On Sequential Topogenic Graphs

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 36, On Sequential Topogenic Graphs Bindhu K. Thomas, K. A. Germina and Jisha Elizabath Joy Research Center & PG Department of Mathematics Mary Matha Arts & Science College Vemom P.O., Mananthavady , India bindhukthomas@gmail.com srgerminaka@gmail.com jishaelizabathjoy@gmail.com Abstract Given any simple graph G =(V,E), not necessarily finite, and a ground set X, a set-valued function f : V 2 X is a set-indexer of G if f is injective and the induced edge function f L : E(G) 2 X, defined by f L (uv) =(f(u) f(v)) (f(v) f(u) for each uv E, is also injective. A set-indexer f of G is called a segregation of X on G if the sets {f(u) :u V (G)} and {f L (e) :e E(G)} are disjoint and if, in addition their union is the set Y (X) =τ for some topology τ on X, then f is called a sequential topogenic labeling of G. A graph is called sequential topogenic if it admits a sequential topogenic labeling with respect to some set X. In this paper we initiate an investigation on sequential topogenic labeling of a graph. Keywords: sequential topogenic labeling, sequential topogenic graphs, sequential topogenic index 1 Introduction For all terminology and notation in graph theory, not specifically defined in this paper, we refer the reader to the text-book by F. Harary [6]. Unless mentioned otherwise, all graphs considered in this paper are simple and selfloop-free. For a (p, q)-graph G =(V,E) and a nonempty set X of cardinality n, Acharya [1] defined a set-indexer of G as an injective set-valued (or, equivalently, setlabeling ) function f : V (G) 2 X such that the induced edge-function f L : E(G) 2 X defined by f L (uv) =f(u) f(v) for all uv E(G) is also injective, where 2 X is the set of all subsets of X and denotes the binary operation of taking the symmetric difference of pairs of subsets of X. He defined G to be set-sequential if G admits a set-sequential labeling, which

2 1800 B. K. Thomas, K. A. Germina and J. E. Joy is a bijection f : V E 2 X such that f(uv) =f(u) f(v) for all uv E(G). The notion of topogenic set-indexer is defined as a set-indexer f such that f(v ) f L (E) is a topology on X. A wide discussion of topogenic labeling and topogenic graphs have been done in [3], [4] and [5]. As a generalization of the notion of set-sequential labeling we define the sequential topogenic labeling as follows. Definition 1.1. Let G =(V,E) be any graph and X be any set. A setindexer L f of G is called a segregation of X on G if the sets {f(u) :u V (G)} and {f (e) :e E(G)} are disjoint and if, in addition their union is the set Y (X) =τ for some topology τ on X, then f is called a sequential topogenic labeling of G. A graph is called sequential topogenic if it admits a sequential topogenic labeling with respect to some set X. 2 Preliminary Analysis Observation 2.1. then, / f(v E). If f :(V E) 2 X is a sequential topogenic labeling Proof. If f(u) = for some u V (G), then for any edge x = uv, we have f(x) =f(v), a contradiction to the injectivity of f. On the other hand if f(x) = for some x = uv E(G), then = f(u) f(v) which is true if and only if f(u) =f(v). This again leads to a contradiction to the injectivity of f. Remark 2.2. Due to the absence of in f(v ) f(e) a sequential topogenic labeling f can never be a topogenic labeling. Since f(v ), a topogenic labeling f can never be a sequential topogenic labeling. Observation 2.3. Every set-sequential graphs are sequential topogenic. Proof. Let G =(V,E) be a set-sequential graph. Then there exists a bijection f :(V E) 2 X for some set X. Now, f(v E) =2 X, the discrete topology on X. Observation 2.4. A sequential topogenic graph need not be set-sequential as C 4 which is not set-sequential is sequential topogenic by assigning the sets {1}, {3}, {1, 2}, {3, 4} to the vertices v 1,v 2,v 3,v 4 respectively. 3 Sequential Topogenic Graphs The concepts of topogenic graphs and sequential topogenic graphs are closely related. Hence we recall some results from [3] which are useful in this paper.

