EQUITABLE DOMINATING CHROMATIC SETS IN GRAPHS. Sethu Institute of Technology Kariapatti, Tamilnadu, INDIA 2 Department of Mathematics

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1 Iteratioal Joural of Pure ad Applied Mathematics Volume 104 No , ISSN: (prited versio); ISSN: (o-lie versio) url: doi: PAijpam.eu EQUITABLE DOMINATING CHROMATIC SETS IN GRAPHS L. Muthusubramaia 1, S.P. Subbiah 2, V. Swamiatha 3 1 Departmet of Mathematics Sethu Istitute of Techology Kariapatti, Tamiladu, INDIA 2 Departmet of Mathematics Maar Thirumalai Naicker College Madurai, Tamiladu, INDIA 3 Ramauja Research Ceter i Mathematics Saraswathi Narayaa College Madurai, Tamiladu, INDIA Abstract: Let G = (V,E) be a simple graph. A subset D of V(G) is said to be a equitable domiatig set of G if for every vertex v V D there exists a vertex u D such that uv E(G) ad d(u) d(v) 1. A subset D of V(G) is said to be a equitable domiatig chromatic set of G if D is a equitable domiatig set of G ad χ(< D >) = χ(g). Sice V is a equitable domiatig chromatic set of G, the existece of equitable domiatig chromatic set i a graph is guarateed. The miimum cardiality of such a set is called the equitable domiatig chromatic umber of G ad is deoted by γch e (G). Thepropertyof equitable domiatigchromatic set issuperhereditary. Hece equitable domiatig chromatic set is miimal if ad oly if it is 1- miimal. Characterizatio of miimal equitable domiatig chromatic sets is derived. The values of γch e (G) for may classes of graphs have bee foud. It is established that 1 γch e (G). Iterestig results are proved with respect to the ew parameters. Received: April 29, 2015 Correspodece author c 2015 Academic Publicatios, Ltd. url:

2 194 L. Muthusubramaia, S.P. Subbiah, V. Swamiatha AMS Subject Classificatio: 05C17, 05C69, 05C70 Key Words: equitable domiatig set, equitable domiatig chromatic set 1. Itroductio The cocept of Chromatic preservig sets was extesively studied by [3]. Further Chromatic preservig sets with specific property were cosidered. For example, Chromatic preservig strog(weak) domiatig sets were defied ad studied i [1]. Chromatic preservig trasversals were itroduced ad studied by [5]. Degree Equitability is a cocept itroduced by Prof. E. Sampathkumar. Equitable domiatig set was defied i [2]. I this paper, Equitable domiatig chromatic sets are defied ad studied. Defiitio 1.1. A vertex ad a edge are said to cover each other if they are icidet. A set of vertices which covers all the edges of a graph is called a vertex cover of G. The smallest umber of vertices i ay vertex cover of G is called the vertex coverig umber of G ad is deoted by α o (G). Defiitio 1.2. Let G = (V,E) be a simple graph. A subset D of V(G) is said to be a equitable domiatig set of G if for every vertex v V D there exists a vertex u D such that uv E(G) ad d(u) d(v) 1. Defiitio 1.3. Let G = (V,E) be a simple graph. A subset D of V(G) is said to be a dom-chromatic set or (dc-set) if D is a domiatig set of G ad χ(< D >) = χ(g). The miimum cardiality of dom-chromatic set i a graph G is called the dom-chromatic umber or dc-umber ad is deoted by γ ch (G). Defiitio 1.4. Let G be a simple graph ad let D be a subset of V(G). D is said to be a equitable domiatig chromatic set of G if D is a equitable domiatig set of G ad χ(< D >) = χ(g). The miimum cardiality of such a set is called the equitable domiatig chromatic umber of G ad is deoted by γ e ch (G). Sice V is a equitable domiatig chromatic set of G, the existece of equitable domiatig chromatic set i a graph is guarateed. γch e (G) for stadard graphs 1. γ e ch (K ) =

