International Mathematical Forum, Vol. 7,, no. 35, 737-749 Special Classes of Divisor Cordial Graphs R. Varatharajan Department of Mathematics, Sri S.R.N.M. College Sattur - 66 3, Tamil Nadu, India varatharajansrnm@gmail.com S. Navanaeethakrishnan Department of Mathematics V.O.C. College, Tuticorin - 68 8 Tamil Nadu, India snk.voc@gmail.com K. Nagarajan Department of Mathematics Sri S.R.N.M.College, Sattur - 66 3 Tamil Nadu, India k nagarajan srnmc@yahoo.co.in Abstract A divisor cordial labeling of a graph G with vertex set V is a bijection f from V to {,,..., V } such that if each edge uv is assigned the label if f(u) divides f(v) or f(v) divides f(u) and otherwise, then the number of edges labeled with and the number of edges labeled with differ by at most. If a graph has a divisor cordial labeling, then it is called divisor cordial graph. In this paper, we proved some special classes of graphs such as dragon, corona, wheel, full binary trees, G K,n and G K 3,n are divisor cordial. Mathematics Subject Classification: 5C78 Keywords: Cordial labeling, divisor cordial labeling, divisor cordial graphs Introduction By a graph, we mean a finite, undirected graph without loops and multiple edges, for terms not defined here, we refer to Harary [5].
738 R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan First we give the some concepts in Number Theory [3]. Definition.. Let a and b be two integers. If a divides b means that there is a positive integer k such that b = ka. It is denoted by a b. If a does not divide b, then we denote a b. Graph labeling [4] is a strong communication between Number theory [3] and structure of graphs [5]. By combining the divisibility concept in Number theory and Cordial labeling concept in Graph labeling, we introduced a new concept called divisor cordial labeling [8]. In paper[8], we proved the standard graphs such as path, cycle, wheel, star and some complete bipartite graphs are divisor cordial and complete graph is not divisor cordial. In this paper we are going to prove some special classes of graphs such as full binary trees, dragon, corona, wheel, G K,n and G K 3,n are divisor cordial. A vertex labeling [4] of a graph G is an assignment f of labels to the vertices of G that induces for each edge uv a label depending on the vertex label f(u) and f(v). The two best known labeling methods are called graceful and harmonious labelings. Cordial labeling is a variation of both graceful and harmonious labelings []. Definition.. Let G = (V, E) be a graph. A mapping f : V (G) {,} is called binary vertex labeling of G and f(v) is called the label of the vertex v of G under f. For an edge e = uv, the induced edge labeling f : E(G) {,} is given by f (e)= f(u) f(v). Let v f (), v f () be the number of vertices of G having labels and respectively under f and e f (), e f () be the number of edges having labels and respectively under f. The concept of cordial labeling was introduced by Cahit []. Definition.3. A binary vertex labeling of a graph G is called a cordial labeling if v f () v f () and e f () e f (). A graph G is cordial if it admits cordial labeling. Cahit proved some results in []. Main Results Sundaram, Ponraj and Somasundaram [6] have introduced the notion of prime cordial labeling. Definition.. [6] A prime cordial labeling of a graph G with vertex set V is a bijection f from V to {,,..., V } such that if each edge uv assigned the label if gcd(f(u), f(v)) = and if gcd(f(u), f(v)) >, then the number of edges labeled with and the number of edges labeled with differ by at most.