3 On sequential topogenic graphs 1801 Theorem 3.1. [3] The complete graph K p is topogenic for p {1, 2, 3, 6}. Theorem 3.2. [3] K 4,K 5 are not topogenic. Theorem 3.3. [3] K 1,m,n is topogenic for any m, n. Conjecture 3.4. [3] {1, 2, 3, 6}. The complete graph K n is topogenic if and only if n Remark 3.5. The path P 2 is sequential topogenic with the labeling f(u 1 )= {1},f(u 2 )={2} where u 1,u 2 V (P 2 ) and P 3 is sequential topogenic with the labeling f(u 1 )={1},f(u 2 )={2},f(u 3 )={1, 3} where u 1,u 2,u 3 V (P 3 ). But in general paths are not sequential topogenic. Theorem 3.6. P 4 is not sequential topogenic. Proof. If possible let P 4 be sequential topogenic with respect to a nonempty set X. Then there exists a sequential topogenic set-indexer f such that f(v E) =τ for some topology τ on X. Let f(v )={A 1,A 2,A 3,A 4 }. Then, f(e) ={A 1 A2,A 2 A3,A 3 A4 }. Clearly, X f(v E). Then, there arises two cases viz., X = A i for some i or X = A i Aj for some i, j. Case 1: X = A i for some i. Then, X is the assignment of either a pendant vertex or an internal vertex. (i) Without loss of generality, let X = A 1. Then, A 2,A 3,A 4 can be such that A 2 A 3 A 4 = A 1 or A 2 A 3 A 4 A 1. If A 2 A 3 A 4 = A 1, then A 2,A 3,A 4 cannot be mutually disjoint, if so then A 1 A2 = A 2 A3. Hence, at least two of them must have a nonempty intersection. Without loss of generality, let A 2 A 4. But, A 2 A 4 / f(v E) asa 2 A 4 A i for any i [ since if A 2 A 4 = A 2 or A 4, then (A 2 A3 ) (A 3 A4 ) / f(v E). Now, A 2 A 4 = A 3 = A 2 = A 3 T 1,A 4 = A 3 T 2, where T 1,T 2 are nonempty subsets of X. Then, (A 2 A3 ) (A 4 A3 )=T 1 T 2 / f(v E).] and A 2 A 4 A i Aj, for distinct i, j {1, 2, 3, 4} [since if A 2 A 4 = A 2 A3 = A 3 = A 2 (A 2 A 4 )= A 3 A 2 and A 3 A 4 = = A 3 A4 = A 1. Similar contradiction arises when A 2 A 4 = A 3 A4.] Therefore, A 2 A 3 A 4 A 1. As seen before A 2,A 3,A 4 cannot be mutually disjoint. Hence at least two of them must have a nonempty intersection. If adjacent labelings A 2,A 3 have nonempty intersection, then A 2 A 3 must be in f(v E). Proceeding as before we get that, A 2 A 3 A i for any i and A 2 A 3 A i Aj, a contradiction to our assumption. In a similar manner we can show that if nonadjacent labelings A 2,A 4 have nonempty intersection, then A 2 A 4 / f(v E). Proceeding as before we can prove that X is not an assignment of any internal vertices. Thus we have X/ f(v ). Case 2: Let X = A i Aj for some i, j. Then, without loss of generality, (a) X = A 1 A2 or (b) X = A 2 A3.

4 1802 B. K. Thomas, K. A. Germina and J. E. Joy (a) If X = A 1 A2, then A 1 A 2 = X and A 1 A 2 =. Then A 3 and A 4 should be subsets of A 1 or A 2 or contains elements from both A 1 and A 2. (i) If A 3 A 1 and A 4 A 1, then A 3 A 2 = A 4 A 2 =. Therefore, A 2 A 4 / f(v E). (ii) If A 3 A 1 and A 4 A 2, then A 1 A 4 / f(v E). (iii) If A 3 A 1 and A 4 A 1,A 4 A 2, then A 1 A 4 / f(v E). (iv) If A 3 A 1,A 3 A 2,A 4 A 1 and A 4 A 2, then A 1 A 3 / f(v E). Similar contradiction arises for all other choices of X and A i, where i {1, 2, 3, 4}. Therefore, X / f(e). Hence X / f(v E) and the results follows by contraposition. Theorem 3.7. A graph G =(V,E) is sequential topogenic if and only if G + K 1, V (K 1 )={v} has a topogenic labeling F such that F (v) =. Proof. Let G =(V,E) be sequential topogenic. Then, there exists a sequential topogenic set-indexer f such that f(v E) =τ for some topology τ on X. Let G = G + K 1. Define F as an extension of f such that F (u) =f(u) for all u V (G) and F (v) =. Clearly, F is a topogenic set-indexer. Conversely if G = G+K 1, has a topogenic labeling F with F (v) =, then the removal of v from G obviously results in a sequential topogenic labeling. Since K 4,K 5 are not topogenic K 3 and K 4 are not sequen- Corollary 3.8. Corollary 3.9. topogenic. Remark tial topogenic. K 5 is sequential topogenic since K 6 = K 5 +K 1 is topogenic. K m,n is sequential topogenic since K 1,m,n = K m,n + K 1 is Remark The complete graph K 1 is sequential topogenic with any non-empty set assigned to its only one vertex and K 2 is sequential topogenic by Remark 3.5. By invoking Conjecture 3.4 and Theorem 3.7 we strongly believe that the only sequential topogenic complete graphs are K 1,K 2 and K 5. Thus, we conjectured the following. Conjecture The complete graph K n is sequential topogenic if and only if n {1, 2, 5}. 4 Sequential Topogenic Index Definition 4.1. The sequential topogenic index γ(g) of a graph G is the least cardinality of a set X with respect to which G has a sequential topogenic labeling. Further, if f : V E 2 X is a sequential topogenic labeling of G with X = γ(g) we call f an optimal sequential topogenic labeling of G.

