J. Comp. & Math. Sci. Vol. 1(7), 921-924 (2010)
The Strong Domatic Numbers of Factors of Graphs PUTTASWAMY Department of Mathematics, PES College of Engineering Mandya-571401, India Email: prof.puttaswamy@gmail.com ABSTRACT The maximum cardinality of a partition of the vertex set of a graph G into strong dominating sets is the strong domatic number of G, denoted ds(G). We consider Nordhaus-Gaddum type results involving the strong domatic number of a graph, where a Nordhaus-Gaddum type results is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. Therefore we investigate the upper bounds on the sum and product of the strong domatic numbers ds(G1), ds(G2) & ds(G3), where G1 ⊕ G 2 ⊕ G 3 = K n .
INTRODUCTION In a graph G=(V,E) the open neighborhood of a vertex v ∈ V is N(v)={x ∈ V; vx ∈ E}, the set of vertices adjacent to v. The closed neighborhood is N[v]=N(v) ∪ {v}. A set S ⊆ V is a dominating set if every vertex in V is either in S or is adjacent to a vertex in S that is V= U s∈S N[S]. The domination number
γ (G ) is the minimum cardinality of a dominating set. A domatic partition is a partition of V into dominating sets and the domatic number d (G) is the largest number of sets in a domatic partition2. The domatic number of a graph has been extensively studied, see1,2,6. Let G = (V , E ) be a graph and u , v ∈ V . Then u strongly dominats v if
(i )uv ∈
E and (ii ) deg (u ) ≥ deg (v) . A set S ⊆ V is a strong dominating set of G If every vertex in V − S is strongly dominated by atleast one vertex in S . The strong domination number γ s (G ) of G is the minimum cardinality of such a set5. Haynes and Henning3 have obtained the domatic number of factors of graphs. Similarly we study the concept of strong domatic numbers of factors of graphs. A strong domatic partition is a partition of V in to strong dominating sets and the strong domatic number ds(G) is the largest number of sets in a strong domatic partition. The following result is similar to that of cockayne and Hedetniemi2. Theorem 1. For any graph G,
d s (G ) ≤ δ (G ) + 1 .
Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)
Puttaswamy., J. Comp. & Math. Sci. Vol. 1(7), 921-924 (2010)
922 If
G1, G2, ..., Gt are graphs on the
same vertex set V with disjoint edge sets, then G = G1 ⊕ G2 ⊕ ...,⊕Gt denotes the
for all d s (G ) ≥ 1 graphs G , d s1 + d s 2 ≥ 2 and G, d s1 d s 2 ≥ 1 . Since
That these lower bounds are sharp, may be
≅ K1UK t −1, t . Then
seen
graphs G1, G2 ,..., Gt are called a t-factoring
each of G1 and G2 contain an isolated vertex. The upper bounds are more interesting.
of G. The special case of a 2-factoring of the complete graph K n is simply a factoring of K n into a graph G and its complement G . A Nordhaus-Gaddum type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In 1956 the original paper4 by Nordhaus and Gaddum appeared. In this paper we consider the strong domatic number and two variations of Nordhaus-Gaddum type inequalities. First, we extend the concept of a NordhausGaddum type result by considering G1 ⊕ G2 = Kt ,t . We establish sharp lower and upper bounds on the sums and products of ds(G1) and ds (G2). Further we investigate upper bounds on the sum and product of the strong domatic numbers ds(G1), ds(G2) and ds(G3), where G1 ⊕ G 2 ⊕ G3 = K 3 and n ≥ 3 . Here we consider G1 ⊕ G2 = Kt , t , where t ≥ 2, and lower and upper bounds on the sums and products of ds(G1) and ds(G2). We use notation by letting
d si = d s ( Gi ) , γ si = γ s ( Gi ) , δ i = δ ( Gi ) and ∆ i = ∆ ( Gi )
for i=1,2.
by
taking G1
and edge set E (G ) = E (G1 )U E (G 2 )... U E (Gt ) and the
graph with vertex set V
(
)
d s1 = 1 and d s 2 = d s K i ,t U K t −1 = 1 . Since
Theorem 2. Let t ≥ 2 be an integer and let G1 ⊕ G2 = K t ,t Then
d s1 + d s 2 ≤ t + 2 Further, the equality holds if G1 or G2 is isomorphic to tΚ 2 Proof. Let V be any vertex of G1. Then, by Theorem 1,
d s1 ≤ δ ( G1 ) + 1 ≤ deg G1 ( v ) + 1 and d s 2 ≤ deg G 2 (ν ) + 1
Thus,
d s1 + d s 2 ≤ deg G1 (ν ) + deg G 2 (ν ) + 2 =t+2 This proves the Upper bound, Assume, G1 ≅ tΚ 2 . Then, G2, The complement of G1 relative to Kt,t, may be obtained from Kt,t by removing the edges of a 1-factor Thus, ds1=2 and ds2=t, so ds1+ds2=t+2 Theorem 3. Let t ≥ 2 be an integer and let
G1 ⊕ G2 = Κ t ,t if Then
γ s1 = 2 ,
d s 1 d s 2 ≤ 2t
Further more, d s1d s 2 = 2t if and only If either G1 or G2 is isomorphic to tK2.
Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)
Puttaswamy., J. Comp. & Math. Sci. Vol. 1(7), 921-924 (2010)
Proof. If G2 has isolated vertex, then
d s1 ≤ t and d s 2 = 1 whence d s1d s 2 ≤ t < 2t. Hence we may assume that δ ≥ 1 and δ ≥ 1 Thus ∆1 ≤ t − 1and 2
1
923
t
t ≤ ∑ ≤ d s (Gi ) ≤ n + t − 1. i =1
Proof. The proof of the lower bound is trivial. Since d s (G ) ≥ 1 for all graphs G.
∆ 2 ≤ t − 1. In particular, If t = 2, then G1 ≅ 2K2 and G2 ≅ 2K2 So we may Assume that t ≥ 3 since γ s1 = 2 it
That this lower bound is sharp for t ≥ 3 , may be seen by taking G1 ≅ Κ 1 ∪ Κ n −1 ,
follower, as is the above.Theorem 2, that d s1 ≤ t and d s 2 ≤ 2 whence d s1 d s 2 ≤ 2t
if t ≥ 4, G i ≅ Κ n for i=4,5,….,t. Then Gi contains an isolated vertex for i=1,2,…,t, so
with equality if and only if d s1 = t and
d s 2 = 2 if and only if G2 ≅ tΚ 2 . Proposition 4. Let t ≥ 4 be an thteger and let G1 ⊕ G2 = Κ t ,t if γ s1 ≥ 4, then
d s1 d s 2
t ≤ 2
2
And this bound is sharp. Proof. Since γ sj ≥ 4 . We know that
d sj ≤ 2t / γ sj = t / 2 for j=1,2. Hence d sj ≤ t / 2 for j=1,2 and thus d s1 d s 2 ≤ t / 2
2
G2 ≅ Κ 2 ∪ Κ n−2 , G 3 ≅ Κ 1 ∪ Κ n − 2 , and
t
ds(Gi)=1. Hence,
∑d
s
(Gi)=t. To prove the
i=1
upper bound, we note that for each factor Gi , d s (Gi ) ≤ δ (Gi ) + 1 . Since
G1 ⊕ G 2 ⊕ ... ⊕ Gt = K n , t
t
∑ d (G ) ≤ ∑ (δ (G ) + 1) ≤ n − 1 + t. i =1
s
i
i =1
i
The upper bound is sharp may be seen by taking G1 ≅ K n , so Gi ≅ K n for i=2,3,…,t. In particular, it t=2, then we have the following result of cockayne and Hedetniemi2. Corollary 6. For any graph G,
We consider first the sum of the strong domatic numbers ds (G1), ds (G2) and ds (G3). We show that the upper bound on the sum is n+2. Theorem 5. Let n ≥ 3 be an integer and let G1 ⊕ G2 ⊕ .... ⊕ Gt = Κ n be a t-factoring of Kn. Then
d s (G ) + d s (G ) ≤ n + 1. Corollary7. Let n ≥ 3 be an integer and let G1 ⊕ G 2 ⊕ G3 = K 3 . Then
3 ≤ d s (G1 ) + d s (G 2 ) + d s (G 3 ) ≤ n + 2 ,
and these bounds are sharp.
Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)
Puttaswamy., J. Comp. & Math. Sci. Vol. 1(7), 921-924 (2010)
924 REFERENCES
1. G. J. Chang, the domatic number problem. Discrete Math. 125, 115-122 (1994). 2. E. J Cockayne and S. T.
Hedetniemi, Towards a theory of domination in graphs. Networks 7, 247-261 (1997). 3. T. W. Haynes and M. A. Henning, The domatic numbers of factors of
graphs (preprint). 4. E. A. Nordhaus and J. W. Gaddum, on complementary graphs. Amer. Math. Monthly 63, 175-177 (1956). 5. E.Sampathkumar and L.pushpa Latha. Strong, Weak domination and domination balance is a graph. Discrete Math. 161, 235-242 (1996). 6. B. Zelinka, Regular totally domatically full graphs. Discrete Math. 86, 81-88 (1990).
Journal of Computer and Mathematical Sciences Vol. 1, Issue 7, 31 December, 2010 Pages (769-924)