# On Integer Additive Set-Indexers of Graphs

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1 On Integer Additive Set-Indexers of Graphs arxiv: v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A set-indexer of a graph G is an injective set-valued function f : V (G) 2 X such that the function f : E(G) 2 X { } defined by f (uv) = f(u) f(v) for every uv E(G) is also injective, where 2 X is the set of all subsets of X and is the symmetric difference of sets. An integer additive set-indexer is defined as an injective function f : V (G) 2 N 0 such that the induced function g f : E(G) 2 N 0 defined by g f (uv) = f(u) + f(v) is also injective. A graph G which admits an IASI is called an IASI graph. An IASI f is said to be a weak IASI if g f (uv) = max( f(u), f(v) ) and an IASI f is said to be a strong IASI if g f (uv) = f(u) f(v) for all u, v V (G). In this paper, we study about certain characteristics of inter additive set-indexers. Key words: Set-indexers, integer additive set-indexers, uniform integer additive set-indexers, compatible classes, compatible index. AMS Subject Classification : 05C78 1 Introduction For all terms and definitions, not defined specifically in this paper, we refer to [9] and for more about graph labeling, we refer to [5]. Unless mentioned otherwise, all graphs considered here are simple, finite and have no isolated vertices. Department of Mathematics, Vidya Academy of Science & Technology, Thalakkottukara, Thrissur , Department of Mathematics, School of Mathematical & Physical Sciences, Central University of Kerala, Kasaragod, 1

2 For a (p, q)- graph G = (V, E) and a non-empty set X of cardinality n, a set-indexer of G is defined in [1] as an injective set-valued function f : V (G) 2 X such that the function f : E(G) 2 X { } defined by f (uv) = f(u) f(v) for every uv E(G) is also injective, where 2 X is the set of all subsets of X and is the symmetric difference of sets. Theorem 1.1. [1] Every graph has a set-indexer. Let N 0 denote the set of all non-negative integers. For all A, B N 0, the sum of these sets is denoted by A + B and is defined by A + B = {a + b : a A, b B}. The set A + B is called the sumset of the sets A anb B. Using the concept of sumsets, an integer additive set-indexer is defined as follows. Definition 1.2. [6] An integer additive set-indexer (IASI, in short) is defined as an injective function f : V (G) 2 N 0 such that the induced function g f : E(G) 2 N 0 defined by g f (uv) = f(u) + f(v) is also injective. A graph G which admits an IASI is called an IASI graph. Definition 1.3. [7] The cardinality of the labeling set of an element (vertex or edge) of a graph G is called the set-indexing number of that element. Definition 1.4. [6] An IASI is said to be k-uniform if g f (e) = k for all e E(G). That is, a connected graph G is said to have a k-uniform IASI if all of its edges have the same set-indexing number k. Definition 1.5. An element (a vertex or an edge) of graph G, which has the set-indexing number 1, is called a mono- indexed element of that graph. In particular, we say that a graph G has an arbitrarily k-uniform IASI if G has a k-uniform IASI for every positive integer k. In [8], the vertex set V of a graph G is defined to be l-uniformly setindexed, if all the vertices of G have the set-indexing number l. In this paper, we intend to introduce some fundamental notions on IASIs and establish some useful results. 2 Integer Additive Set-Indexers of Graphs Analogous to Theorem 1.1 we prove the following theorem. Theorem 2.1. Every graph G admits an IASI. 2