5 On sequential topogenic graphs 1803 Lemma 4.2. q +1). For any sequential topogenic (p, q)-graph G, γ(g) log 2 (p+ Proof. Let G be a sequential topogenic (p, q)-graph with respect to a set X. Then, 1 p + q 2 X 1= p + q 2 X 1= p + q +1 2 X = log 2 (p + q +1) X. Thus, we have γ(g) log 2 (p + q +1). Remark 4.3. p + q forms an upper bound for γ(g). Theorem 4.4. For a complete bipartite graph K m,n, log 2 ((m + 1)(n + 1)) γ(k m,n ) m + n Proof. The result follows from Lemma 4.2 and Theorem 3.7. Remark 4.5. We have the sequential topogenic index γ(k 1 )=1, γ(k 2 )= 2. From Lemma 4.2 we have γ(k 5 ) 4. But, K 5 is sequential topogenic by assigning the sets {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4} to the vertices v 1,v 2,v 3,v 4,v 5 respectively. Therefore, γ(k 5 )=4. 5 Embedding and NP-Completeness A graph G is said to be embedded in a graph H, written G H, if there exists an induced subgraph G of H such that G is isomorphic to G (written, G = G ); then we call H a host of G. Theorem 5.1. Every (p, q)-graph G can be embedded into a connected sequential topogenic graph. Proof. Let G be a (p, q)-graph and V (G) ={u 1,u 2,...,u p }. Case I: Let G has at least one pendant vertex say u p. Let X = {1, 2,...,p} and X = {1, 2,...,p 1}. Label u i by the singleton set {i}, where 1 i p 1 and u p by X. Define S j as the subsets of X of cardinality j which contains the element 1. If u i is not adjacent to u j where i, j 1,p introduce a new vertex, label it by {1,i,j} and join it to u 1. If u 1 is not adjacent to u k, introduce a vertex, label it by {1, 2,k} and join to the vertex u 2 [if{1, 2,k} f(v ), then join that vertex to u 2 ]. Now introduce S 4 + S S n new vertices and label them by the elements of S 4,S 5,...,S n in an injective manner. The subsets of cardinality three which was not a vertex label is obtained as edge labels by joining the vertices which was labeled by elements of S 4 to u 1 and appropriate elements of S 4 to u i, 1 i p 1 as required. Then join the vertices which was labeled by the elements of S 5,S 6,...S n to u 1. The resulting graph is sequential topogenic. Case II: If G has no pendant vertex, attach a pendant vertex u p+1 to u 1 and take X = {1, 2,...p+1} and X = {1, 2,...p}. Then proceed as in Case I.

6 1804 B. K. Thomas, K. A. Germina and J. E. Joy Corollary 5.2. Let G be any (p, q)-graph and H be its sequential topogenic host. Then, the chromatic number χ(h) =χ(g) +1 and the clique number ω(h) is either ω(g) or ω(g) +1. If the chromatic number and the clique number of G are greater than or equal to three then so is that of H. Therefore, the problems of determining the chromatic number and clique number of a connected sequential topogenic graph is NP- Complete. Remark 5.3. We have shown that every graph G has a sequential topogenic host H, in the sense that G is contained as an induced subgraph in H which is sequential topogenic, implying there by that there is no forbidden subgraph characterization of sequential topogenic graphs. 6 Conclusion and scope Characterizing sequential topogenic graphs in general is a challenging problem. Finding good bounds for γ(g) for a sequential topogenic graph is also seems to be a challenging problem. Acknowledgements The authors are thankful to the Department of Science & Technology, Government of India for supporting this research under the Project No. SR/S4/MS:277/06. References [1] B.D. Acharya, Set-valuations of graphs and their applications, MRI Lecture Notes in Applied Mathematics, No.2, Mehta Research Institute of Mathematics and Mathematical Physics, Allahabad, [2] B.D. Acharya, S. M. Hegde Set-sequential graphs, Nat. Acad. Sci. Letters, 8(12): , [3] K.A. Germina, Jisha Elizabath Joy, Topogenic graphs, Submitted. [4] K.A. Germina, Jisha Eizabath Joy, Topogenic graphs II: Embedding,Indian Journal of Mathematics, 51(3), , 2009 [5] K.A. Germina, Jisha Elizabath Joy, Enumeration of Graphical Realization, International Journal of Algorithm, Computing and Mathematics, 3(1),31-46,2010.

7 On sequential topogenic graphs 1805 [6] F. Harary, Graph Theory, Addison-Wesley Pub. Comp., Reading, Massachusetts, Received: March, 2010