THE DOMATIC NUMBER PROBLEM

The domatic number problem*

Gerard J. Chang

Institute of Applied Mathematics, National Chiao Tung University, Hsinchu 30050, Taiwan, ROC

Received 12 July 1991 Revised 28 March 1992

Abstract

A dominating set of a graph G =( P’, E) is a subset D of Vsuch that every vertex not in D is adjacent to some vertex in D. The domatic number d(G) of G is the maximum positive integer k such that V can be partitioned into k pairwise disjoint dominating sets. The purpose of this paper is to study the domatic numbers of graphs that are obtained from small graphs by performing graph operations, such as union, join and Cartesian product.

1. Introduction

A dominating set of a graph G = (V, E) is a subset D of V such that every vertex not in D is adjacent to some vertex in D. The domatic number d(G) of a graph G = (V, E) is the maximum positive integer k such that V can be partitioned into k pairwise disjoint dominating sets D,, D2, . . . , Db A partition of V into pairwise disjoint dominating sets is called a domatic partition. The concept of a domatic number was introduced in [S]. The word ‘domatic’ was created from the words ‘dominating’ and ‘chromatic’ in the same way the word ‘smog’ was created from the words ‘smoke’ and ‘fog’. In a certain sense a domatic number is analogous to the chromatic number of a graph, which is the minimum positive integer k such that the vertex set can be partitioned into k pairwise disjoint stable sets.

Lower bounds, upper bounds and many propositions of domatic numbers were studied extensively in [336, 8-10, 12, 13, 15-231. In particular, in [S] it was proved that for any graph G there is a natural primal dual weak inequality

d(G) <6(G) + 1,

where 6(G) is the minimum degree of a vertex of G. Motivated from this, a graph G is called domaticully full if d(G)=&G)+ 1. For instance, the complete graph K, of

Correspondence CO: Gerard J. Chang, Institute of Applied Mathematics, National Chiao Tung University, Hsinchu 30050, Taiwan, ROC.

*Supported in part by the National Science Council of the Republic of China under grant NSC8(M208-M009-26.

0012-365X/94/$07.00 0 1994-Elsevier Science B.V. All rights reserved SSDI 0012-365X(92)00188-U

n vertices, the complement E, of K,, the cycle CJn of 3n vertices, trees and maximal outerplanat+ graphs are all domatically full.

On the algorithmic side, the domatic number problem is NP-complete for general graphs [7] and circular arc graphs [2]. The problem has been solved in 0(n2 log n)’ time for proper circular arc graphs [2], O(~Z’.~) time for interval graphs [l], and O(n log n) time for proper interval graphs [ 11, and has been improved by linear-time algorithms for interval graphs [ll, 141.

The purpose of this paper is to study the domatic numbers of graphs that are obtained from small graphs by performing graph operations, such as union, join and Cartesian product. In particular, Section 2 gives solutions to the domatic number of the union of two graphs and the domatic number of the join of two or more graphs. Section 3 gives partial results of the domatic number of the Cartesian product of paths.

2. Graph union and join

Suppose G1 =(V,E,) and G2 =(V,E,) are two graphs with disjoint vertex sets V, and Vz and disjoint edge sets El and E2. The union of G1 and G2 is the graph GluGz =(V,uV,, E1uE2). The join of G1 and G2 is the graph G1 +G2 that consists of GluG2 and all edges joining VI and V,.

Proposition 2.1. d(G1uG2)=min{d(G1), d(G,)) for any two graphs G1 and G2.

Proof. The proposition follows from the fact that D is a dominating set of GluG2 if and only if D is the union of a dominating set of G1 and a dominating set of G2. 0

A dominating vertex is a vertex which forms a dominating set, i.e. a vertex adjacent to all other vertices. If x is a dominating vertex of a nontrivial graph G, then G is isomorphic to (G - x) + K 1.

Proposition 2.2. If x is a dominating vertex of a graph G, then d(G)=d(G-x)+ 1.

