A Note on Chromatic Weak Dominating Sets in
Graphs
P. Selvalakshmia and S. Balamuruganb
aSrinivasa Ramanujan Research Center in Mathematics,
Sethupathy Government Arts College, Ramanathapuram, Tamilnadu, India
bPG Department of Mathematics,
Government Arts College, Melur - 625 106, Tamilnadu, India
Email:[email protected], [email protected]
Abstract
In this paper chromatic weak dominating sets and chromatic weak domination number are defined. Chromatic weak domination number for standard graphs are found, bounds for chromatic weak domination number are obtained.
AMS Subject Classification: 05C69
Keywords: Domination, Weak domination, Chromatic weak domination, Chromatic number.
1
Introduction
LetG= (V, E) be a finite, simple and undirected graph with neither loops nor multiple edges. The order and size ofGare denoted bynandm respec-tively. One of the fastest growing areas within graph theory is the study of domination and related problems. A subset D of V is said to be a domi-nating set of G if every vertex inV −D is adjacent to a vertex in D. The minimum cardinality of a dominating set is called the domination number of
Gand is denoted byγ(G). A comprehensive treatment of the fundamentals of domination is given in the book by Haynes et al [7]. Prof. E. Sam-pathkumar and L. Pushpa Latha introduced the concept of strong (weak) domination in graphs in [9]. A setD⊆V is called aweak dominating setif for every vertexv∈V−D, there exists a vertexu∈Dsuch thatuv ∈E(G) anddeg(u)≤deg(v). The minimum cardinality of a weak dominating set is called theweak domination number of Gand is denoted by γw(G). A weak
dominating set of cardinality γw(G) is called as γw-set of G. For D ⊆ V,
0
the subgraph induced by D is denoted by< D >. A set D ⊆V is said to be a chromatic preserving set ifχ(< D >) =χ(G). It was defined by T. N. Janakiraman and M. Poobalaranjani [8]. A set D⊆V is said to be a dom-chromatic set ifDis a dominating set andχ(< D >) =χ(G). The minimum cardinality of a dom-chromatic set is called the dom-chromatic number and is denoted byγch(G). This concept is also defined by T. N. Janakiraman and
M. Poobalaranjani [8]. A split graph is a graph G = (V, E) whose vertices can be partitioned into two setsV1 andV2, where the vertices inV1 form a
complete graph and the vertices inV2form a null graph. A clique of a graph
G is a maximal complete subgraph. The cardinality of a maximum clique is called the clique number and is denoted byω(G). A graph Gis aperfect graph if χ(H) = ω(H) for all induced subgraph H of G. The length of a smallest cycle (if any) of a graph G is called girth and is denoted by g(G) and the length of a smallest odd cycle (if any) is denoted byg0(G). A graph
G is planar if it can be embedded in a plane, a plane graph has already been embedded in the plane. A setD⊆V is said to be achromatic strong dominating set if D is a strong dominating set and χ(< D >) = χ(G). The minimum cardinality of a chromatic strong dominating set is called thechromatic strong domination number of G and is denoted by γsc(G). A chromatic strong dominating set of cardinality γsc(G) is called as γsc-set of
G. This concept was defined by V. Swaminathan and S. Balamurugan [2]. A graph G is called a vertex χ-critical graph if χ(G−v) < χ(G) for all
v ∈ V(G). A vertex v said to be a weak vertex if deg(v) < deg(u) for all
u∈N(v).
Definition 1.1. [8] A subset D of V is said to be a chromatic preserv-ing set (or cp-set) if χ(< D >) = χ(G). The minimum cardinality of a chromatic preserving set in a graphGis called the chromatic preserving number(or cpn-number) of Gand is denoted by cpn(G).
2
Chromatic Weak Dominating Sets
Definition 2.1. Let G= (V, E)be a graph. A subset Dof V is said to be a chromatic weak dominating set(or cwd-set) ifDis a weak dominating set and χ(< D >) =χ(G). The minimum cardinality of a chromatic weak dominating set in a graph G is called the chromatic weak domination number(or cwd-number) and is denoted byγwc(G).
