ON DIFFERENCE CORDIAL GRAPHS

Transcription

1 Mathematica Aeterna, Vol. 5, 05, no., 05-4 ON DIFFERENCE CORDIAL GRAPHS M. A. Seoud Department of Mathematics, Faculty of Science Ain Shams University, Cairo, Egypt Shakir M. Salman Department of Mathematics, Basic Education College Diyala University, Diyala, Iraq Abstract In this paper we introduce some results in dierence cordial graphs and the dierence cordial labeling for some families of graphs as: ladder,triangular ladder,grid,step ladder and two sided step ladder graph. Also we discussed some families of graphs which may be dierence cordial or not,such as diagonal ladder and some types of one-point union graphs. Mathematics Subject Classication: 05C78 Keywords: Dierence cordial graphs. Introduction In this paper we will deal with nite,simple and undirected graphs. By the expression G = (V, E) we mean a simple undirected graph with vertex set V, V is called the order of graph and edge set E, E is called its size. Graph labeling connects many branches of mathematics and is considered one of important blocks of graph theory, for more details see [3]. Cordial labeling was rst introduced in 987 by Cahit [], then there was a major eort in this area made this topic growing steadily and widely,see[]. In [4] Ponraj,Shathish Naraynan and Kala introduce the notions of dierence cordial labeling for nite undirected and simple graph,as in the following denition :

2 06 Seoud and Salman Denition. Let G = (V, E) be a (p, q) graph,and f be a map from V (G) to,,..., p. For each edge uv assign the label f(v) f(u), f is called a dierence cordial labeling if f is one to one map and e f (0) e f () where e f () denotes the number of edges labeled with while e f (0) denotes the number of edges not labeled with. A graph with a dierence cordial labeling is called a dierence cordial graph [4]. Ponraj et al. show every graph is a subgraph of a dierence cordial graph and any r regular graph with r 4 is not dierence cordial graph,every path and cycle are dierence cordial graphs,the star graph K,n is dierence cordial if and only if n 5,the graph K n is dierence cordial only when n 4 while the bipartite graph K m,n is not dierence cordial if m 4 and n 4,the bistar B m,n is not dierence cordial when m + n 9 but the wheel W n,the fan F n,the gear G n,the helm H n and all webs are dierence cordial graphs for all n [4]. In [5] the authors investigated the dierence cordial labeling behavior of G P n, G mk (m =,, 3) where G is either unicyclic or a tree and G G are some more standard graphs. Some graphs obtained from triangular snake and quadrilateral snake were investigated with respect to the dierence cordial labeling behavior. Also the behavior of subdivision of some snake graphs is investigated in [5]. Proposition. If G is a (p, q) dierence cordial graph,then q p [4]. Denition.3 The number δ(g) = min {d(v) v V } is the minimum degree of the vertices in the graph G, the number (G) = max {d(v) v V } is the maximum degree of the vertices in the graph G, the number d(g) = V d(v) is the average degree of the vertices in the graph G [7] v V Denition.4 A fan graph is obtained by joining all vertices of a path P n to a further vertex,called the center. Thus F n contains n + vertices say c, v, v, v 3,..., v n and n edges,say cv i, i n, and,v i v i+, i n. Notation.5 The maximum number of edges labelled that is related with a specic vertex,equals. Main Results Proposition. The graph G(p, q) is not dierence cordial graph if δ(g) 4. Proof. Let G(p, q) be any graph with δ(g) 4 ; then the minimum value of q is p; but p p, this contradicts Proposition..

3 On Dierence Cordial Graphs 07 Proposition. The graph G(p, q) is not dierence cordial if d(g) 4. Proof. Let G(p, q) be any graph with d(g) 4;then the value of q is more than or equal to p, but p p, which is contradicts Proposition.. Remark The value of e f (0) is not exceeding p in any dierence cordial graph G(p, q). Proof. Direct consequence of Proposition.. Proposition.3 Let G(p, q) be a graph with two vertices of degree (p ) then G is not a dierence cordial graph for all p 8. Proof. Let G(p, q) be a graph with p vertices, p 8 and has two vertices v i, v j of degree (p ) then there are p 3 dierent edges incident with them, If there are more than two additional edges then G is not dierence cordial since q p. If there are only two additional edges then q = p, then we have two cases: Case : the edge connecting v i and v j is labelled 0, then there are at most 6 edges labelled : two passing through v i, two are passing through v j and the two additional edges. In this case p 7 6 = p 3 where p 8 i.e., G is not dierence cordial. Case : the edge connecting v i and v j is labelled, then there are at most 5 edges labelled : one passing through v i and v j, two edges are: one is incident with v i and other is incident with v j and the two additional edges. In this case p 6 5 = p where p 7 i.e., G is not dierence cordial. In case there is one additional edge, other than those incident with v i, v j, similar argument is used. Example.4. Figure : The graph G = (8, 5) deg(v 8 ) = 7, deg(v 7 ) = 7 and G cannot be a dierence cordial graph.