3 Proof. Let G be a graph with vertex set V (G) = {v 1, v 2,..., v n }. Let A 1, A 2,..., A n be distinct non-empty subsets of N 0. Let f : V (G) 2 N 0 defined by f(v i ) = A i. We observe that f is injective. Define g f : E(G) 2 N 0 as g f (v i v j ) = {a i + b j : a i A i, b j A j }. Clearly, g f (v i v j ) is a set of nonnegative integers. For suitable choices of the sets A i, we observe that g f is injective. Hence, f is an IASI on G. Figure 1 depicts an IASI graph. Figure 1 The following results verify the admissibility of IASIs by certain graphs that are associated with a given graph G. Theorem 2.2. A subgraph H of an IASI graph G also admits an IASI. Proof. Let G be a graph which admits an IASI, say f. Let f H be the restriction of f to V (H). Then, g f H is the corresponding restriction of g f to E(H). Then, f H is an IASI on H. Therefore, H is also an IASI graph. We call the IASI f H, defined in Theorem 2.2 as the induced IASI of the subgraph H of G. By edge contraction operation in G, we mean an edge, say e, is removed and its two incident vertices, u and v, are merged into a new vertex w, where the edges incident to w each correspond to an edge incident to either u or v. We establish the following theorem for the graphs obtained by contracting the edges of a given graph G. The following theorem establishes the admissibility of the graphs obtained by contracting the edges of a given IASI graph G. Theorem 2.3. Let G be an IASI graph and let e be an edge of G. Then, G e admits an IASI. 3

5 f (u i ) = g f (e i ), 1 i n. Clearly, f is injective. Since each f (u i ) is a set of non-negative integers, the associated function g f : E(L(G)) 2 N 0, defined by g f (u i u j ) = f (u i ) + f (u j ), is also injective and each g f (u i u j ) is a set of non-negative integers. Therefore, f is an IASI of L(G). Figure 2 depicts the admissibility of an IASI by the line graph L(G) of a weakly IASI graph G. Figure 2 Now we recall the notion of the total graph of a given graph G and hence verify the admissibility of IASI by the total graph of an IASI graph. Definition 2.8. [2] The total graph of a graph G is the graph, denoted by T (G), is the graph having the property that a one-to one correspondence can be defined between its points and the elements (vertices and edges) of G such that two points of T (G) are adjacent if and only if the corresponding elements of G are adjacent (either if both elements are edges or if both elements are vertices) or they are incident (if one element is an edge and the other is a vertex). Theorem 2.9. If G is an IASI graph, then its total graph T (G) is also an IASI graph. Proof. Since G admits an IASI, say f, by the definition of an IASI f(v), v V (G) and g f (e), e E(G) are sets of non-negative integers. Define a map f : V (T (G)) 2 N 0 which assigns the same set-labels of the corresponding elements in G under f to the vertices of T (G). Clearly, f is injective and each f (u i ), u i V (T (G)) is a set of non-negative integers. Now, define the associated function g f : E(T (G)) 2 N 0 defined by g f (u i u j ) = f (u i ) + f (u j ), u i, u j V (T (G)). Then, g f is injective and each g f (u i u j ) is a set of non-negative integers. Therefore, f is an IASI of T (G). This completes the proof. 5

6 Figure 3 illustrates the admissibility of an IASI by the the total graph T (G) of a weakly IASI graph G. Figure 3 3 Some Certain Types of IASIs 3.1 Cardinality of the Set-Labels of Graphs To determine the set-indexing number of an edge of a graph G, if the setindexing numbers of its end vertices are given, is an interesting problem. All the sets mentioned in this paper are sets of non-negative integers. We denote the cardinality of a set A by A. Then, we recall the following lemma. Lemma 3.1. [7] If A, B N 0, then max( A, B ) A + B A B. Let f be an IASI defined on G and let u and v be two adjacent vertices of G labeled by two non-empty sets A and B respectively. Then, the setlabel for the edge uv in G is A + B. That is, f(u) = A and f(v) = B and g f (uv) = A + B. Definition 3.2. Two ordered pairs (a, b) and (c, d) in A B compatible if a + b = c + d. If (a, b) and (c, d) are compatible, then we write (a, b) (c, d). Definition 3.3. A compatible class of an ordered pair (a, b) in A B with respect to the integer k = a + b is the subset of A B defined by {(c, d) A B : (a, b) (c, d)} and is denoted by [(a, b)] k or C k. 6