Proof. Since a domatic partition of G-x together with {x} forms a domatic partition of G, d(G)>d(G-x)+ 1. On the other hand, suppose D1,D2, . . ..Dk is a domatic partition of G, where k=d(G). Assume XED~. Note that DIuD2-{x}, D3, . . ..Dk is a domatic partition of G-x. So d(G-x)ak-l=d(G)-1. Thus, d(G)=D(G-x)+ 1. 0

In the rest of this section, we give results for the domatic number of the join of graphs. By Proposition 2.2, from now on, we need only consider graphs without a dominating vertex. Let r be a positive integer greater than or equal to 2. If

Gl,Gz,...,

G, are

graphs

without

a dominating

vertex,

then

their

join

G1 +Gz+ ... +G, also has no dominating vertex. For the domatic number of this

join, there are two possible cases, which are solved in Theorem 2.3 and Corollary 2.6.

Theorem 2.3. Suppose ra2 and G1, Gz, . . . . G, are graphs with nI, n2, . . . . n, vertices,

respectively, and without a dominating vertex. If

1

<n2 < ... <

nl + ...+r~,_~>n,, thend(G,+G2+...+G,)=L(n,+n2+...+n,)/2J

Proof.

Since G1 +G2+ ... + G, has no dominating vertex, each dominating set con-

tains at least two vertices and so

On the other hand, we prove that G1 +G, + ... +G, has a domatic partition of

L(n,+n2+...+n,)/2Jd

ominating sets such that each dominating set has exactly two

vertices, except possibly one dominating set has three vertices. This assertion clearly

implies the theorem. We prove the assertion by induction on

Note that the following argument is valid even when some Gi has a dominating vertex.

The assertion is clearly true for

3. It is also true for

since

and any

vertex of G1 together with any vertex of G2 is a dominating set of G, + G2. Now,

suppose

and the assertion is true for

Choose a vertex x in G,_ i

and a vertex yin G,. Consider the graph G’=G1+~..+G,_2+(Gr_1-~)+(G,-y).

For the case of

<

have

n, < ... <n,_,<n,-1,

n,_l-l<n,-l

and

nl +

...

For the case of

have

n, 6...

<nr_3<n,_2,

n,_1-1=n*-1<n,_2

and

ni+ . ..+n.-,+(n,-,-l)+(n,-l)~n,-2.

To see the last inequality: when

=nr_2 22, the left-hand side

when

1,

implies that

and so the left-hand side >

n, _ 3 = 1 = n,_ 2.

In either case, by the induction hypothesis, G’ has a domatic partition

ofL(n,+n,+...+(n,-,-l)+(n,-1))/2Jd

ominating sets such that each dominating

set has exactly two vertices, except possibly one dominating set has three vertices.

These dominating sets together with {x, y} form the desired domatic partition of

G1+G2+...+G,.

0

For the case of

cannot get results similar to Theorem

2.3. To solve the problem for this case, we need a slightly more general concept, as

follows. For any nonnegative integer m, an

of a graph G =

is

a collection

of k pairwise disjoint dominating

sets such that

ID1uDzu... uDkl <m. The m-domatic number d(G 1 m) of G is the maximum k such that an m-domatic partition of k dominating sets exists. Note that d(G)=d(G\ n) for any graph G of n vertices.

Proposition 2.4. d(G ( m) d d(G I m’) f or any graph G and any nonnegative integers

m<m’.

Theorem 2.5. Suppose n1 < n2 and Gi = (Vi, Ei) is a graph

of

ni

ting vertex for i = 1,2. Then

d(G,+G,Im)=

Cm/21

nl+d(G21m-2n,) if 2n,<mdn,+nz.

Proof. For the case when 0 d m < 2nl, there exist L m/2 J pairs of vertices, each of them containing one vertex in Gi and the other in Gz. So, each such pair is a dominating set of G, + G2 and d(G, + G2 Irn)aL m/2]. On the other hand, since each Gi has no dominating vertex, neither does G, + Gz. Consequently, each dominating set is of a size of at least two, and so d(G1+G2/m)<Lm/2J Thus, d(G,+GZIm)=Lm/2J.

For the case when 2n,<m<nl+nz, first of all, choose an (m- 2n,)-domatic partition D1, D2, . . . , Dk of G2. These k dominating sets are also dominating sets of G1 +Gz. Note that Gz has at least n,-(m-2n,)>nl vertices not in D,uD2u...uDk. By an argument similar to that in the first paragraph, n, of these vertices together with the n, vertices of G, form n, dominating sets of G1 +Gz. Thus, d(G1 +G21m)>nl +d(G*/m-2nl). On the other hand, suppose D1,D,, . . ..D. is an m- domatic partition of G1 + G2, where r = d(G, + G2 I m). Note that each Di contains at least two vertices, since GI + G, has no dominating vertex. A dominating set is called standard if it contains exactly one vertex in VI and exactly one vertex in V2. We claim that among these r dominating sets, there are exactly n, standard ones and the other r-n, sets are all subsets of V, by considering the following cases.