4 08 Seoud and Salman Proposition.5 Let G(p, q) be any graph with two vertices of degrees (p ) and (p ); then G is not a dierence cordial graph for all p 9. Proof. Similar to the proof of Proposition.3. Example.6. Figure : The graph G = (9, 7) deg(v 8 ) = 7, deg(v 9 ) = 8 and G cannot be a dierence cordial graph. In [6] theorem.4,r. Ponraj,S. Sathish Narayanan and R. Kala state that "Let G be a (p, q) dierence cordial graph with k(k > ) vertices of degree p. Then p 7 ". However : Corollary.7 The graph G(p, q) is not a dierence cordial graph if there exist three vertices of degree (p ) for all p 6. Proof. Let G(p, q) be a graph with three of its vertices of degree p then there exist at least 3p 6 edges in the graph, by Proposition if the graph is a dierence cordial graph then A contradiction when p 6. Example.8. 3p 6 p Figure 3: The graph G = (6, ) 6 and G cannot be a dierence cordial graph.

5 On Dierence Cordial Graphs 09 Proposition.9 Let G be a (p, q) graph with one vertex of degree (p ) then G is not a dierence cordial if there exists a set of non adjacent vertices S with v i S (deg(v i) 3) 4. Proof. Let G be a (p, q) graph with p vertices and have a vertex v k of degree p and there exists a set of non adjacent vertices S with v i S (deg(v i) 3) 4. Then there are at least p 3 edges passing through v k labelled 0, hence e f (0) p = p +, i.e.,g is not a dierence cordial graph. Example.0. Figure 4: The ower graph F l 8 p = 7, q = 3 deg(v) = 6 S = {v, v 3, v 5, v 7 } then v i S (deg(v i) 3) = = 4 there are at least = 8 edges labelled 0 then the graph is not a dierence cordial. Proposition. Let G be a (p, q) graph then G is not dierence cordial graph if there exists a set of non adjacent vertices S with (deg(v) ) = p +. v S Proof. Let S be a set of non adjacent vertices with v i S (deg(v i) ) = p+ Since the maximum number of edges labelled that are incident with a specic vertex equals,the number of edges labelled 0 that are incident with vertices of S are at least v i S (deg(v i) ) this means the minimum value for e f (0) in the graph G is p +,therefor the graph cannot be a dierence cordial graph. Proposition. The complement graph of a dierence cordial graph is not dierence cordial when the number of its vertices is more than eight. Proof. Let G be a (p, q) dierence cordial graph with p 9 then by proposition.. q p ()

6 0 Seoud and Salman G c, the complement of graph G contains p(p ) q edges and p vertices, let G c be dierence cordial then by adding () and () we get A contradiction for all p 9 p(p ) q p () p(p ) 4p p 9p 4 3 Dierence cordial labeling for Some graphs: In This section we will discuss the ability of applying dierence cordial labeling for some graphs and the functions which make it dierence cordial graphs. The Proposition. consider necessary condition for dierence cordial labeling but it is not sucient. 3. Ladder graphs L n The ladder graph is a planner undirected graph denoted by L n with n vertices and 3n edges [3]. The ladder graph L n can be expressed as L n = Pn P Figure 5: Ladder Graph L n Proposition 3. Every ladder graph L n is dierence cordial for all n. Proof. Let L n be a ladder graph, then it has n vertices and 3n edges. Let the vertices be v, v,..., v n such that v n v n+ is an edge in this graph. Dene the mapping f : L n {,,..., n} by :

7 On Dierence Cordial Graphs i if i E 3 f(v i ) = E + n + i if E < i E + n 4 (i n) if E + n < i n and n is odd 4 (i n) if E + n < i n and n is even 4 From the rst part of denition notice that there are E of edges labelled, in the second part we notice that f(v i+ ) f(v i ) = 3 E +, so n + (i + ) 3 E n f(vi ) f(v n (i+ ) = 3 E + n + i n + (i + ) = 3 E + n + 3 i n = 3 (3n ) + n + 3 i n = 3n i > which means all these edges are labelled 0. In the third part of denition we notice when n is even : f(v i+ ) f(v i ) = (i + n) (i n) = + i = and f(vi ) f(v n (i+) ) = (i n) n + (i + ) = 3i 4n > 3( E + 4 n ) 4i > 3( (3n + 4 n ) 4n > n n 3 5m 3 if n = 4m > 5m + if n = 4m > this means all the edges v i v n (i+) in this third part are labelled 0. But if n is an even number then the number of the total edges of the ladder L n is

