Product Cordial Sets of Trees

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1 Product Cordial Sets of Trees Ebrahim Salehi, Seth Churchman, Tahj Hill, Jim Jordan Department of Mathematical Sciences University of Nevada, Las Vegas Las Vegas, NV Abstract A binary vertex coloring (labeling) f : V (G) Z 2 of a graph G is said to be friendly if the number of vertices labeled is almost the same as the number of vertices labeled. This friendly labeling induces an edge labeling f : E(G) Z 2 defined by f (uv) = f(u)f(v) for all uv E(G). Let e f (i) = {uv E(G) : f (uv) = i} be the number of edges of G that are labeled i. Product-cordial index of the labeling f is the number pc(f) = e f () e f (). The product-cordial set of the graph G, denoted by P C(G), is defined by P C(G) = {pc(f) : f is a friendly labeling of G }. In this paper, we determine the product-cordial sets of certain classes of trees. Key Words: friendly coloring, product-cordial index, product-cordial set, fully cordial. AMS Subject Classification: 5C78 Introduction In this paper all graphs G = (V, E) are connected, finite, simple, and undirected. For graph theory notations and terminology not described in this paper, we refer the readers to [6]. Let G be a graph and f : V (G) Z 2 be a binary vertex coloring (labeling) of G. For i Z 2, let v f (i) = f (i). The coloring f is said to be friendly if v f () v f (). That is, the number of vertices colored is almost the same as the number of vertices colored. Any friendly coloring f : V (G) Z 2 induces an edge labeling f : E(G) Z 2 defined by f (xy) = f(x)f(y) xy E(G). For i Z 2, let e f (i) = f (i) be the number of edges of G that are labeled i. The number pc(f) = e f () e f () is called the product-cordial index (or pc-index) of f. The product-cordial set (or pc-set) of the graph G, denoted by P C(G), is defined by Congressus Numerantium 22 (24),

2 P C(G) = {pc(f) : f is a friendly vertex coloring of G }. To illustrate the above concepts, consider the graph G of Figure, which has 9 vertices. The condition v f () v f () implies that four vertices be labeled and the other five or vice versa. Figure : An example of product-cordial labeling of G. Figure also shows the associated edge labeling of G, where all edges have label. Therefore, the product-cordial index (or pc-index) of this labeling is e() e() = 8 = 8. It is easy to see that P C(G) = {, 2, 4, 6, 8}. The friendly colorings of G that provide the other four pc-indices are presented in Figure 2. Figure 2: Four friendly labelings of G with pc-indices 6, 4, 2 and. In 978, I. Cahit [2, 3, 4] introduced the concept of cordial labeling as weakened version of the less tractable graceful and harmonious labeling. A graph G is said to be cordial if it admits a friendly labeling with index or. M. Hovay [9], later generalized the concept of cordial graphs and introduced A-cordial labelings, where A is an abelian group. A graph G is said to be A-cordial if it admits a labeling f : V (G) A such that for every i, j A, v f (i) v f (j) and e f (i) e f (j). Cordial graphs have been studied extensively. Interested readers are referred to a number of relevant literature that are mentioned in the bibliography section, including [, 5, 8,,, 4, 2]. 84

3 Product cordial labeling of a graph was introduced by Sundaram, Ponraj and Somasundaram [23]. They call a graph G product-cordial if it admits a friendly labeling whose product-cordial index is at most. Then Sundaram, Ponraj and Somasundaram [23, 24, 25] investigated whether certain graphs such as trees, cycles, complete graphs, wheels, etc. are product-cordial. Later E. Salehi [5] introduced the concept of product-cordial set (or pc-set) of a graph and determined the pc-sets of certain classes of graphs such as: complete graphs, complete bipartite graphs, stars and double stars, cycles, and wheels. In this paper we determine the product cordial sets of certain classes of trees. In what follows, whenever there is no ambiguity, we suppress the index f and denote e f (i) by simply e(i). For a graph G = (p, q) of size q, and a friendly labeling f : V (G) Z 2 of G, we have pc(f) = e f () e f () = q 2e f () = q 2e f (). (.) Therefore, to find the pc-index of f it is enough to find e f () ( or e f () ). Moreover, to determine the pc-set of G it is enough to compute e f () for different friendly colorings of G. Another immediate consequence of (.) is the following useful fact: Observation.. For a graph G of size q, P C(G) {q 2k : k q/2 }. Definition.2. A graph G of size q is said to be fully product-cordial (fully pc) if P C(G) = {q 2k : k q/2 }. For example, the graph G of Figure is not fully pc. However, P n, the path of order n, is fully pc. In case of P n, it is easy to see that there are friendly colorings of P n such that e() =,,, n, which proves the following theorem: 2 Theorem.3. For any n 2, the graph P n is fully product-cordial. That is, P C(P n ) = {n 2k : k n 2 }. The different friendly labelings of P 7 that provide its pc-set are illustrated in Figure 3. Definition.4. A matching in a graph is a set of edges with no shared endpoints. A matching M in a graph G is said to be a perfect matching if every vertex of G is incident with an edge in M. Theorem.5. [6] Any tree T of order p with a perfect matching is fully product-cordial. That is, P C(T ) = {, 3, 5,, p }. 85

