EDGE DOMATIC NUMBERS OF COMPLETE N-PARTITE GRAPHS

Graphs and Combinatorics (1994) 10:241-248

Graphs and

Combinatorics

Edge Domatic Numbers of Complete n-Partite Graphs*

i Department of Information Engineering, Feng Chia University, Taichung 40724, Taiwan 2 Institute of Applied Mathematics, National Chiao Tung University, Hsinchu 30050, Taiwan.

Abstract. An edge dominating set of a graph is a set of edges D such that every edge not in D is adjacent to an edge in D. An edge domatic partition of a graph G = (V, E) is a collection of pairwise disjoint edge dominating sets of G whose union is E. The maximum size of an edge domatic partition of G is called the edge domatic number of G. In this paper we study the edge domatic numbers of complete n-partite graphs. In particular, we give exact values for the edge domatic numbers of complete 3-partite graphs and balanced complete n-partite graphs with odd n.

1. Introduction

In this paper all graphs are simple, i.e., finite, undirected, loopless, and without multiple edges. An edge dominating set of a graph is a set of edges D such that every edge not in D is adjacent to an edge in D. An edge domatic partition of a graph G = (V, E) is a collection of pairwise disjoint edge dominating sets of G whose union is E. The edge domatic number problem is to determine the edge domatic number ed(G) of G, which is the maximum size of an edge domatic partition of G. Zelinka [8] showed that tS(G) _< ed(G) < tSe(G ) + 1 where cS(G) is the minimum degree of a vertex in G and re(G) is the minimum degree of an edge in G. He also determined the values ofed(G) when G are circuits, complete graphs, complete bipartite graphs, and trees. Algorithmic results on domatic numbers are also extensively studied in [1,2,3,5,6].

The purpose of this paper is to study the edge domatic number of a complete n-partite graph K I , ... whose parts are M 1 , . . . , M, of size ml . . . m,, respectively. F o r simplicity, we assume mt < ... < m,. F o r 1 < i < j < t, we denote by Eij the set of all edges between Mi and Mj, i.e.,

Eij=

It is well k n o w n that ed(K . . . . 2) = m2" In general, the exact value of the edge domatic n u m b e r of a complete n-partite graph with n > 3 is not easy to find. This

* Supported in part by the National Science Council of the Republic of China under grant NSC79-0208-M009-31.

242 Shiow-Fen Hwang, G.J. Chang

p a p e r gives exact values for the edge domatic numbers of complete 3-partite graphs and balanced complete n-partite graphs with odd n.

2. Edge Domatic Numbers of Complete 3-Partite Graphs

This section determines the exact value of the edge domatic n u m b e r ed(Kmz,m2,,n3)

of a complete 3-patite graph Km,.m2,m3 with rn~ < m 2 < m 3.

In a graph G, an edge set A is said to cover a vertex v i times if v is incident to

at least i edges in A. An edge set A is said to cover a vertex set B if every vertex in

B is incident to some edge in A. N o t e that an edge set D is an edge dominating set of Kml ... if and only i f D covers U My for some i.

First of all, we shall establish the lower b o u n d of ed(Km~,m~,m~). T o do this, we

want to partition E(Km,m~,m) into as m a n y edge dominating sets as possible. F o r

convenience, we need the following notation. Suppose r, s, t are integers a n d

1 < t < m~mJd~y where d~y is the greatest c o m m o n divisor of m~ and my. Denote b y Eiy(r,s,t) the set { ( a m o d m i , (a + r ) m o d m i) ~ Eo: a = s + 1,s + 2 . . . s + t} and E~y(r) = Eiy(r, O, rn~mJd~y), where "x m o d y " always results a positive integer between

1 a n d y.

L e m m a 2.1. [Eij(r, s, t) l = t. Also, Eiy(r, s, mi) covers Mi and Eiy(r, s, my) covers M i U My. Proof. Suppose (a rood m/, (a + r) rood rn~) = (b m o d m/, (b + r) m o d %) for some

s + 1 < a _< b _< s + t. Then b - a is a c o m m o n multiple of rn~ and m r. H o w e v e r

0 < b - a <_ t - 1 < m~my/diy where m~my/d~y is the least c o m m o n multiple ofm~ and

%. So a = b and hence IEiy(r,s,t)[ = t.

Eiy(r, s, mi) covers Mi since a m o d m/ranges over { 1, 2 . . . mi} when a ranges over

{s + 1, s + 2, : . . , s + m~}. Similar arguments and the fact that m~ < my imply that

Eiy(r, s, n~) covers M i

[.J

L e m m a 2.2. E o can be partitioned into Eij(r), r = 1, 2, ..., di~.