(1) Suppose some Di contains at least one vertex x in VI and at least one vertex in Vz. We can replace Di by a standard dominating set

{x,

(2) Suppose some Di contains vertices only in V,, say x and y, and some Dj contains only vertices in V,, say z and w. We can replace Di and Dj by two standard dominating sets (x, z} and {y, w}.

(3) Suppose all nonstandard dominating sets are subsets of VI. Since n, dn2, we can replace each nonstandard dominating set by a standard one by taking a vertex from this set and a vertex of Vi, which is not in any Di.

(4) Suppose there is a vertex x of VI not in any Di. We can choose a vertex y in V2, which either is in some nonstandard dominating set DjC V2 or is not in any Di. In the former case, we can replace Dj by

{x,

The discussion of the above cases shows that the domatic partition has exactly ni standard dominating sets and r-n, nonstandard dominating sets that are subsets of V,. These r - n1 nonstandard dominating sets form an (m - 2n,)-domatic partition of

Gz. Therefore, d(GzIm-2n,)~r-n,, i.e. d(G1+GZIm)~n,+d(GzIm-2n,). Thus, d(G1+Gz(m)=n,+d(G,(m-2n,). 0

Corollary 2.6. Suppose r>2 and Cl, GZ, . . . . G, are graphs with nl, n2, . . . . n, vertices,

respectively, and without a dominating vertex. Zf n1 +nz + ... + n,_ 1 <n,, then d(G1 + Gz + . ..+G.)=n,+n2+...+n,_1+d(G,In,-nl-...-n,_,).

Proof. The corollary follows from Theorem 2.5 by considering G1 + . . . + G,_ I as Gr, G, as GZ, and m =nl+nz+...+n,. 0

Corollary 2.7. Zf r 3 2 and n1 + n2 + ... + n,_ 1 <n,, then d(l?,, + I?,, + . . + I?,,) =

Proof. The corollary

a>b. 0

3. Cartesian product

follows from Corollary 2.6 and the fact that d(K, 1 b) = 0 for

The Cartesian product of two graphs G1 =( Vi, E,) and Gz =( Vz, E,) is the graph G1 x Gz=(V1 x I’*/,,E) where

E = {{(a, 4, (a, 4): aE Vi and {c, d}EE2} u{{(a,c), (b,c)}: {a,b}EE1 and CEV,}.

Denote by P, the path of n vertices, i.e. P, has vertex set { 1,2, . . . , n} and edge set {{i, i + l} : 1~ i < n - 1). The purpose of this section is to determine the domatic number of a r-dimensional grid P,, x Pn2 x ... x P,F, where all ni>,2. Note that P,, x P,,z x ... x P,r has n1 .n2 “‘II, vertices of the form (a1,a2, . . ..a.), where 1 <ai<ni for 1 did r. Vertex (al, a2, . . , a,) is adjacent to vertex (b,, bZ, . . . , b,) if and only if there is exactly one laj- bjJ = 1 and all other ai= bi. Also d(P,, x P,, x ... x P,,)< &P,, x P,, x ... x P,,)+ 1 =r+ 1.

It is clear that P, is domatically full for any n3 1.

For any 2-dimensional grid P,, x Pnz, D1 = {(a, b): a is odd} and D2 = ((a, b): a is even} form a domatic partition. So 2 6 d(Pnl x P,,) < 3. It is clear that d(Pz x P2) = 2 since P, x Pz has only four vertices and no dominating vertex. It is also the case that d(Pz x P4) = d(P4 x P2) = 2. However, d(P,, x P,,) = 3 for all other 2-dimensional grids. To establish this result as well as others, we employ Propositions 2.1 and 3.1. Proposition 3.1. d(H) < d(G) for any spanning subgraph H = (V, E’) of G = (V, E).

Proof. The proposition follows from the fact that a dominating set of H is also a dominating set of G. 0

Theorem 3.2. d(P,, x P,,)= 3 for any 2-dimensional grid P,, x P,, except that d(P, x Pz)=d(Pz x P4)=d(P4x P,)=2.