4 Figure 3: P C(P 7 ) = {, 2, 4, 6}. In general, for a friendly coloring f : V (G) Z 2 of a graph G, it is not necessarily true that e f () e f (). For example, let n > 3 and consider the coronation [7] of the complete graph K n with K, as indicated in Figure u 2 u K n u n Figure 4: A friendly coloring with e() > e(). If we color all vertices of K n by and the end-vertices by, then e() = n(n )/2 while e() = n. However, for certain graphs one can prove that the number of edges labeled is not less than the number of edges labeled. Trees are among such graphs as we will see in the following theorem: Theorem.6. For any tree T and any friendly coloring of T, e() e(). Proof. The statement is true for trees of order n =, 2, 3. Let T be a tree of order n 4 and observe that at least e()+ vertices of T are labeled with. Since the coloring is friendly, at least e() vertices of T are labeled with. This implies that n 2e() + or E 2e(). Therefore, 2e() E = e() + e(), or e() e(). One immediate consequence of Theorem.6 and equation (.) is that for any coloring f : T Z 2 of a tree T, pc(f) = q 2e f (). As a result, to show that a tree T is fully product cordial, one needs to to present different friendly colorings of T such that e() =,,, q 2, where q is the number of edges of T. This observation leads us to the following useful fact: 86

5 Theorem.7. Consider the tree T = (p, q) and for any j =,,, q 2 let S j be a subtree of T with j + vertices. Label all vertices of S j with. If we can extend this labeling to a friendly labeling of T such that e() = j, then T is fully product cordial. This theorem is applied to P 7, as illustrated in Figure 3. Observation.8. If T is a tree of odd (even) order, then there is a friendly coloring of T with pc index (). Proof. Let T be tree of order n and choose a subtree S of T of order n/2. If we label all the vertices of S by and the remaining vertices of T by, then this is a friendly coloring of T with n n e() e() =. 2 2 Next, we show that any full binary tree is fully cordial. Definition.9. Full binary tree of depth d, denoted by F B d, is defined inductively as follows: F B is the path P 3 with the middle vertex being its root and for n 2, F B n is a binary tree of root r n whose left and right children are F B n. Note that in a full binary tree of depth d, there are 2 d+ vertices, hence 2 d+ 2 edges. Such a tree contains 2 d leaves (end-vertices) all of them in the d th row and there are 2 d vertices with degrees bigger than. Theorem.. Any full binary tree is fully product cordial. Moreover, every index can be obtained by a friendly coloring that labels half of the leaves and the other half. Proof. We proceed by induction on d, the depth of the full binary tree. For convenient, for any non-root vertex u, its parent vertex will be denoted by p u. Clearly the statement is true for d =, 2 as illustrated in Figure 5. N= N=2 N= 4 N= 6 Figure 5: Four friendly colorings of F B 2 with four different indices. Suppose the statement is true for F B d. That is, P C(F B d ) = {, 2, 4,, 2 d 2} and every member of this set can be obtained by a friendly coloring that labels 2 d leaves and the other 87

6 2 d leaves. We wish to show that P C(F B d ) = {, 2, 4,, 2 d 2, 2 d, 2 d+ 2}. Let j be any non-negative even integer less than 2 d+. We consider two cases: A. j 2 d 2. In this case, let f be the friendly coloring of F B d that produces index j and extend it to φ : V (F B d ) Z 2 by defining φ(u) = { f(u) if deg u > ; f(u p ) if deg u =. Then φ is a friendly coloring of F B d with e φ () = e f () + 2 d, e φ () = e f () + 2 d. Therefore, pc(φ) = e φ () e φ () = pc(f) = j. B. 2 d j 2 d+ 2. In this case, let g be the friendly coloring of F B d that produces index j 2 d and extend it to ψ : V (F B d ) Z 2 by defining ψ(u) = { f(u) if deg u > ; f(u p ) if deg u =. Then ψ is a friendly coloring of F B d with e ψ () = e f (), e ψ () = e f () + 2 d. Therefore, pc(ψ) = e ψ () e ψ () = pc(f) + 2 d = j. Moreover, not only φ, ψ are friendly colorings of F B d, but they also label 2 d leaves and the other 2 d leaves. 2 Near Perfect Matching Trees In [6], Salehi-Mukhin showed that any tree with perfect matching is fully product cordial. However, there are many other fully pc trees that do not have perfect matchings. Paths of odd orders P 2n+ are the most obvious examples. In this section we introduce another class of fully pc trees. Namely, near perfect matching trees. Definition 2.. A matching of a graph G is called near perfect matching if it covers all the vertices of G but one. G is called a near perfect matching graph if any maximum matching of G is near perfect matching. Observation 2.2. A tree T with near perfect matching M contains at least a P 3 pendent u u 2 u 3 such that deg u =, deg u 2 = 2 and u u 2 M. 88