Proposition 2.2. Let G be a graph of order n. Then 1≤γwc(G)≤n.
Proposition 2.3. Let G be a graph. Then γc
Observation 2.4. (i). Chromatic weak dominating set exists for all graphs.
(ii). Vertex setV is a trivial chromatic weak dominating set.
(iii). For a vertex-color-critical graph, V is the only chromatic weak domi-nating set.
(iv). For any graph G, cpn(G)≤γwc(G).
(v). IfDis a chromatic weak dominating set ofG, then each vertex ofV−D
is not adjacent to at least one vertex of D.
(vi). Acwd-set of a graph G is a global dominating set.
Proof. (i) to (iv) follows trivially.
(v). SupposeDis a chromatic weak dominating set such that x∈V −D is adjacent to each vertex ofD. SinceDcontains acp-set ofG, letD1 ⊆Dbe a cp-set of D. Since x is adjacent to each vertex ofD1,χ(G) ≥χ(< D1 >
) + 1 =χ(G) + 1, a contradiction.
(vi). LetDbe acwd-set of a graph G. From (v), inGeach vertex ofV−D
is adjacent to at least one vertex ofD. Hence D is a dominating set ofG. Since D is a dominating set of G, it follows that D is a global dominating set.
3
Results for Standard Graphs
Proposition 3.1. Let G=Pn. Then
γwc(Pn) =
(n
3
+ 1 if n≡1 or 2(mod 3)
n
3
+ 2 if n≡0( mod 3)
Proof. LetD be a γw-set of G. Case 1: Letn≡0( mod 3).
Let n= 3k. Since γw(Pn) =
n
3
+ 1 ifn ≡0 or 2( mod 3). Therefore
γw(Pn) =
3k
3
+ 1 =k+ 1. Therefore D={u1, u4, u7, . . . , u3k−2, u3k} is a
uniqueγw-set ofP3k. Thenχ(< D >) = 1, butχ(Pn) = 2. LetD 0
=D∪{u}, whereu∈N(ui),ui ∈D. Thenχ(< D
0
>) = 2. This impliesD0 is aγc w-set
of P3k. Thus, γwc(Pn) = |D 0
|= |D∪ {u}|= |D|+ 1 = n3+ 1. Therefore
γwc(Pn) =
n
3
+ 2 ifn≡0( mod 3).
Case 2: Letn≡2( mod 3).
Letn= 3k+ 2. Sinceγw(Pn) =
n
3
+ 1 ifn≡0, 2( mod 3),
γw(Pn) =
3k+2
3
+ 1 =k+ 2. LetD={u1, u4, u7, . . . , u3k−2, u3k+1, u3k+2}
be a unique γw-set of P3k+2. Since D is not independent, χ(< D >) = 2.
This impliesDis a γc
w-set of P3k+2. Thereforeγwc(P3k+2) =|D|=
n
3
Thereforeγwc(P3k+2) =
n
3
+ 1 ifn≡2( mod 3).
Case 3: Letn≡1( mod 3).
Letn= 3k+ 1. Sinceγw(Pn) =
n
3
ifn≡1( mod 3), therefore γw(Pn) =
3k+1
3
=k+ 1. Let D={u1, u4, u7, . . . , u3k−2, u3k+1}
be a uniqueγw-set ofP3k+1. SinceDis not independent, χ(< D >= 1, but χ(Pn) = 2. LetD
0
=D∪ {u}, whereu∈N(ui),ui∈D. Thenχ(< D 0
>) = 2. This impliesD0 is aγwc-set ofPn. Thusγwc(Pn) =|D
0
|=|D|+1 =n 3
+1. Thereforeγwc(Pn) =
n
3
+ 1 ifn≡1( mod 3). Hence the proof.
Proposition 3.2. Let G=Cn. Then
γwc(Cn) =
n if nis odd
n
3
+ 1 if n≡0,2( mod 3)and n is even
n
3
if n≡1( mod 3)and n is even
Proof. LetCn be a cycle ofnvertices. Case 1: Letnbe odd.
