAs in the 30◦-60◦-90◦triangles in
order to get rid of loops. This reduces the resulting (n−1) 30◦-60◦-90◦triangles.
The stripped triangles are split into two halves by the paths η(i), each half leading
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Rainbow Tetrahedra in Cayley Graphs 753
to isomorphic D-modeled subgraphs, with the vertex numbers in the two halves,
for y≥1, equal to: |V′−
y|= 2 Py
i=1 iand |V′+
y|=−2 + 6 Py
i=1 i, if k= 5 + 2y;
respectively, |U′−
y|=|V′−
y| − yand |U′−
y|=|V′+
y| − 3y, if k= 4 + 2y. By removing
from U′±
y(respectively, V′±
y) the isolated vertices in lower-left (respectively, lower-
; upper-right) corners in the τn(1) in Figure 7, tagged 2 (respectively, k;n/3 if
n≡0(mod 3)), a maximal connected D-modeled subgraph U±
y(respectively, V±
y)
is obtained.
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subgraphs in edge-colorings with local constraints, J. Random Structures Algorithms
23 (2003) 409–433.
doi:10.1002/rsa.10102
[3] J. Bar´at and I.M. Wanless, Rainbow matchings and transversals, Australas. J. Com-
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Received 6 November 2014
Revised 27 February 2015
Accepted 27 February 2015
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