Proof. Suppose there exist 1 _< r < s < dij such that E~j(r)fq Eo(s ) ~ ~, i.e., a - b

(rood mi) and a -b r =- b + s (mod my) for some 1 < a, b < raimJd O. Since diy is a

c o m m o n divisor of m~ and my, a - b (mod dij) a n d a + r - b + s (mod d~y). These imply that r = s (mod d~y), in contradiction to 1 < r < s < d~j. [ ]

L e m m a 2.3. ed(K,,l,m2,m3) > f m l + 2m2 + [m2(ml - m2)/msJ,

- [ m l + m3 + [ml(m2 - ml)/msJ-

Proof. By L e m m a 2.2, we can partition E12 into E12(r ), 1 < r _< d12. By L e m m a 2.1,

we can further partition each Ela(r) into E12(r, (i - 1)m 1, ml), 1 < i < mz/d12, each

of size m~ and covering M~ but covering only m~ vertices of M 2. F o r simplicity, we denote these mE sets by Ax . . . A , , . The idea is for e a c h j to find a subset Bj of E23 such that IBjl = m2 - ml and

AjUBj

Edge Domatic Numbers of Complete n-Partite Graphs 243

fl is a multiple of m 2 . Let B = g E23 (r) U E 23 (ct q- 1, 0, fl). Then B is of size m 2 (m 2 - - m 1) r = l

Ftl 2

a n d covers each vertex o f M 2 exactly m 2 -- m 1 times. Since El2 = U Aj covers each

j = l

vertex o f M 2 exactly m 1 times a n d each Aj covers each vertex of M 2 at m o s t once, we c a n p a r t i t i o n B into B~ . . . Bm~ each o f size m 2 - - m 1 such that A~U Bj covers M 2 for 1 _<j < m 2. T h e n each A j U B j covers M i U M 2 , i.e., E 1 E U B can be parti- tioned into m 2 edge d o m i n a t i n g sets of K,I.,~,,~.

F o r the first lower b o u n d , we shall partition the remaining edges into d o m i n a t i n g d2a

s e t s o f s i z e m a . N o t e t h a t E 2 3 - B = E23(ct + 1,fl, m2m3/d23 - fl)U U E2a(r) a n d

r = ~ t + 2 dla

El3 = U Ela(r)" By L e m m a 2.1, E2a(Ct + 1,fl, m2m3/d2a - fl) can be partitioned r = l

into L ( m 2 m a / d 2 3 - fl)/maJ = L m 2 / d 2 3 - fl/maJ edge d o m i n a t i n g sets a n d each E 2 3 ( r ) (resp. Eia(r)) c a n be p a r t i t i o n e d into m2/d23 (resp. m l / d l 3 ) edge d o m i n a t i n g sets. H e n c e E ( K . . . . ~.,,~) c a n be p a r t i t i o n e d into m 2 q- L m 2 / d 2 3 - fl/maJ + (d2a - ot - 1)m2/d23 + d l a m l / d l 3 = m 1 + 2m 2 + [-o~m2/d23 - fl/maJ = m 1 + 2m 2 +

I.mz(m 1 - m2)/ra3J edge d o m i n a t i n g sets. These give the first lower b o u n d in the lemma.

F o r the s e c o n d lower b o u n d , b y L e m m a 2.1, E23 -- B = E23(g + 1,fl, m2m3/ d2a

d2a - fl)O U E23(r ) can be p a r t i t i o n e d into (m2ma/d23 - fl)/m 2 + ( d 2 3 - g -

r = a + 2

1)(m2ma/d23)/m2 = m a - m2 + mx sets Cx . . . . , C,,3-r~+m~ each of size m 2 a n d

coveting M2. Suppose m l ( m a - m 2 + m l ) = 2(mlma/di3 ) + # where 0 < # < reims~

l

d~ 3. N o t e t h a t # is a multiple o f m~. Let D = U E13(/') U g 13 (2 -~- 1, 0 , # ) . Then, by I"=1