Thenχ(Cn) = 3 =χ(< D >). ThereforeDis not a proper subset ofV(Cn).
Let D be a γwc-set of Cn. Therefore γwc(Cn) = |V(Cn)|= |D| =n. Hence γwc(Cn) =n.
Case 2: Letnbe even.
Subcase 2(a): Letn≡1( mod 3).
Let n = 3k + 1, k ≥ 1. Let D = {u2, u5, u8, . . . , u3k−1, u3k} is a γw
-set and D is not independent. Therefore D is a γwc-set of Cn. Therefore γwc(Cn) =γw(Cn) =
n
3
.
Subcase 2(b): Let n≡0( mod 3).
Letn= 3k,k≥1. LetD1 ={u2, u5, u8, . . . , u3k−1},D2 ={u3, u6, u9, . . . , u3k}
andD3={u1, u4, u7, . . . , u3k−2}. ThenD1,D2 andD3 are the onlyγw-sets
which are also independents. Therefore, χ(< Di >) = 1, i = 1,2,3 but χ(Cn) = 2 and γwc(Cn) > γw(Cn). Now, D1 ∪ {u1} is a chromatic weak
dominating set ofCn. Therefore,γwc(Cn)≤ |D|+ 1 =γw(Cn) + 1 =
n
3
+ 1.
Subcase 2(c): Let n≡2( mod 3). Let n = 3k+ 2, k ≥ 1. Then γw(Cn) =
n
3
= 3k3+2 = k+ 1. Let D
be a γw-set. Suppose D is not independent. Let vjvj+1 ∈ D. Consider
the path P1 : vj+3, vj+4, . . . , vn, v1, v2, . . . , vj−2 of length n−4. Therefore, γw(P1) =
n−4
3
=3k+23−4=3k3−2=k.
Therefore, γw(Cn) = γw(P1) + 2 = k + 2, which is a contradiction.
Therefore D is independent and χ(G) = 2. Then D∪ {x} is a chromatic weak dominating set ofCn,x∈N(y) for somey∈D. Therefore,γwc(Cn) = γw(Cn) + 1 =
n
3
+ 1. Hence γc
w(Cn) =
n
3
+ 1.
Proposition 3.3. For n≥1,γwc(Kn) =n.
Proof. Let D be any proper subset of V(Kn). Then χ(< D >) < n.
Therefore D is not a chromatic weak dominating set. Therefore V(Kn) is
the only chromatic weak dominating set. Hence γc
w(Kn) =n. Proposition 3.4. Let G=nK1. Then γwc(nK1) =n.
Proof. LetGbe totally disconnected graph. Thereforeγw(nK1) =n. Thus
n=γw(nK1)≤γwc(nK1)≤n.
Thereforeγwc(nK1) =n.
Proposition 3.5. Let G=K1,n−1,n≥2. Then γwc(K1,n−1) =n.
Proof. LetDbe a γw-set of K1,n−1. Then |D|=n−1 andχ(K1,n−1) = 2,
but χ(< D >) = 1. Since each vertex of D has degree 1 and let u be the vertex ofK1,n−1 with degreen−1.
LetD0 =D∪{u}. This implies|D0|=|D|+ 1 =n−1 + 1 =n. Therefore
|D0| = n and χ(< D0 >) = 2. Therefore D0 is a γc
w-set of K1,n−1. Hence γwc(K1,n−1) =n,n≥2.
Proposition 3.6. For 2≤m≤n,
γwc(Km,n) = (
2 if m=n
max{m, n}+ 1 if m6=n.
Proof. LetG=Km,n. LetD be a weak dominating set ofG. Case 1: Letm=n.
Thenγw(G) = 2. LetD={u, v}whereuv∈E(G). SinceGis bipartite, χ(G) = 2. Since uv ∈ E(G), D is a chromatic weak dominating set of G
and hence 2 =χ(G)≤γwc(G)≤ |D|= 2. Therefore γwc(G) = 2.
Case 2: Letm6=n.
Let us consider two subcases.
Subcase 2(a): Letn > m.