Special classes of divisor cordial graphs 739 In [6], they have proved some graphs are prime cordial. Motivated by the concept of prime cordial labeling, we introduced a new special type of cordial labeling called divisor cordial labeling as follows. Definition.. [8] Let G =(V, E) be a simple graph and f : V {,,... V } be a bijection. For each edge uv, assign the label if either f(u) f(v) or f(v) f(u) and the label if f(u) f(v). f is called a divisor cordial labeling if e f () e f (). A graph with a divisor cordial labeling is called a divisor cordial graph. Example.3. Consider the following graph G. 4 5 6 F ig We see that e f () = 3 and e f () = 4. Thus e f () e f () = and hence G is divisor cordial. 3 In this paper we prove some special classes of graphs are divisor cordial. Theorem.4. Given a positive integer n, there is a divisor cordial graph G which has n vertices. Proof. Suppose n is even. Construct a path containing n + vertices v, v,..., v n + which are labeled as,,..., n + respectively. Note that the label of the edge v v is and the other edges v i v i+ ( i n + ) have the labels. Attach n vertices v n +3, v n +4,..., v n which are labeled as n +3,... n respectively, to the vertex v. We see that the labels of the edges v v i ( n + 3 i n) are. v v v v n 3 + v n + n 3 4 + n + v n +3 n + 3 n v n +4 + 4 n v n Fig
74 R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan So, we have e f () = n and e f () = + n = n and hence e f () e f (). Thus, the resultant graph G is divisor cordial., Similarly, we can construct a graph for n is odd. Theorem.5. If G is a divisor cordial graph of even size, then G e is also divisor cordial for all e E(G). Proof. Let q be the even size of the divisor cordial graph G. Then it follows that e f () = e f () = q. Let e be any edge in G which is labeled either or. Then in G e, we have either e f () = e f () + or e f () = e f () + and hence e f () e f (). Thus G e is divisor cordial. Theorem.6. If G is a divisor cordial graph of odd size, then G e is also divisor cordial for some e E(G). Proof. Let q be the odd size of the divisor cordial graph G. Then it follows that either e f () = e f () + or e f () = e f () +. If e f () = e f () + then remove the edge e which is labeled as and if e f () = e f () + then remove the edge e which is labeled from G. Then it follows that e f () = e f (). Thus G e is divisor cordial for some e E(G). Note.7. Similarly we can prove for the divisor cordiality of G + e. Definition.8. An ordered rooted tree is a binary tree if each vertex has atmost two children. Definition.9. A full binary tree is a binary tree inwhich each internal vertex has exactly two children. Next we will prove that a full binary tree is divisor cordial. Theorem.. Every full binary tree is divisor cordial. Proof. We note that every full binary tree has odd number of vertices and hence has even number of edges. Let T be a full binary tree and let v be a root of T which is called zero level vertex. Clearly, the i th level of T has i vertices. If T has m levels, then the number of vertices of T is m+ and the number of edges is m+. Now assign the label to the root v and assign the labels 3 and to the first level vertices. Next, we assign the labels i, i +,..., i+ to the i th level vertices for i m In zeroth, first and second levels, we have e f () = e f () = 3. Clearly, i + j i+ + j for j i and i + j i+ + j for j i. Then after the second levels, we have e f () = e f () = m 3. Thus e f () e f () = and hence T is divisor cordial.
Special classes of divisor cordial graphs 74 The next example shows that the divisor cordiality of full binary tree upto 4 levels. Example.. Here we see that e f () = e f () = 5. 3 4 5 6 7 8 9 3 4 5 6 7 8 9 3 4 5 6 7 8 9 3 3 F ig 3 Theorem.. Let G be any divisor cordial graph of size m and K,n be a bipartite graph with the bipartition V = V V with V = {x, x } and V = {y, y,... y n }. Then the graph G K,n obtained by identifying the vertices x and x of K,n with that labeled and the largest prime number p such that p m respectively in G is also divisor cordial. Proof. Let G be any divisor cordial graph of size m. Let v k and v l be the vertices having the labels and the largest prime number namely p such that p m. Let V = V V be the bipartition of K,n such that V = {x, x } and V = {y, y,... y n }. Now assign the labels m +, m +,... m + n to the vertices y, y,..., y n respectively. Now identify the vertices x and x of K,n with that labeled and the largest prime number p such that p m respectively in G. Case (i) p < m + n < p Then the multiples of p are not available in the labels of y i ( i n). Then we see that the edges of K,n incident with the vertex v k have the label and with the vertex v l have the label. Thus the edges of K,n contribute equal numbers namely n to both e f () and e f () in G K,n. Hence G K,n is divisor cordial. Case (ii) m + n p.