L e m m a 2.1, D c a n be partitioned into 2 ( m l m 3 / d i a ) / m 1 + #/m i = m a - m 2 q- m i sets D 1 . . . . , Dm3-m2+m~ each of size m~ a n d covering M 1. T h e n each CiUDz, 1 _< i _< m a -- m 2 + m l , covers M 2 U M i . S o (E23 - B) U D can be partitioned into m3 - m2 - k m i edge d o m i n a t i n g sets. Finally, by L e m m a 2.1, E13 - D = E13(2 + 1,

dxa

lz, m l m 3 / d l 3 - #)U ~ Ela(r) can be partitioned into I.(mlm3/dl3 - # ) / m 3 J +

r =,;.+2

( d i s - 2 - 1)(mima/d13)/m a = [ m l ( m 2 - ml)/maJ edge d o m i n a t i n g sets. Hence

e d ( K m t , m 2 , m 3 ) >_ m 2 -b (m 3 - - m 2 -I.- / ' h i ) + kmx(m2 - mi)/maJ = m i + m 3 --k

Lmi (m 2 - mx)/maJ. These give the s e c o n d lower b o u n d of the theorem. [ ]

N o t e t h a t for the two lower b o u n d s in L e m m a 2.3, ml + 2m2 -t- Lm2(mi - m2)/ m3] < ml + m3 + Lml(m2 - ml)/rn3J if a n d only if or mi + m2 < m3.

W e n o w give a n example to d e m o n s t r a t e L e m m a 2.3, as follows. C o n s i d e r K . . . . 2,m3 with m 1 = 3, m 2 = 8 a n d m 3 = 12. Let M 1 = {1,2,3}, M 2 = {1',2', . . . . 8'} a n d M 3 = { 1", 2 " , . . . , 12"}. I n Fig. 2.1 a n d Fig. 2.2, an entry (a, b) represents an edge (a, b) of K . . . . 2.,~3 a n d entry (a, b) is n u m b e r e d by i when edge (a, b) is in the i-th edge d o m i n a t i n g set X~.

I n Fig. 2.1, E12 is partitioned i n t o A 1 . . . A a, each of size 3 a n d covering M 1 but c o v e t i n g o n l y 3 vertices of M2. G r a y entries in E23 f o r m the set B of size 40. B is partitioned into B 1 . . . B s such t h a t X 1 = A~ U B 1 . . . X s = A a U B s are edge

244 Shiow-Fen Hwang, G.J. Chang

Fig. 2.1. An edge domatic partition of K s . s , 12, which has 15 edge dominating sets

Fig. 2.2. An edge domatic partition of K 3 . s , 12, which has 16 edge dominating sets dominating sets. E 2 3 - - B is then partitioned into 4 edge dominating sets X 9 . . . X12 with 8 edges unused. Finally, Ea3 is partitioned into 3 edge dominating sets Xx3, Xa4 and X~5. So, there are a total of ml + 2m2 + [.m2(ml -

rn2)/m3J

In Fig. 2.2, we also partition Ea2 into A ~ , . . . , A 8 and B into B 1 .. . . . B 8 to form 8 edge dominating sets X 1 = A a t.J Bx . . . X 8 = A 8 t.J B a. Then, E23 -- B is parti- tioned into 7 sets Ca . . . C7, each of size 8 and covering M2. White entries in E13 f o r m the set D of size 24. D is partitioned into 7 sets D 1 . . . D 7, each of size 3 and

Edge Domatic Numbers of Complete n-Partite Graphs 245 c o v e r i n g M1, w i t h 3 edges unused. So, X 9 = C1 U D 1 . . . X~5 = C7 U D7 a r e 7 edge d o m i n a t i n g sets. F i n a l l y , E l 3 - D is p a r t i t i o n e d into 1 edge d o m i n a t i n g set X~6. So t h e r e a r e a t o t a l of m 1 + m 3 q- Lm~(m2 - m~)/m3J = 16 edge d o m i n a t i n g sets a l t o g e t h e r .

F o r t h e u p p e r b o u n d o f e d ( K . . . . ~,m~), a s s u m e P is a n edge d o m a t i c p a r t i t i o n o f K . . . . ,,m3 a n d x i = I{D e P : [DI = i}l for e a c h i. N o t e t h a t P c o n t a i n s ~ xi edge

i

d o m i n a t i n g sets o f Kml,m2,m~. Since the size o f each edge d o m i n a t i n g set o f Km,,~2,m3 is a t l e a s t m2, we h a v e

2 ixi = m l m 2 "-F mira 3 q" m2m 3 (2.1)

i ~ m 2

L e m m a 2.4. ~ (m i + m 2 - - i ) x i <_ m i r a 2 where c~ = m i n { m 1 + m 2 - - 1 , m 3 - - 1}.