8 Seoud and Salman even and thus there must exist additional edge labelled, which we may get it from the label of the last vertex in part two and the rst label in part three. Notice that if i = E + n then 4 f(v E + 4 n ) = 3 E + n ( ) + E + 4 n (3) and if i = E + n +, then 4 ( f(v E + n +) = ( 4 E + ) 4 n + ) n (4) by subtracting (4 ) from (3 ) we get f(v E + n ) f(v 4 E + n +) 4 ( ) ( = 3 E + n + E + 4 n ( E + = E n 4 n = (3n ) n 4 n = 3 n n 4 4 n = n n = { if n = 4m if n = 4m + ) 4 n + ) n thus the edge v E + n v 4 E + n + is labelled, then the graph is 4 dierence cordial. Now if n is an odd number then E is an odd number and then from the rst part we get E edges labelled and all other edges in the second and third part are labelled 0, similarly when n is even, and f(v E + n ) f(v 4 E + n +) 4 ( ) = 3 E + n + E + 4 n = E + n 4 4 n + n = ( ( (3n ) E + 4 n + n 4 } ) + ) n 4 n + + n (3(4m + ) ) + (4m + ) 4 (4m + ) + (4m + ) if n = 4m + 4 = (3(4m + 3) ) + (4m + 3) 4 (4m + 3) + (4m + 3) if n = 4m { } 0 if n = 4m + = if n = 4m + 3

9 On Dierence Cordial Graphs 3 then e f () = e f (0) if n is even e f () = e f (0) if n is odd & n = 4m + e f () = e f (0) + if n is odd & n = 4m + 3 Hence G is dierence cordial. Example 3. Consider the graph L 0 n = 0, E = 8, E = 4, n = 5, n = 3 then 4 f(v i ) = i if i 4 49 i if 4 < i 7 (i 9) if 7 < i 0 f(v ) =, f(v ) =,..., f(v 4 ) = 4, f(v 5 ) = 9, f(v 6 ) = 7, f(v 7 ) = 5, f(v 8 ) = 6, f(v 9 ) = 8, f(v 0 ) = 0. e f (0) = 4, e f () = 4 Figure 6: A dierence cordial labeling for L 0 Example 3.3 Consider the graph L n =, E = 3, E = 6, n = 6, n = 3 then 4 f(v i ) = f(v ) =, f(v ) =,..., f(v 6 ) = 6, f(v 7 ) =, f(v 8 ) = 0, f(v 9 ) = 8, f(v 0 ) = 7, f(v ) = 9, f(v ) =. i if i 6 56 i if 6 < i 9 (i n) if 9 < i Figure 7: A dierence cordial labeling L e f (0) = 5, e f () = 6

10 4 Seoud and Salman 3. Triangular ladder graph T L n A triangular ladder T L n, n, is a graph obtained from the ladder L n = P n P by adding the edges u i v i+ for i n. Such graph has n vertices with 4n 3 edges Figure 8: Triangle ladder graph T L n Proposition 3.4 The triangular ladder graph T L n, n is a dierence cordial graph for all n. Proof. Let G = T L n, n be a triangular ladder graph,then G = (n, 4n 3). Dene the function f(v i ) = i and f(u i ) = i ; i n (5) It is clear that e f () = n hence e f (0) = (4n 3) (n ) = n, then e f (0) e f (0) =, thus G = T L n, n is a dierence cordial graph. Example 3.5 Consider the graphs T L 6 and T L 7 Figure 9: A dierence cordial labeling for T L 6 Figure 0: A dierence cordial labeling for T L 7

11 On Dierence Cordial Graphs The Grid graph P m P n In this subsection we will investigate the dierence cordial labeling for every grid graph of the form P m P n for all m, n. Let the vertices of the grid graph be arranged as a sequence in certain order as in the gure Figure : The grid graph P m P n This kind of graphs contains mn vertices and mn (m + n) edges. Proposition 3.6 Every grid graph is P m P n is a dierence cordial graph for all integers m, n greater than. Proof. Let G be a graph P m P n then G = (mn, mn (m + n)) Case : If m = n then V = n and E = (n n), dene the function f for labeling vertices of G by : f(v ij ) = (i ) n + j in each row of the grid graph there exist n edges labelled this leads to e f () = n(n ) and the number of edges labelled 0 is equal to : n(n ) n(n ) = n(n ), thus G is a dierence cordial graph. Case : If m n =, then V = mn and E = mn (m + n). Let n = m + then E = m. Now using the same functions in Case we will get e f () = m(n ) = m(m + ) = m and e f (0) = (m )(m + ) = m which means the graph is a dierence cordial graph. Similarly if m = n + Case 3 :If m n. Let n > m and let k = (n m) we dene the mapping : (j ) m + i if j k f(v ij ) = k (m ) + n (i ) + j if j = k +,..., n