LetD={u1, u2, . . . , un}be a minimal weak dominating set of G. Then χ(< D >) = 1, since {u1, u2, . . . , un} is an independent set. But χ(G) = 2.
LetD0 =D∪ {vi}, wherevi ∈ {u1, u2, . . . , um}. Thenχ(< D 0
>) = 2. This implies D0 is a γwc-set of G. Therefore, γwc(G) = |D0| = |D|+ 1 = n+ 1. Thereforeγc
w(G) =γwc(Km,n) =n+ 1 ifn > m. Subcase 2(b): Let n < m.
Proposition 3.7. Let G=Dr,s. Then γcw(Dr,s) =n−1, for all 2≤r ≤s, where n=r+s+ 2.
Proof. LetD be aγw-set ofDr,s. Thenγw(Dr,s) =r+sand χ(Dr,s) = 2.
Butχ(< D >) = 1. Letdeg(u) =r+ 1 anddeg(v) =s+ 1. Since 2≤r≤s,
deg(u)≤deg(v). LetD0 =D∪{u}. Thenχ(< D0 >) =χ(< D∪{u}>) = 2. This impliesD0 is aγwc-set of Dr,s. Therefore, γcw(Dr,s) =|D
0
|=|D|+ 1 =
r+s+ 1 =n−1. Thereforeγwc(Dr,s) =n−1.
Proposition 3.8. Let G=Wn. Then
γwc(Wn) = (
n if nis even
3 if nis odd
Proof. Let G=Wn. Let V(G) ={u, v1, v2, . . . , vn−1}. Let D be aγwc-set
ofG. Letu be a vertex with deg(u) =n−1.
Case 1: Letnbe odd.
Then hV(G)− {u}i is an even cycle. Therefore χ(hV(G)− {u}i) = 2. Therefore χ(G) = 3. Then D = {u, v1, v2, . . . , vn−1} is a chromatic weak
dominating set. Therefore 3 = χ(G) ≤ γwc(G) ≤ χ(< D >) = 3. Hence
γc
w(G) = 3 ifn is odd. Case 2: Letnbe even.
Then < V(G)− {u} > is a cycle on n−1 vertices. Since n is even and
n−1 is odd. Therefore χ(< V(G)− {u} >) = 3 and χ(G) = 4. Since
χ(< V(G)− {x} >) ≤ 3 for all x ∈ V(G) since D is a γwc-set. Therefore
γwc(G) =γwc(Wn) =n.
Proposition 3.9. For n≥3,γwc(Fn) =
n
3
+ 2where Fn denotes a fan.
Proof. Since γw(Fn) =
(n
3
ifn≡1, 2( mod 3)
n
3
+ 1 ifn≡0( mod 3) . LetDbe a weak dominating set.
Case 1: Letn= 3k+ 1 and n= 3k+ 2.
Here χ(< D >) = 1 because no two vertices of Dare adjacent. LetD1=D∪ {u1} whereu1u2 ∈E(Fn) andu2∈D.
Then χ(< D1 >) = χ(< D∪ {u1} >) = 2. But χ(Fn) = 3. Then u
be the vertex of Fn with degree n−1. Then D2 = D1 ∪ {u3} contains
a triangle and χ(< D2 >) = 3. Thus D2 is a γwc-set of Fn. Therefore, γc
w(Fn) =|D2|=|D∪ {u1, u3}|=
n
3
+ 2.
Case 2: Letn= 3k.
Here χ(< D >) = 2 because induced graph of Dcontains an edge. But
χ(Fn) = 3. Let D1 = D∪ {u1} where u1 is the vertex of Fn with degree
n−1. Then< D1 >contains a triangle and χ(< D1 >) = 3. ThereforeD1
is aγwc-set ofFn. Thusγwc(Fn) =|D1|=|D∪ {u1}|=
n
3
+ 1 + 1 =n 3
+ 2. Thereforeγc
w(Fn) =
n
3
+ 2. Hence the proof.
Theorem 3.10. Given a positive integer k≥2, there exists a graphGsuch thatγwc(G) =k.
Proof. LetG be a complete graphKk. Thenγcw(Kk) =k.