i = m 2

Proof. L e t D be a n y edge d o m i n a t i n g set of K~l,m2,m 3 with m 2 _ ID[ ---- i _< C(. I t suffices to p r o v e t h a t D f] E12 h a s at least mx + m 2 - - i edges. S u p p o s e not,

i.e., D f') E l 2 has a t m o s t ml + m2 - - i - 1 edges a n d so it c o v e r s at m o s t ml + m2 - -

i - - 1 vertices o f M1. Since IOl < ~ < m3 - 1, D c a n n o t c o v e r M 3. T h e n , since D is a n e d g e d o m i n a t i n g s e t o f K . . . . 2,m3, D c o v e r s M~ U M 2. Therefore, D fl E~ 3 c o v e r s a t l e a s t i + 1 - m 2 vertices o f M i a n d so ID f] E131 _> i d- 1 -- m 2. H o w e v e r , since D c o v e r s M2, IO t3 ( E l 2 U E23)t ~ 19l 2. H e n c e IO] = ID f] E13l + IO t] ( E l 2 IJ E23)1 >_ i + 1, in c o n t r a d i c t s to IDI = i. H e n c e ~ (ml + m2 - - i ) x l <-- mira2. []

i = r a 2

L e m m a 2.5. ~ xi < m l + m3

m 2 < i < m 3

Proof. L e t D be a n y e d g e d o m i n a t i n g set o f K . . . ~ w i t h m 2 _< ]D[ : i < m 3. Since D c a n n o t c o v e r M3, D covers M~ U M 2. By the fact t h a t D c o v e r s M 2 , [D f') ( E l 2 I J E23)]

~_ m 2. T h e r e f o r e , m 2 Z xi ~- m2(rrtl q- m3), i.e., ~ x i _< m 1 + m 3. [ ]

r a 2 < i < r a 3 m 2 < i < r a 3

ml + 2m2 + Lm2(ml - m2)/raaJ i f m i + m2 >- m3, L e m m a 2.6. e d ( K . . . . 2,m3) <

- mi + ma + Lmi(m2 - m i ) / m a J i f m l + m 2 <_ m 3.

Proof. I n the case o f m 1 + m 2 >_ ma, t h e c~ in L c m m a 2.4 is e q u a l to m 3 - 1. By

m u l t i p l y i n g the i n e q u a l i t y o f L c m m a 2.4 b y (m a - m2)/rn i a n d a d d i n g it to (2.1), we h a v e

Y, (i + (m~ + m~ - i)(m~ - m~)/m~)x, + Y, ix,

ra2 < i <ra 3 i>_m 3 <_ m3(m i + 2m2) + m2(mi -- m2) F o r m 2 _ i < m a, i + (m i + m 2 - - i)(m a -- m2)/m 1 = m 3 -}- (i - m2)(m 1 q- m 2 -- m3) / m i _> m 3. Therefore, m 3 ~ xl < m3(ml + 2m2) -I- m 2 ( m I - - m 2 ) a n d so e d ( K . . . . 2.m3) i>_ ra 2 < ~ xl < ml + 2m2 + Lm2(ml - mE)/maJ. i>_m 2 I n t h e case o f m 1 + m E _< m 3, ~ = ml + m2 - - 1. By m u l t i p l y i n g the i n e q u a l i t y in L e m m a 2.5 b y m 3 - m l - m E a n d a d d i n g it to (2.1) a n d the i n e q u a l i t y in L e m m a

246

2.4, we have

m3xi +

rn2 <_ i < m ! + m 2

Shiow-Fen Hwang, G.J. Chang

E (i + m 3 -- m 2 + m i ) x i + ~ ixi

m I + m 2 < i < m 3 i>_m 3

__< m3(m 1 + m3) + mi(m2 -- mt).

N o t e that each coefficient of x~ in the left hand side is greater than or equal to m 3. Then ed(Kml . . . 3) <- ~, xi <- ml + m3 + Lml(m2 - ml)/maJ. []

i>_m 2

Theorem 2.7. ed(Kml m 2 m 3 ) = ~ ml + 2m2 + Lm2(mt - m2)/maJ i f mr + m2 >- m3, ' ' /.ml + m 3 + Lml(m2 - ml)/m3J i f m l + m 2 <-- m 3 .

Proof. This theorem follows directly from Lemmas 2.3 and 2.6. [ ]

3. Balanced Complete n-Partite Graphs

N o w we consider the edge domatic number problem for the balanced complete n-partite graph K(r, n) =- K ... in which every part has exactly r vertices. Exact values for ed(K(r, n)) with odd n and ed(K(r, 4)) with even r are given in this section. N o t e that ed(K(r, 2)) = r. So we only consider ed(K(r, n)) for n > 3. Also, K(1, n) = K , and it was showed in [8] that ed(K,) = n for odd n > 3 a n d ed(K,) = n - 1 for even n.