7 Proof. Let P : u u 2 u 3 u k 2 u k u k be the longest path in T. Clearly, deg u = deg u k =. Also, deg u 2 = 2 or deg u k = 2. Otherwise, any maximum matching of T would miss at least two vertices. If deg u 2 = deg u k = 2, then u u 2 M or u k u k M. Suppose (wlog) deg u 2 = 2 and deg u k > 2. Then u u 2 M. Otherwise, any maximum matching of T would miss at least two vertices. Theorem 2.3. Any tree T with near perfect matching is fully product-cordial. Proof. Note that T is a tree of odd order, T = 2n +. We proceed by induction on n. Clearly, the statement of theorem is true for n =. Suppose the statement is true for any tree of order 2n + and let T be a tree of order 2n + 3 with near-perfect matching M. By 2.2, T contains vertices u v w such that deg w =, deg v = 2 and the edge vw is in M. Now consider the tree S = T {v, w} which has order 2n + and has near perfect matching M Therefore, by the induction hypothesis P C(S) = {, 2, 4,, 2n}. = M {vw}. We need to show that P C(T ) = {, 2,, 2n, 2n+2}. Consider a friendly coloring f : V (S) Z 2 of S and extend it to φ : V (T ) Z 2 by defining f(x) if x v, w; φ(x) = f(u) if x = w; f(u) if x = v. Then φ is a friendly coloring of T with e φ () = e f (), e φ () = e f () + 2. Therefore, pc(φ) = pc(f) + 2. This implies that 2 + P C(S) = {2, 4,, 2n + 2} P C(T ). It only remains to show that P C(T ), which follows from.8. Definition 2.4. Fibonacci Trees, denoted by F T n, are defined inductively as follows: F T is the trivial tree with one vertex, F T 2 is the path P 2, and for n 3, F T n = (V n, E n ) is the binary tree of root r n, whose left and right children are F T n and F T n 2, respectively. There is a relationship between the order of F T n and Fibonacci numbers. In fact, the number of vertices of F T n is A n+2, where A n is the n th Fibonacci number. Theorem 2.5. For n, every Fibonacci tree F T n is fully product cordial. Proof. Note that every Fibonacci tree has either a perfect matching or is a near perfect matching tree. In fact, if n (mod 3), then F T n is a near perfect matching tree; Otherwise, it has a 89

8 r 4 FT FT 2 FT 3 FT 4 r 4 r 5 FT 5 Figure 6: Graphs of the first five Fibonacci trees. perfect matching. We prove this statement by induction on n. Clearly the statement is true for n =, 2, 3. Now suppose the statement is true for all positive integers less than n (3 < n) and let F T n be the Fibonacci tree of order n. We consider the following cases: (A) n (mod 3). In this case, by the induction hypothesis, both the left and right children have perfect matchings. Let M and M 2 be perfect matchings of F T n and F T n 2, respectively. Then M M 2 is a maximum matching of F T n that covers all the vertices but its root. Therefore, F T n is a near perfect matching tree. (B) n 2 (mod 3). In this case, by the induction hypothesis, the left child F T n is near perfect matching while the right child F T n 2 has a perfect matching. Let M be a maximum matching of F T n (we may assume that M leaves the root r n out) and M 2 be a perfect matching of F T n 2. Then M M 2 {r n r n } will form a perfect matching of F T n. (C) n (mod 3). The argument is similar to the previous case. 3 Lucas Trees Definition 3.. Lucas Trees, denoted by LT n, are defined inductively as follows: LT is the trivial tree with one vertex, LT 2 is the path P 3 with the middle vertex being its root, and for n 3, LT n = (V n, E n ) is the binary tree of root r n, whose left and right children are LT n and LT n 2, respectively. 9

9 LT LT 2 LT 3 r 4 r 5 LT 4 LT 5 Figure 7: The first five Lucas trees. Theorem 3.2. For n, every Lucas tree LT n is fully product cordial. Proof. This is an easy consequence of theorem.7. References [] M. Benson and S-M. Lee, On Cordialness of Regular Windmill Graphs, Congressus Numerantium 68 (989), [2] I. Cahit, Cordial Graphs: a weaker version of graceful and harmonious graphs, Ars Combinatoria 23 (987), [3] I. Cahit, On Cordial and 3-equitable Graphs, Utilitas Mathematica 37 (99), [4] I. Cahit, Recent Results and Open Problems on Cordial Graphs, Contemporary Methods in Graph Theory, Bibligraphisches Inst. Mannhiem (99), [5] N. Cairnie and K. Edwards, The Computational Complexity of Cordial and Equitable Labelings, Discrete Mathematics 26 (2), [6] G. Chartrand and P. Zhang, Introduction to Graph Theory, McGraw-Hill, Boston (25). 9

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