3 EQUITABLE DOMINATING CHROMATIC SETS IN GRAPHS γ e ch (K ) = 3. γch e (K 1,) = +1, 1 (+3)/3 if 0(mod 3), 4. γch e (P ) = γ ch (P )= (+2)/3 if 1(mod 3), (+1)/3 if 2(mod 3), (+3)/3 if 0(mod 3), 5. γch e (C ) = γ ch (C )= (+2)/3 if 1(mod 3), (+1)/3 if 2(mod 3), { 6. γch e (W 3 if is odd, )= if is eve, { 7. γch e (K 2 if m 1, m,)= m+ if m 2, r +s+1 if r s 1,r (or) s 2, 8. γch e (D r,s)= r +s+2 if r s 2, r +s if r = s = 1, 3 +2 if 1,4, 9. γch e (F )= 2 if = 1, 3 if = 4, 10. γ e ch (P) = 5 = γ ch(p). Remark 1.5. Sice ay equitable domiatig set is a domiatig set, γ ch (G) γ e ch (G). Theorem 1.6. Let D be a equitable domiatig chromatic set of G. Every vertex i V D is ot adjacet to at least oe vertex of D. Proof. Let u V D. Suppose u is adjacet with every vertex of D. The χ(< D {u} >) = χ(< D >)+1. Sice D is a equitable domiatig chromatic set of G, χ(< D >) = χ(g). Therefore, χ(< D {u} >) = χ(< G >)+1. Sice for ay subgraph S of G, χ(s) χ(g), we get a cotradictio. Therefore, u is ot adjacet with some vertex of D.

4 196 L. Muthusubramaia, S.P. Subbiah, V. Swamiatha Remark 1.7. Let G be a simple graph. The 1 γch e (G). Remark 1.8. γ e ch (G) = 1 if ad oly if G = K 1. Theorem 1.9. Let G be a equitable graph without isolates. (That is, Give u,v i V(G) with uv E(G) the deg(u) deg(v) 1). The γch e (G) = if ad oly if G is a χ-critical graph. Proof. Suppose G satisfies the hypothesis. Suppose G is a χ-critical graph. The for ay equitable domiatig chromatic set D of G, χ(< D >) = χ(g). Sice G is χ-critical graph, D = V(G). Therefore γch e (G) = D =. Coversely, Suppose γch e (G) =. Let D be a miimum equitable domiatig chromatic set of G. The D =. If G is totally discoected the γch e (G) = ad G is χ-critical. Suppose G is ot totally discoected. The there exist vertices u,v which are adjacet. Sice G is equitable d(u) d(v) 1. Therefore, G {u} is a equitable domiatig set of G. Sice γch e (G) =, G {u} is ot a domiatig chromatic set. Therefore, χ(< G {u} >) < χ(g). Let H be a proper subgraph of G. Let u / V(H). The H G {u} ad χ(h) χ(g {u}) < χ(g). Suppose u V(H). Let w H. If w is ot adjacet with ay vertex i H, the w is adjacet with some vertex x i G, x / H. H G {x}. Sice G {x} is equitable domiatig set of G, χ(g {x}) < χ(g). Therefore, χ(h) < χ(g). Suppose N(H) H. The H is ot a domiatig set of G, sice H is a proper subgraph of G ad N(H) H. Let y V V(H). The y is adjacet with some z V V(H). Therefore, G {y} is a equitable domiatig set of G. If G {y} is a equitable domiatig chromatic set of G, the γch e (G) γe ch (G {y}). Sice γe ch (G) =, γch e (G {y}) =. But V(G {y}) 1, a cotradictio. Therefore, χ(h) χ(g {y}) < χ(g). Therefore, G is a χ-critical graph. Remark SupposeGis a discoected graph with γch e (G) =. The either G is K or G has exactly oe o trivial compoet which is χ-critical, equitable ad without isolates. Let G be a discoected graph with γ e ch (G) =. If every compoet of G is trivial, the G = K. Suppose there exist two or more compoets of G which are o trivial. Let G 1 ad G 2 be two compoets of G which are o trivial. Let χ(g 1 ) χ(g 2 ). Let D 1 be a γ e ch -set of G 1 ad D 2 be a γ e -set of G 2. The D 1 D 2 is a γ e ch -set of G 1 G 2. For: D 1 D 2 is a equitable domiatig set

