odd number, we count the number of different colored triangles in X.. 1. .... The first graph having scalene-color-triangles for n different from a prime number is ...
On Unitary Cayley Graphs Italo J. Dejter University of Puerbo Rico Department of Mathematics Rio Piedras PR 00931 Reinaldo E. Giudici
Universidad Sim6n Bolivar Departamento de Matem6ticas Caracas, Venezuela
AgsrRAcr. We deal with
a family of undirected Cayley graphs
X* which are unions of disjoint Hamilton
cycles, and some of It is
their properties, where n runs over the positive integers.
proved that X* is a bipartite graph when n is even. If n is a"n odd number, we count the number of different colored triangles
in X..
1. Introduction The graphs considered in this paper are undirected, simple and without loops. Given a graph X"r, we denote its vertex set and edge set by V(X) and E(X), respectively. Given a positive integer ro and an element r of the additive cyclic group zn of integer residue classes mod 2, there is only one integer representative y of c such that 0 < y < n; we use the same symbol c to denote gr and define lrl :, if a < nl2 and lcl : rt - a if t > nf2The group Z* possesses the subset Un : U(Zn) of units modulo n, constituted by those t in Zn with integer representatives relatively prime to n. It is known that [/, is a multiplicative group ' Let Wn: UnlZzbe given by the classes {*, -r}, where z varies in U'. Notice thatWn inherites a group structure ftom (Jn. We represent each {r, -t} € Wn again by t, where 0 < r ( [n/2], unless confusion arises. We deal with a class {X'} of undirected Cayley graphs X', that we call "Unitary Ctyley graphs", defined as follows: V(X*) : Zn and any two vertices ?.r1 and a2 are adjacent if and only if lr, - ozl e [/'. Each edge
JCMCC
l8
(1995), pp.
t2t-t24
of Xn is attributed a color from the ,"t {J, 2, ...,k}, where k : [n/Z] according to the following rule: If e is an edge with e'nivertices u1 a,'d. ai and if o1 *i (mod ?), i e {!,2,... , lc},-then ." it" edge e witt Laz: i' Accordingtothe definition of xn, we have that i €"otor [/,n. Therefore, x, i1tng_ylairected Cayley graph of Zn with generator v':t, Wn. Written Caj l!",w^l,hence, we may say that x, has an edge coloring ,uith o.ru hatf oj the unit element of Zn. For intance, if n. : g th; U* : g} and Wn: {1,2,4}. Thus, Xs: Cay [Zg,Wsl is represented {'7,2,;:;,2, in the Figure 1.1. In [1] Dejter dealt with Cayley graphs of the form Cay where 1, is the generator s9J {1,2,...,i}, n is odd and ft : [Zn,In], jlt2, i.e., 6,': the complete graphs 1(* edge-colo."d i., a symmetric fashion, and studied some induced subgraphs K" of these cayley graphs that were called totally multicolored (TMC) subgraphs, motivated by a questio' of E.dO", pyber and T\rza [2].
65 Figure 1.1 In the present paper, we deal with some properties of the X,". In partic_ ular, besides proving that if , is even, then x, is bipartite wL count the number of different colored triangles of Xn, when n is odd.
2. Basic Properties of Unitary Cayley Graphs we now state some properties of the unitary cayley graphs. The
are easy and left to the reader.
X, is a regular graph Euler g-tunction, and is a union of 66) any positive integer n. Proposition 2.L.
of
degree
d(n),
u"iitt"n'iyiin
proofs
g(n) is the or teigtn n, for
where
Proposition 2.2. Xn is isomorphic to a complete graph with n ver_ ticest il n is,a prime number and. isomorphic to a comprete bipafiite graph K(2t-t,2t-r) if n-2t. 122
The graphs X* for n' :2t are interesting from the chromatic point of view. They are regular of degree one half of the number of vertices and the number of edges turns out to be the square ofthe degree ofthe graph. These graphs are a special type studied in [3]; they are chromatically unique. Unitary Cayley graphs have different properties according to the parity of n, as will be seen in the next proposition.
Proposition 2.3. Xn is a bipartite
graph
if n is
ant
even number.
of Znrnust be odd. Thus, no two even labeled vertices are adjacent. This implies that the even labeled vertices and the tr odd labeled vertices form a bipartition of the vertex set.
Proof:
Since n is even, units
Corollary 2.3.1. If n is even, then Xn particular, Xn is triangle-free.
has no odd length cycles. In
3. Number of tiangles in X. since X* is triangle-free for n even, we will
consider ru to be an odd number in this section. Let us denote by {o, b,c} a triangle in X' with vertices a, b and c;therefore t(b-a), *(c-b), +(a-c) are elements inUn' Without loss of generality we may assume that our triangles have vertices {0, 1,u}, u € (Jn.If we denote by ?or the set of all the triangles having the common vertices 0 and 1 i.e., Tor : {{0, 1,r} I u e Un}, then the cardinality procedures it lTorl : {u e Un I (" - t) € U^}. Using elementary counting can be proved that
l?orl:
"y^(t -:)
To count the number of triangles in x, let us consider the action of the group G : (In x Zn on the set of triangles of X' i.e., if. (u,a) e G, then (u,z){0,1, u}: {ur,o(1+r), u(u*r)}' Each orbit of the triangles corresponding to the pair (u, r) may have at most six different elements (0,1,;), (0,tl--r,1), (1 -r)-1,0,1), (1,(r^,,- 1)u-1,0), (u(u- 1)-t,1,0) in Tor. It can happen that some orbits have exactly 3,2 or even 1 elements. We have l"orl : Dotedha, where h6 is the number of orbits with exactly d elements in Tor. fhe orbits with d difigr.ent elements have a number of : d(nli(nt. Thus, if ? stands for the triangles given by lohdl : W number of triangles of X', then
": D
haloal:
h6nQ(n) +
h"?9P + hrryP *
nrgP,
d/6
r:
n4@) (ne +
+.+.+) :d@)Y:nd@)*, r23
where dz(n) : nDp/n (t - *) by its resemblance to the Euler ffunction. Note that $2(n) is the number of consecutive units modulo n. Finally, if we denote by IT the number of triangles in X, that have two sides painted with equal colors, let us call these "isosceles-color-tria.ngles,' and denote by ET the corresponding number of triangles with their three sides painted in different colors, called totally multicolored triangles. With obvious meaning of "scalene-color-triangles" we obtain the following result.
Proposition 3.1. If X^
is a Unitary Cayley graph with
i t ')
n an odd number
then
rc:?gp
ad
nr:n6!n)
(dz(n) -s).
Proof:
Since n, is odd {0, 1, -1} is a isosceles color triangle; its orbit under the group action gives all the others. It can easily be verified that d:3. Thus the formula for IT follows. For the scalene-color-triangles let us observe that there is no "equilateralcolor-triangles", i.e. triangles with its three sides painted with the same color. Therefore, ET : T - IT. D
Remark: Xg has 27 triangles and all of them
are isosceles-color-triangles.
The first graph having scalene-color-triangles for n different from a prime number is Xzt. It has 126 isosceles-color-triangles, 84 scalene-color-triangles and a total of270 triangles. The authors are grateful for the suggestions ofthe referee which shortened and improved the paper and for the help given by Professor P. Berrizbeitia and the corrections of English given by our colleague Professor T. Berry. Thanks to all of them.
References [1] I.J. Dejter, Totally Multicolored, Subgraphs of Complete Cayley Graphs, Congressus Num. 7O (1990). pp. 53-64. [2] P. Erd