7 From Definition 3.3, we note that (a, b) [(a, b)] k, each compatibility classes [(a, b)] k is non-empty. That is, the minimum number of elements in a compatibility class is 1. If a compatibility class contains exactly one element, then it is called a trivial class. All the compatibility class need not have the same number of elements but can not exceed a certain number. The compatibility classes which contain the maximum possible number of elements are called saturated classes. The compatibility class that contains maximum elements is called a maximal compatibility class. Hence, it can be observed that all saturated classes are maximal compatibility classes, but a maximal compatibility class need not be a saturated class. The following result provide the maximimum possible number of elements in a saturated class. Proposition 3.4. The cardinality of a saturated class in A B is n, where n = min( A, B ). Proof. Let A = {a 1, a 2, a 3,..., a m } and B = {b 1, b 2, b 3,..., b n } be two sets with m > n. Arrange the elements of A + B in rows and columns as follows. Arrange the elements of A + {b j } in j-th row in such a way that equal sums in each row comes in the same column. Hence, we have n rows in this arrangement in which each column corresponds to a compatibility class in (A, B). Therefore, a compatibility class can have a maximum of n elements. Definition 3.5. The number of distinct compatibility classes in A B is called the compatibility index of the pair of sets (A, B) and is denoted by (A,B). Proposition 3.6. The compatibility relation on A B is an equivalence relation. The relation between the set-indexing number of an edge of a graph G and the compatibility index of the pair of set-labels of its end vertices is established in the following lemma. Lemma 3.7. Let f be an IASI of a graph G and u, v be two vertices of G. Then, g f (uv) = f(u) + f(v) = {a + b : a f(u), b f(v)}. Then, the set-indexing number of the edge uv is g f (uv) = (f(u),f(v)). Proof. By Definition 3.3, each (a, b) C k contributes a single element k to the set g f (uv) = f(u) + f(v). Therefore, the number of elements in the set g f (uv) is the number of compatibility classes C k in f(u) f(v). Hence, g f (uv) = (f(u),f(v)). 7

8 Let r k be the number of elements in a compatibility class C k. From Lemma 3.7, we note that only one representative element of each compatibility class C k of f(u) f(v) contributes an element to the set-label of the edge uv and all other r k 1 elements are neglected and hence we may call these elements neglecting elements of C k. The number of such neglecting elements in a compatibility class C k may be called the neglecting number of that class. Then, the neglecting number of f(u) f(v), denoted by r, is given by r = r k1, where the sum varies over distinct compatibility classes in f(u) f(v). Hence, we have the following theorem. Theorem 3.8. Let f be an IASI of a graph G and let u and v be two vertices of G. Let f(u) = m and f(v) = n. Then, g f (uv) = mn r, where r is the neglecting number of f(u) f(v). We rewrite Lemma 3.1 as follows and provide an alternate proof for it. Lemma 3.9. [7] Let f be an IASI on G, then for the vertices u, v of G, max( f(u), f(v) ) g f (uv) = f(u) + f(v) f(u) f(v). Proof. Let f(u) = m and f(v) = n. Since 0 r < mn, by Observation 3.8, g f (uv) = f(u) + f(v) mn = f(u) f(v). Now, without loss of generality, let g f (uv) = m. Then, by Observation 3.8, mn r = m mn = m + r n r. (3.1.01) Also, mn m = r m(n 1) = r (n 1) r. (3.1.02) Since r can assume any non-negative integers less than mn and n is a positive integer such that n r and (n 1) r, we have r = 0. Hence, by Equation , m(n 1) = 0 (n 1) = 0, since m 0. Hence, n = 1. Hence, we have max( f(u), f(v) ). Let g f (uv) = n. Therefore, proceeding as above, m = 1. Combining the above conditions, max( f(u), f(v) ) g f (uv) f(u) f(v). Now, we recall the following definitions. Definition [7] An IASI f is said to be a weak IASI if g f (uv) = max( f(u), f(v) ) for all u, v V (G). A graph which admits a weak IASI may be called a weak IASI graph. A weak IASI is said to be weakly uniform IASI if g f (uv) = k, for all u, v V (G) and for some positive integer k. Definition [8] An IASI f is said to be a strong IASI if g f (uv) = f(u) f(v) for all u, v V (G). A graph which admits a strong IASI may be called a strong IASI graph. A strong IASI is said to be strongly uniform IASI if g f (uv) = k, for all u, v V (G) and for some positive integer k. 8