74 R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan Let q be the largest prime number such that q m + n which is labeled to some y i. Then interchange the labels of v l and y i, that is p and q. We observe that the largest prime number q does not divide the labels of y, y,..., y n. So, again the edges of K,n incident with the vertex v l have the labels and hence G K,n is divisor cordial. The following example explains the construction developed in the Theorem.. Example.3. Consider the following graph G. 6 3 4 Fig 4 5 Here e f () = 4 and e f () = 3 and hence e f () e f () =. Thus G is divisor cordial. Case(i): Now consider the bipartite graph K,3. Then the graph G K,3 and its labels are gives as follows.
Special classes of divisor cordial graphs 743 6 4 3 5 7 8 9 Fig 5 Here e f () = 7 and e f () = 6 and e f () e f () =. Thus G K,3 is divisor cordial. Case (ii): Now consider the bipartite graph K,5. The graph G K,5 and its labels are given as follows.
744 R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan 6 3 4 7 8 9 5 Fig 6 Here we interchanged the labels 5 and and e f () = 9 and e f () =8 and so e f () e f () = Thus G K,5 is divisor cordial. Theorem.4. Let G be any divisor cordial graph of size m and K 3,n be a bipartite graph with the bipartition V = V V with V = {x, x, x 3 } and V = {y, y,... y n }, where n is even. Then the graph G K 3,n obtained by identifying the vertices x, x and x 3 of K 3,n with that labeled, labeled and the largest prime number p such that p m respectively in G is also divisor cordial. Proof. Let G be any divisor cordial graph of size m. Then e f () e f (). Let v k, v l and v r be the vertices having the labels, and the largest prime number namely p such that p m. Let V = V V be the bipartition of K 3,n such that V = {x, x, x 3 } and V = {y, y,... y n }. Now assign the labels m +, m +,... m + n to the vertices y, y,..., y n respectively. Now identify the vertices x, x and x 3 of K 3,n with that labeled, labeled and and the largest prime number p such that p m respectively in G. Case (i) p < m + n < p Then the multiples of p are not available in the labels of y i ( i n). Then we see that the edges of K 3,n incident with the vertex v k have the label and with the vertex v r have the label. Since n divides only even numbers, n edges of K 3,n incident with the vertex v l have the label and remaining n edges have the label. Thus the edges of K 3,n contribute equal numbers
Special classes of divisor cordial graphs 745 namely 3n to both e f () and e f () in G K 3,n. Hence G K 3,n is divisor cordial. Case (ii) m + n p. Let q be the largest prime number such that q m + n which is labeled to some y i. Then interchange the labels of v r and y i, that is p and q. We observe that the largest prime number q does not divide the labels of y, y,..., y n. So, again the edges of K 3,n incident with the vertex v r have the labels and hence G K 3,n is divisor cordial. Note.5. Theorem.4 is also valid for m is even and n is odd. If both m and n are odd, the above labeling pattern is not valid. The following example illustrates the construction developed in the Theorem.4. Example.6. Consider the following divisor cordial graph G. 6 4 3 5 Fig 7 The divisor cordiality of G K 3,4 is shown below.