4
Bounds on
γ
wc-sets
Proposition 4.1. If G is an even cycle, then γwc(G) = n2 iff G is C4, C6, C8.
Proof. Necessary condition is trivial. Conversely, suppose γwc(G) = n2.
Case 1: Letn≡0( mod 3) andn be even. From Proposition 3.2, it is known thatγc
w(G) = n3 + 1. Thenn= 6. Case 2: Letn≡1( mod 3) andn be even.
From Proposition 3.2, it is known thatγwc(G) = n3. Then n= 4.
Case 3: Letn≡2( mod 3) andn be even.
From Proposition 3.2, it is known thatγwc(G) = n+43 . Then n= 8.
Proposition 4.2. If G is a path, then γc
w(G) = γw(G) + 1 iff n ≡1(mod
3).
Proof. Necessary condition is trivial.
Conversely, supposen≡1( mod 3) and n is even. Let n= 3k+ 1. Let
Pn:u1, u2, u3, ..., u3k+1 be a path onn vertices.
Sincen≡1( mod 3), thenγw(Pn) =
n
3
. D={u1, u4, u7, ..., u3k−2, u3k+1}
is a uniqueγw-set ofPn.
This implies D is independent and χ(< D >) = 1 but χ(Pn) = 2. Let D0 = D∪ {ui}, where uiuj ∈ E(Pn) and uj ∈ D. Then χ(< D
0
>) = 2. ThereforeD0 is aγwc-set ofPn. Thusγwc(Pn) =|D
0
|=|D∪ {ui}|=|D|+ 1 =
n
3
+ 1 =γw(Pn) + 1. Henceγwc(Pn) =γw(Pn) + 1. Hence the proof.
Proposition 4.3. Let D be any cwd-set of G. Then
|V −D| ≤ X u∈D
deg(u).
Proposition 4.4. LetDbe anycwd-set ofG. Then|V−D|=P
u∈Ddeg(u) iff G=pK1,p≥1.
Proof. IfG=pK1, thenD=V anddeg(u) = 0 for each u∈D. Then the
equality holds. Now, suppose |V −D|=P
u∈Ddeg(u) =k. Claim: k= 0.
Supposek≥1. Then two cases arise.
Case 1: Gis connected.
Then χ(G) ≥2. LetV −D={u1, u2, ..., uk}. Since D is a dominating set
eachuiis adjacent to a vertex ofDand hence, contributes at least one degree
toD. Since χ(< D >)≥2,D contains at least one edge which contributes 2 degrees toD. Hence,P
u∈Ddeg(u)≥k+ 2, which is a contradiction. Case 2: Gis disconnected.
IfG is totally disconnected, V =D and hence, |V −D|=k= 0, a contra-diction.
Hence G has non trivial component and then < D > contains at least one edge. Then by a similar argument as in case (1), contradiction arises.
In both the cases, contradiction arises. Therefore k= 0. Then |V −D|= P
u∈Ddeg(u) = 0. Therefore V =D and hence, for each u ∈ V, deg(u) = 0. Thus, G is a totally disconnected graph and hence
G=pK1.
Corollary 4.5. For any non trivial connected graph with a cwd-set D,
X
u∈D
deg(u)≥ |V −D|+ 2.
Proof. If G is vertex color critical, then V = D and P
u∈D2V ≥ 2 ≥ |V −D|+ 2. Suppose G is not vertex color critical. As G is non-trivial,
χ(G) ≥ 2. By similar argument as in case (1) of the above proposition,
P
u∈Ddeg(u)≥ |V −D|+ 2.
Theorem 4.6. If G is a planar graph with diam(G) = 2, χ(G) = 3 and
γw(G) = 2, then 3≤γwc(G)≤5.
Proof. Lower bound is trivial. Let S={a, b}be a γw-set ofG.
Sincediam(G) = 2,g0(G) = 3 or 5.
Case 1: g0(G) = 3.