12 Theorem 4.1. Let X be a finite set of non-negative integers and let f : V (G) 2 X { } be an IASI on G, which has n vertices. Then, X has at least log 2 (n + 1) elements, where x is the ceiling function of x. Proof. Let G be a graph on n vertices. Then, there must be n non-empty subsets of X to label the vertices of G. Therefore, X must have at least n+1 subsets including. That is, 2 X (n + 1). Hence, X log 2 (n + 1). Theorem 4.2. Let X be a finite set of non-negative integers and let f : V (G) 2 X { } be an IASI on G, which has n vertices, such that V (G) is l-uniformly set-indexed. Then, the cardinality of X is given by ( ) X l n. Proof. Since V (G) is l-uniformly set-indexed, f(u) = l u V (G). Therefore, X must have at least n subsets with l-elements. Hence, ( ) X l n. 4.1 Open Problems We note that an IASI with a finite ground set X is never a topological setindexer. Is an IASI set-graceful? We note that all IASIs are not set-graceful in general. The following is an open problem for further investigation. Problem 4.3. Check whether an IASI f can be a set-graceful labeling. If so, find the necessary and sufficient condition for f to be set-graceful. Problem 4.4. Check whether an IASI f can be a set-sequential labeling. If so, find the necessary and sufficient condition for f to be set-sequential. 5 Conclusion In this paper, we have discussed about integer additive set-indexers of graphs and the admissibility of IASI by certain associated graphs of a given IASI graph. Some studies have also been made certain IASI graphs with regard to the cardinality of the labeling sets of their elements. The characterisation of certain types IASIs in which the labeling sets have definite patterns, are yet to be made. Questions related to the admissibility of different types of IASIs by certain graph classes and graph structures are to be addressed. All these facts highlights a wide scope for future studies in this area. 12

13 References [1] B D Acharya,(1983). Set-Valuations and Their Applications, MRI Lecture notes in Applied Mathematics, The Mehta Research Institute of Mathematics and Mathematical Physics, New Delhi. [2] M Behzad, (1969). The connectivity of Total Graphs, Bull. Austral. Math. Soc, 1, [3] M Capobianco and J Molluzzo, (1978). Examples and Counterexamples in Graph Theory, North-Holland, New York. [4] N Deo, (1974). Graph theory With Application to Engineering and Computer Science, PHI Pvt. Ltd., India. [5] J A Gallian, A Dynamic Survey of Graph Labelling, The Electronic Journal of Combinatorics (DS 16), (2011). [6] K A Germina and T M K Anandavally, (2012). Integer Additive Set- Indexers of a Graph:Sum Square Graphs, Journal of Combinatorics, Information and System Sciences, 37(2-4), [7] K A Germina, N K Sudev, (2013). On Weakly Uniform Integer Additive Set-Indexers of Graphs, Int. Math. Forum., 8(37-40), [8] K A Germina, N K Sudev, Some New Results on Strong Integer Additive Set-Indexers, Communicated. [9] F Harary, (1969). Graph Theory, Addison-Wesley Publishing Company Inc. [10] K D Joshi, Applied Discrete Structures, New Age International, (2003). [11] N K Sudev and K A Germina, (2014). A Characterisation of Weak Integer Additive Set-Indexers of Graphs, ISPACS Journal of Fuzzy Set Valued Analysis, 2014, Article Id: jfsva-00189, 7 pages. [12] D B West, (2001). Introduction to Graph Theory, Pearson Education Inc. 13

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