746 R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan 4 3 7 5 8 9 Fig 8 Here e f () = e f () =. Now we present divisor cordial labeling for the graphs obtained by joining apex vertices of two stars to a new vertex. We extend this result for three copies of stars. Definition.7. Consider two stars K (),n and K (),n. Then G =< K (),n, K (),n > is the graph obtained by joining apex (central) vertices of stars to a new vertex x. Note that G has n + 3 vertices and n + edges. Definition.8. Consider t copies of stars namely K,n, () K (),n,..., K (t),n. Then G =< K (),n, K (),n,..., K (t),n > is the graph obtained by joining apex vertices of each K (m ),n and K (m),n to a new vertex x m where m t. Note that G has t(n + ) vertices and t(n + ) edges. First we will prove that < K (),n, K (),n > is divisor cordial. Theorem.9. The graph G =< K (),n, K (),n > is divisor cordial. Proof. Let v (), v (),..., v n () be the pendant vertices of K (),n and v (), v (),..., v n () be the pendant vertices of K (),n. Let c and c be the apex
Special classes of divisor cordial graphs 747 vertices of K (),n and K (),n respectively and they are adjacent to a new common vertex x. Now assign the label to c and the largest prime number p such that p n+3 to c and the remaining labels to the vertices of G. Since divides any integer the edges incident to c contribute n + to e f () and since p does not divide any labels of the vertices adjacent to c, the edges incident to c also contribute n + to e f (). Hence e f () e f () =. Thus G is divisor cordial. Next we will extend this result to 3 stars as follows. Theorem.. The graph G =< K (),n, K (),n, K (3),n > is divisor cordial. Proof. Let Let v (i), v (i),..., v n (i) be the pendant vertices of K (i),n and let c i be the apex vertex of K (i),n for i =,, 3. Now c and c are adjacent to x and c and c 3 are adjacent to x. Note that G has 3n + 5 vertices and 3n + 4 edges. Now assign the label to c, to c and p to c 3 where p is the largest prime number such that p 3n + 5. Then assign the odd and even labels equally to the vertices v (), v (),..., v n (), x, x if n is even. Suppose n is odd, then assign n+ odd labels and n+3 even labels to these vertices. Then assign remaining labels to the remaining vertices. Since divides any integer, the edges incident to c contribute n + to e f (). Since p does not divide any labels of the vertices adjacent to c 3, the edges incident to c 3 also contribute n + to e f (). Since divides even labels and does not divide odd labels, the edges incident to c contribute n+ ( n+) to e f() and n+ ( n+3) to e f() if n is even (odd). Thus, if n is even, then e f () = e f () = n + + n+ = 3n+4 and if n is odd then e f () = n + + n+ =n+ and e f () = n + + n+3 = n + and hence e f () e f (). Thus G is divisor cordial. The Theorems.9 and. are explained using the following example. Example.. Consider the graph G =< K (),8, K (),8 >.
748 R. Varatharajan, S. Navanaeethakrishnan and K. Nagarajan 6 7 5 x 8 8 6 9 5 4 9 3 3 4 7 Fig 9 Here e f () = e f () = 9. Example.. Consider the graph G =< K (),7, K (),7, K (3),7 >. 8 7 6 5 3 4 x x 9 6 8 5 5 4 4 3 Fig 7 9 Here e f () = and e f () = 3 and e f () e f () =. References [] I. Cahit, Cordial graphs: A weaker version of graceful and harmonious graphs, Ars combinatoria, 3(987), -7. [] I. Cahit, On cordial and 3-equitable labelings of graph, Utilitas Math, 37(99), 89-98. [3] David M. Burton, Elementary Number Theory, Second Edition, Wm. C. Brown Company Publishers, 98.
Special classes of divisor cordial graphs 749 [4] J. A. Gallian, A dynamic survey of graph labeling, Electronic Journal of Combinatorics, 6(9), DS6. [5] F. Harary, Graph Theory, Addison-Wesley, Reading, Mass, 97. [6] M. Sundaram, R. Ponraj and S. Somasundaram, Prime cordial labeling of graphs, Journal of Indian Academy of Mathematics, 7(5) 373-39. [7] S. K. Vaidya, N. A. Dani, K. K. Kanani and P. L. Vihol, Cordial and 3- equitable labeling for some star related graphs, International mathematical Forum, 4, 9, no. 3, 543-553. [8] R. Varatharajan, S. Navaneethakrishnan and K. Nagarajan, Divisor cordial labeling of graphs, International Journal of Mathematical Combinatorics, Vol. 4(), 5-5. Received: November,