LetCbe a 3-cycle xyzx. Ifa, b /∈C, then 2 vertices ofC are adjacent to
aand one vertex is adjacent to b or vice versa, otherwise K4 is induced, a contradiction. Letxand y be adjacent tob. Then axyais a 3-cycle. Hence
{a, x, y, a} is a cwd-set of G. If a or b is in the 3-cycle together with the
remaining vertex ofS is acwd-set ofG.
Case 2: g0(G) = 5.
LetC be a 5-cycleuvwxyu. Ifa, b /∈C, then asSis dominating, vertices ofC are adjacent toaand one vertex is adjacent toaorband not to both, otherwise 3-cycle is induced. Also no two consecutive vertices of C can be both adjacent to a or b. Otherwise a 3-cycle is induced. Then S can dominate at most 4 vertices ofC, a contradiction.
Hence a or b ∈ C. Let a ∈ C and b /∈ C. Let u = a. Then x and w
are adjacent toband hence a 3-cycle is induced, a contradiction. Therefore both a, b ∈ C and hence C is a γwc-set of G. From case (1) and (2), the upper bound is proved.
Theorem 4.7. A chromatic weak dominating setD is minimal iff for each
u∈D one of the following condition hold:
(i). χ(< D− {u}>)< χ(G) (ii). u is a weak isolate of D
(iii). there exists s∈V −D such that Nw(s)∩D={u}.
Proof. Let a chromatic weak dominating setDbe minimal. ThenD− {u}
is not a chromatic weak dominating set. This implies either D− {u} is not a chromatic weak dominating set or χ(< D − {u} >) 6= χ(G). If
χ(< D− {u}>)6=χ(G), clearlyχ(< D− {u}>)< χ(G). SupposeD− {u}
is not a chromatic weak dominating set, there existss∈V −(D− {u}) such thatsis not weak dominated by any vertex of D− {u}.
If w = u, Nw(u) ∩D = φ, that is, u is a weak isolate of D. Let w /∈ u. Then w ∈ V −D. This implies w is strong dominated by u.
Nw(w)∩D={u}. Hence the proof.
Theorem 4.8. If Gis a vertex-color-critical, then α0(G)< γwc(G).
Proof. Supposeγwc(G) =n. Let u∈V(G). Let S=V − {u}. ThenS is a vertex cover ofG. Therefore,α0(G)≤ |S|=|V−{u}|=n−1< n=γwc(G). Thereforeα0(G)< γwc(G).
Proposition 4.9. IfGis vertex-color-critical graph withdiam(G)≥2, then
α0(G) + 2≤γwc(G).
Proof. Given diam(G) ≥ 2. Then there exists u, v ∈ V(G) such that
uv /∈E(G). Let S =V − {u, v}. Then S is a vertex cover ofG. Therefore,
α0(G) ≤ |S|= |V − {u, v}|= n−2 =γwc(G)−2. Therefore α0(G) + 2≤ γc
w(G).
Proposition 4.10. If Gis a triangle free with χ(G)≥3, then γc
Proof. Sinceχ(G)≥3, anycwd-set ofGcontains an odd cycle. Since Gis triangle free,γwc(G)≥5.
Proposition 4.11. If H is a spanning subgraph of G such that χ(H) =
χ(G), then γwc(G)≤γwc(H).
5
Graphs with respect to
∆
Theorem 5.1. LetGbe a graph with∆(G) = 1. Thenγwc(G)+∆(G) =n+1
iff G=K2∪(n−2)K1.
Proof. If G=K2∪(n−2)K1, then ∆(G) = 1 andγwc(G) =n. Therefore γwc(G) + ∆(G) =n+ 1.
Conversely, suppose Gis a graph with ∆(G) = 1 satisfying that
γwc(G) + ∆(G) = n+ 1. Then each components of G is either K1 or K2
with at least one K2 component and γwc(G) = n. If there are more than
oneK2 component, then G can be chromatic weak dominated by less than
nvertices.
Hence G has exactly one K2 component and every other component is K1. ThereforeG=K2∪(n−2)K1.
Theorem 5.2. Let Gbe a graph with ∆(G) = 1. Then γwc(G) + ∆(G) =n
iff G= 2K2∪(n−4)K1.