Proof. Assume (n,, n2) is not (2,2) (2,4) or (4,2). For the case when one of n, and n2 is odd, say nl, let

D, = {(a, b): ar0 (mod 2)),

D2 = ((a, b): a = 1 (mod 4) and b E 1 (mod 2)), u{(a,b): a=3 (mod4) br0 (mod2)), D3 = {(a, b): as 1 (mod 4) and b =_O (mod 2)}

~((a, b): a E 3 (mod 4) b s 1 (mod 2)).

Then Dr, D2, D3 form a domatic partition of PnI x P,,. Thus d(Pnl x P,,)=3. Fig. 1 shows a domatic partition of P5 x P4.

For the case when both n1 and n2 are even: d(P4 x P4)= 3, shown in Fig. 2. Now, suppose at least one niB6, say n1 26. Since (P3 x P,,)u(P,, _3 x P,,) is a spanning subgraph of P,, x Pn2 and d(P, x P,,) = d(P,, _ 3 x P,,) = 3 by the above cases,

@‘,, x J’,,)~W=, x P&V’,l-~ x P,,))

>,min

For results on other grids, we need the concept about identifying two copies of a graph at a vertex set expressed in the following lemmas. More precisely, suppose G =( V, E) is a graph and S a subset of V’. Consider the graph G A S =( 1/*, E*) with v*= Vu{x*: XEV-S) and

E*=Eu{{X*,yj: XEV-&YES, (x,y}~E}u({x*,y*}: x,y~V--S, {x,y}~E}.

Fig. 2. d(P, x P4) = 3.

Lemma 3.3. Z_!- S is a subset of V in a graph G = (V, E), then d(G A S) 2 d(G).

Proof. The lemma follows from the fact that for any dominating set D of G, D* =Du{x*: XED-S} is a dominating set of GA S. 0

Lemma 3.4. If x is an end vertex of P,, then P,A{x} is isomorphic to P2,,- 1.

Lemma 3.5. For any two graphs G1 = ( VI, El) and Gz = ( V2, E2), ifS is a subset of VI

then (G, A S) x G2 is isomorphic to (G, x G,) A (S x V,).

Theorem 3.6. If r and n are positiue integers and (n1,n2, . . ..n.)~{n,2n- l}, then d(P+ x P,, x ... x P,,) 2 d(P, x P, x . . x P,), where the grids are r-dimensional.

Proof. By Lemmas 3.4 and 3.5, Pzn_ I x P,, x ... x P,r is isomorphic to (P, x P,, x ... x P,,) A((x} x V, x ... x V,). The theorem can be proved by induction on the number of ni’s that are equal to 2n- 1. 0

For any positive integer n, since n and 2n - 1 are relatively prime, there exists some no such that for any integer m>n,, we can write m = rn + s(2n - 1) for some non- negative integers r and s. The minimum such no is denoted by M(n). For instance, M(2) = 2 and M(3) = 8.

Theorem 3.7. If r and n are positive integers and nI, n2, . . . , n,b M(n) then

d(P,, x P,, x ... x P,,)>d(P, x P, x ... x P,), where the grids are r-dimensional.

Proof. Since for each ni there exist ri and si such that ni =rin +si(2n - l), Pnl x P,, x ... x P”r has a spanning subgraph which is the union of some grids P,, x P,, x ... x P,r, where ml, m2, . . . , m,E{n,2n- l}. The theorem follows from Propositions 2.1 and 3.1 and Theorem 3.6. 0

Theorem 3.8 (Laborde, Zelinka [9,21]). Zfk is a positive integer and r = 2k - 1, then the r-dimensional grid P, x P, x ... x P, is domatically full.

Colloary 3.9. Zf k is a positive integer and r =2k- 1, then any r-dimensional grid

We close this paper with the following conjecture: all r-dimensional grids, with finitely many exceptions, are domatically full. By Theorem 3.6, this conjecture is true if we can find some n such that the r-dimensional grid P, x P, x ... x P, is domatically full. In fact a slight modification of the above arguments shows that the conjecture is true if we can find a domatically full r-dimensional grid.

Acknowledgment

The author thanks an anonymous referee and Prof. B. Zelinka for many useful suggestions on the revision of this paper.

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