5 EQUITABLE DOMINATING CHROMATIC SETS IN GRAPHS 197 of G 1 G 2 of miimum cardiality, χ(d 1 D 2 ) = max{χ(d 1 ),χ(d 2 )} = max{χ(g 1 ),χ(g 2 )} = χ(g 1 ) (Sice χ(d 2 ) χ(g 2 ) χ(g 1 )) = χ(g 1 G 2 ). Therefore, D 1 D 2 is a γch e -set of G 1 G 2. D 2 V(G 2 ) 1 (sice G 2 has o equitable isolates). Therefore, γch e (G 1 G 2 ) < V(G 1 ) + V(G 2 ), a cotradictio. (sice γch e (G) = ). Therefore, there exists exactly oe compoet of G which is o trivial. Let G 1 be a o trivial compoet of G. Therefore, G = G 1 tk 1 where G 1 is a o trivial compoet. Therefore, = γch e (G) = γch e (G 1 tk 1 ) = t+γch e (G 1). Therefore, γch e (G 1) = t = V(G 1 ). Therefore, G 1 is χ-critical. Therefore, If γch e (G) =, the G is either K or G has exactly oe o trivial compoet which is χ-critical ad which is equitable without isolates. Remark The coverse of the above remark is also true. For: If G = K, γch e (G) =. If G has exactly oe o trivial compoet say G 1 which is χ-critical,equitable ad without isolates. The γch e (G 1) = V(G 1 ). Therefore, γch e (G) = V(G 1) +t where t is the umber of trivial compoets of G. Therefore, γch e (G) = V(G) =. Remark Ay superset of a equitable domiatig chromatic set of G is also a equitable domiatig chromatic set of G. That is, equitable domiatig chromatic property is super hereditary. Therefore, a equitable domiatig chromatic set is miimal if ad oly if it is 1-miimal. Theorem Let D be a equitable domiatig chromatic set of G. D is miimal if ad oly if for each u D, oe of the followig holds: (i) N(u) D = φ (or) d(u) d(v) 2 for all v N(u) D; (ii)there exists a vertex v V D such that N(v) D = {u} ad d(u) d(v) 1; (iii) χ(d {u}) < χ(g). Proof. Suppose D is a miimal equitable domiatig chromatic set of G. Sice the property of equitable chromatic domiatio is super hereditary, D is miimal if ad oly if D {u} is ot a equitable domiatig chromatic set for

6 198 L. Muthusubramaia, S.P. Subbiah, V. Swamiatha ay u D. Therefore, either χ(d {u}) < χ(g) or D {u} is ot a equitable domiatig set (ie) N(u) D = φ or d(u) d(v) 2 for all v N(u) D or there exists a vertex v V D such that N(v) D = {u} ad d(u) d(v) 1. Coversely, Suppose D is a equitable domiatig chromatic set such that for every u D oe of the above three coditios is satisfied. Cosider D {u}. If u satisfies (i) the either u is ot domiated by ay vertex of D {u} or u is ot equitably domiated by ay vertex of D {u}. If u satisfies (ii) the u has a equitable private eighbor i V D ad hece D {u} is ot a equitable domiatig set. If u satisfies (iii) the D {u} does ot preserve the chromaticity of G. Hece D {u} is ot a equitable domiatig chromatic set if satisfies ay oe of the three coditios. Hece the theorem. Theorem Let D be a equitable domiatig chromatic set of G. The V D = deg(u) if ad oly if G = K. Proof. If G = K the D = V ad deg(u) = 0 for every u D. Therefore, V D = V V = 0 = deg(u). Suppose G K. The G has a edge ad hece χ(g) 2. Therefore, χ(< D >) 2. Therefore, < D > has a edge. Therefore, deg(u) 2. Sice D is a equitable domiatig set, each vertex i V D is adjacet to at least oe vertex i D. Therefore, deg(u) V D +2. Hece V D deg(u). Hece the theorem. Corollary For ay o trivial coected graph, D +2 where D is a domiatig chromatic set of G. Theorem For ay graph G, if ad oly if G = K. deg(u) V γch e (G) ad equality holds Proof. Sice γ(g) γch e (G), the lower boud is attaied. If G = K the γch e (G) =. = 0+1 =. Therefore, γch e (G) =. Suppose = γch e (G) = k(say). SupposeG K. Thefrom

7 EQUITABLE DOMINATING CHROMATIC SETS IN GRAPHS 199 the above corollary, V D < Therefore, < k(). Therefore, If G K the deg(u) (i.e) k < < k < k < γch e (G). Hece the theorem. deg(u) k (G). Remark The above result is true eve whe equitability is dropped. Theorem Give a positive iteger k there exists a graph G such that γch e (G) = k. Proof. Suppose k = 1. The γ e ch (K 1) = 1 = k. Let k 2. Let G = K k. The γ e ch (K k) = k. Also, γ e ch (K k 1,1) = k,k 3. Theorem Give a positive iteger k there exists a graph G such that: γ e ch (i) γch e (G) γ(g) = k; (ii)γ e ch (G) γe (G) = k; (iii)γ e ch (G) γ ch(g) = k. Proof. (i) Let G = K k+1. The γch e (G) = k +1. But γ(g) = 1. Therefore, (G) γ(g) = k. (ii) Let G = K + k+1. γe (G) = k+1+1 = k+2. γch e (G) = 2(k+1) = 2k+2. Therefore, γch e (G) γe (G) = 2k+2 (k +2) = k. (iii) Let G = K + k. γe ch (G) = 2k, γ ch(g) = k. Therefore, γ e ch (G) γ ch(g) = 2k k = k. Theorem If G is triagle free with χ(g) 3 the γ ch (G) 5 ad hece γch e (G) 5. Proof. Let D be a γ ch -set of G. The χ(< D >) = χ(g) 3. Sice G is triagle free, < D > is also triagle free. If < D > cotais o odd cycle