Proof. If G = 2K2 ∪(n−4)K1, then ∆(G) = 1 and γwc(G) = n−1. Thereforeγwc(G) + ∆(G) =n−1 + 1 =n.
Conversely, suppose Gis a graph with ∆(G) = 1 satisfying that
γwc(G) + ∆(G) = n. Then each components of G is either K1 or K2 and
γwc(G) =n−1. If there are more than two K2 components then Gcan be chromatic weak dominated byn−2 vertices.
Hence Ghas exactly two K2 components and every other component is K1. ThereforeG= 2K2∪(n−4)K1.
Proposition 5.3. For any positive integer k, there exists a vertex color critical graphG such that γwc(G)−α0(G) =k.
Proof. Let n= 2k+ 1. Consider a graph G=Cn. Then γcw(G) = 2k+ 1
and α0(G) = 2k2+1 =k+ 1. Thus γwc(G)−α0(G) = 2k+ 1−(k+ 1) =k. Hence the proof.
Theorem 5.4. Given a positive integer k≥1, there exist a graph G such that
(i). γwc(G)−γw(G) =k.
(ii). γc
w(G)−γs(G) =k.
Proof. (i) Let G be a complete graph withk+ 1 vertices. Then γwc(G) =
k+ 1 and γw(G) = 1. Therefore, γwc(G)−γw(G) =k+ 1−1 =k.
(ii) Let G = Kk+1. Then γwc(G) = k + 1 and γs(G) = 1. Therefore, γwc(G)−γs(G) =k+ 1−1 =k.
Theorem 5.5. Given a positive integerk≥1, there exists a graphG such that
(i). γwc(G)−γsc(G) =k.
(ii). γc
w(G)−γch(G) =k.
Proof. (i). LetG=K1,k+1. Thenγwc(G) =k+ 2 andγsc(G) = 2. Therefore γwc(G)−γsc(G) =k+ 2−2 =k.
(ii). Let G = K1,k+1. Then γch(G) = 2. Therefore γwc(G) −γch(G) = k+ 2−2 =k.
6
Fractional Chromatic Weak Function
Definition 6.1. Let G= (V, E) be a graph. Let g :V(G) → [0,1] be such thatg(Nw[v]≥1)for all v ∈V(G) and the fractional chromatic number of
({v ∈V :g(v) >0}) is the same as the fractional chromatic number of G. Thengis called a fractional chromatic weak function. The minimum weight of such a function is called fractional chromatic weak number of G and is denoted by γwfc (G).
Observation 6.2. (i). γw(Kn) = 1.
(ii). γwc(Kn) =n.
(iii). γwf(Kn) = 1.
(iv). γwfc (Kn) = 1.
References
[1] S. Balamurugan, P. Aristotle and V. Swaminathan, Chromatic Strong dominating sets in Bipartite graphs, AKCE International Journal of Graphs and Combinatorics, Submitted.
[3] S. Balamurugan, A. Wilson Baskar and V. Swaminathan, Equality of strong domination and chromatic strong domination in graphs, Interna-tional Journal of Mathematics and Soft Computing, Vol. 1. No. 1(2011), 69 - 76.
[4] Dieter Gernert, Inequalities between the domination number and the chromatic number of a graph, Discrete Math . 76 (1989) 151-153.
[5] F. Harary, Graph Theory, Addison Wesley , reading Mass (1972).
[6] J. H. Hattingh and M. A. Henning,On strong domination in graphs, J. Combin.Math. Comb. Comput. 26 (1998) 73 - 82.
[7] T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs, Marcel Decker Inc.. New york, 1998.
[8] M. Poobalaranjani,On Some Coloring and Domination Parameters in Graphs, Ph.D Thesis, Bharathidasan University, India, 2006.
[9] E. Sampathkumar, L. Pushpa Latha,Strong weak domination and dom-ination balance in a graph, Discrete Mathematics, 161 (1996), 235 - 242.
[10] P. Thangaraju,Studies in Domination in graphs and Degree sequences, Thesis, Nov., (2001).