Let G (k) = (V (k), E ~k)) be the graph obtained from a graph G = (V, E) by dupli- cating each vertex k times, i.e.,

and

v r = {Vl . . . v~: v ~ v }

E (k) = {(ui, vj): (u, v) e E and 1 < i,j < k}.

L e m m a 3.1. ed(G (kJ) _> k ed(G)

P r o o f Let P = {D 1 .. . . . Ded(O)} be an edge domatic partition of G. It suffices to prove

that G(k).has k ed(G) pairwise disjoint edge dominating sets. F o r every Dq e P, we construct k edge dominating sets of G (k) as follows:

D ~ j = { ( u i , v~+j):(u,v)eD~,l < i < k } , l <_j<_k,

where index of v~§ is taken modulo k. It is straightforward to check that each Dqj is an edge dominating set of G tk), and so G (k) has ked(G) pairwise disjoint edge

dominating sets. [ ]

Theorem 3.2. ed(K(r,n)) < r2n(n - 1)/2[r(n - 1)/2] < m for n > 3.

Proof. Since every edge dominating set of K(r, n) must cover at least n - 1 parts of K(r, n), every edge dominating set of K(r, n) has at least r r ( n - 1)/21 edges. The

Edge Domatic Numbers of Complete n-Partite Graphs 247

A l t h o u g h we believe that the u p p e r b o u n d r2n(n - 1)/2 [r(n - 1)/2] in T h e o r e m 3.2 is the exact value o f e d ( K ( r , n)) for n > 3, only two cases have been settled: those where n is o d d a n d r is even with n = 4.

Theorem 3.3. ed(K(r, n)) = rn i f n >_ 3 a n d n is odd.

P r o o f . ed(K(r, n)) < rn by T h e o r e m 3.2. O n the o t h e r hand, b y [4], e d ( K , ) = n for o d d n. By L e m m a 3.1, e d ( K ( r , n ) ) = e d ( K ~ )) > r e d ( K , ) = rn. So e d ( K ( r , n ) ) = rn.

[ ] By [4], e d ( K , ) = n - 1 w h e n n is even. Again, b y L e m m a 3.1, e d ( K ( r , n ) ) =

ed(Kt, ")) > r e d ( K , ) = r(n - 1). T h e r e is a g a p between the lower b o u n d r(n - 1) a n d the u p p e r b o u n d in T h e o r e m 3.2. W e n o w consider ed(K(r, 4)) w h e n r is even.

Theorem 3.4. ed(K(r, 4)) = 4r i f r is even.

P r o o f . F o r the case of r = 2, let the vertex set of K(2, 4) be {0 (~ 1(i): i = 1, 2, 3, 4} a n d the edge set of K ( 2 , 4 ) be {(x(i),y~ x , y ~ {0, 1}, 1 < i ~ j < 4}. T h e n ed(K(2,4)) = 8, as it is s t r a i g h t f o r w a r d to check t h a t the following sets A 1 . . . A s are pairwise disjoint edge d o m i n a t i n g sets of K(2, 4):

A1---

{(0(1),0(2)),(!(2),0(3)),(1(3) , 1(1))},

{(0(2),0(3)),(1(3),0(')),(1 (4), 1(2))},

{(1TM, 1(4)),(0(~),0m),(1(1),0(3))},

{(O(*),lm),(Om, l(Z)),(O(Z),l(4))},

{(1 m, l(Z)),(O (2),

1(3)),(0(3),0(1))},

{(1 (2), 1(3)),(0 (3), l(')),(O(*),O(Z))},

{(0(3),0(4)),(1 (*), 1(1)),(0 m, 1(3))},

{(1(*),0m),(1(1),0(2)),(1(2),0(4))}.

In general, let r = 2s. T h e n K ( r , 4) = K ( 2 s , 4) = K(2, 4) (s). By L e m m a 3.1, ed(K(r, 4)) > s e d ( K ( 2 , 4 ) ) > 8s = 4r. O n the o t h e r hand, e d ( K ( r , 4 ) ) < 4r b y T h e o r e m 3.2.

H e n c e ed(K(r, 4)) = 4r. [ ]

W e close this p a p e r by the following conjecture: ed(K(r, n)) = r2n(n - 1)/2 [r(n - 1)// 2] for a n y n > 3. T h e solutions to the d o m a t i c n u m b e r s of general c o m p l e t e n-partite g r a p h s are also desirable.

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