8 200 L. Muthusubramaia, S.P. Subbiah, V. Swamiatha the < D > is bipartite ad hece χ(< D >) = 2, a cotradictio. Therefore, < D > cotais a odd cycle of legth greater tha or equal to 5. Therefore, D 5. γ ch (G) 5. Therefore, γ e ch (G) γ ch(g) 5. Theorem Let G be a equitable graph without isolates ad Let G be χ-critical. The α 0 (G) < γ e ch (G). Proof. For ay graph G, α 0 (G) 1. If G is χ-critical ad equitable without isolates, γ e ch (G) =. Therefore, α 0(G) < γ e ch (G). Theorem If G is χ-critical ad equitable without isolates, ad diam(g) 2. The α 0 (G)+2 γ e ch (G). Proof. Sice diam(g) 2, β 0 (G) 2. α 0 (G) 2. Therefore, α 0 (G)+ 2 = γ e ch (G). Theorem Let G be a perfect graph. The γ e ch (G) γe (G)+ω(G). Proof. Let S be a maximum clique i G ad D a γ e -set of G. Sice G is perfect, ω(g) = χ(g). Therefore, χ(g) = S = χ(< S >) = χ(< S D >). Sice equitable domiatio is super hereditary, S D is a equitable domiatig set of G. Sice χ(< S D >) = χ(g), < S D > is a equitable domiatig chromatic set of G. Therefore, γ e ch (G) S D S + D = ω(g)+γe (G). Remark There exists a graph G, such that γ e ch (G) = γe (G)+ω(G). For: Cosider the Peterse graph P. γ e ch (P) = 5,γe (P) = 3,ω(P) = 2. Therefore, γ e ch (P) = γe (P)+ω(P). Remark Let D be a graph with a full degree vertex. The γ e ch (G) eed ot be equal to χ(g). For example, γ e ch (D r,s) = r +s+1 where r s 1,r(or)s 2. Therefore, γ e ch (D r,s) χ(d r,s ) = 2. Theorem Let G be a perfect graph with a full degree vertex which equitably domiates all other vertices. The γch e (G) = χ(g). Proof. Let u be a full degree vertex of G such that u equitably domiates every other vertex of G. Let S be a maximum clique i G. The u S.

9 EQUITABLE DOMINATING CHROMATIC SETS IN GRAPHS 201 S = ω(g) = χ(g). Sice S is a equitable domiatig chromatic set of G, γch e (G) S = χ(g). But χ(g) γe ch (G). Therefore, γe ch (G) = χ(g). Example K 1, is a perfect graph with a full degree vertex. If 3, the full degree vertex does ot equitably domiate other vertices. Also, γch e (G) = +1 ad χ(g) = 2. Therefore, γe ch (G) > χ(g). Example K 1,2 is a perfect graph with a full degree vertex which equally domiates the other two vertices. Here, χ(k 1,2 ) = 2,γ e ch (K 1,2) = 2. Theorem There exists a coected graph G ad a graph G 1 such that G 1 isobtaied fromgbyaddigexactly oevertex adγ e ch (G1 ) γ e ch (G) = 1. Proof. Let G = P 3(N+2). The γch e (G) = (3(N +2) +3)/3 = N +3. Let G 1 be the graph obtaied from G by addig a ew vertex v ad joiig it to all the vertices of G. γch e (G1 ) = γch e (G)+1. Hece γe ch (G1 ) γch e (G) = 1. Corollary Give a positive iteger N, there exists a graph G ad a graph G! obtaied from G by addig exactly N vertices such that γch e (G1 ) γch e (G) = N. Proof. Take a path of 3(N +2) vertices ad add N vertices ad make them adjacet with every vertex of the path. The the result follows. Theorem Let G be a bipartite graph ad let G K. The γ e (G) γ e ch (G) γe (G)+1. Proof. Sice G is bipartite ad G K, χ(g) = 2. Let D be a miimum equitable domiatig set of G. The D = γ e (G). If < D > cotais a edge the χ(< D >) = 2 ad hece < D > is a equitable domiatig chromatic set of G. Therefore, γch e (G) = γe (G). Suppose < D > is totally discoected for every miimum equitable domiatig sets of G, the D {v} is a γch e -set of G where v V D. Therefore, γch e (G) = D +1 = γe (G)+1. Refereces [1] S. Balamuruga, A. Wilso Baskar, V. Swamiatha, Equality of strog

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