12 6 Seoud and Salman It follows that : and so e f () = k(m ) + m(n k ) = mn (m + k) e f (0) = mn (m + n) mn + (m + k) = mn (n k) e f (0) e f () = mn n + k mn + m + k = n + k + m { 0 if n m is even = if n m is odd Similarly if m > n we apply the same mapping but replacing i by j and m by n, i.e.: k = (m n) and : f(v ij ) = (i ) n + j if i k k (n ) + m (j ) + i if i = k +,..., m Hence the grid graph P m P n is a dierence cordial graph for all m, n. Example 3.7 Let P m P n = P 4 P 3 n = 3, m = 4, V =, E = 7 f(v ij ) = 3(i ) + j } Figure : A dierence cordial labeling for grid graph P 4 P 3 e f () = 8, e f (0) = 9 Example 3.8 Let P m P n = P 5 P 8 n = 8, m { = 5, V = 40, E = 67, k = 5(j ) + j j f(v ij ) = (5 i) + 8(i ) + j j > e f (0) = 34, e f () = 33 }

13 On Dierence Cordial Graphs 7 Figure 3: A dierence cordial labeling for grid graph P 5 P Step ladder graph S(T n ): Denition 3.9 Let P n be a path on n vertices denoted by (, ), (, ),..., (, n) and n edges denoted by e, e,..., e n where e i is the edge joining the vertices (, i) and (, i + ). On each edge e i,i =,,..., n we erect a ladder with n (i ) steps including the edge e i. The graph obtained is called a step ladder graph and is denoted by S(T n ), where n denotes the number of vertices in the base. The following sketch shows the step ladder graph : Figure 4: The step ladder graph S(T n ) The number of vertices and edges are : V = n + n = n(n + ) + (n ) = n + 3n

14 8 Seoud and Salman E = ( V n) = n(n + ) We notice for all step ladder graphs that i + j n + Proposition 3.0 Every step ladder graph S(T n ) is a dierence cordial graph for all n. Proof. Let S(T n ) be a step ladder graph then E = n(n + ) = n + n Dene the function f : S(T n ) {,,..., n(n + ) + (n )} by : j + (i )n i 3 f(v ij ) = j + (i )n (i 3)(i ) i 4 e f () = (3n 4) + (n 3) + (n 4) + (n 5) = (n ) + (n ) + (n ) + (n 3) = (n ) + n(n ) = (n + n ), then e f () = E which means e f() e f (0) = 0. Therefor S(T n ) is a dierence cordial graph for all n 3.5 Double Sided Step Ladder Graph S(T n ): Denition 3. Let P n be a path of length n with n vertices (, ), (, ),..., (, n) with n edges,e, e,..., e n, where e i is the edge joining the vertices (, i)and(, i + ). On each edge e i, for i =,,..., n, we erect a ladder with i + steps including the edge e i and on each edge e i, for i = n +, n +,..., n,we erect a ladder with n + i steps including the edge e i.the double sided step ladder graph S(T n ) has vertices denoted by (, ), (, ),..., (, n), (, ), (, ),..., (, n), (3, ), (3, 3),..., (3, n ), (4, 3), (4, 4),, (4, n ),..., (n +, n), (n +, n + ). In the ordered pair (i, j), i denotes the row number (counted from bottom to top) and j denotes the column number (from left to right) in which the vertex occurs. Example 3. The gure 5 is the S(T 0 ) Proposition 3.3 The double sided step ladder graph S(T m ) is a dierence cordial graph, where m = n denotes the number of vertices in the base.

15 On Dierence Cordial Graphs 9 Figure 5: Double sided step ladder graph S(T n ) Proof. Let G = (V, E) be the double sided step ladder graph S(T m ) where m = n then V = n + 3n and E = n + 3n Dene f : V {,,..., n + 3n} by : u if i = and j n f(v i,j ) = j + n (i ) if i = and j n + j + n (i ) if i = j + n (i ) (i ) if i = 3, 4,..., n + where u = j (mod n + ) if n = 3 or n 0 (mod 4 ) j (mod n j + ) + if n, (mod 4 ) n + (j ) (mod n + ) + j n + if n 3 (mod 4 ) from the last three parts of the denition of f we will get n + n n edges give e f (), while in the rst part all edges give e f (0) except when n 4 we will get an edge in e f () since u 3. Case : If n = then e f () = n + n n + = 7 and e f (0) = 6, if n = 3 then e f () = n + n n + = 3 and e f (0) = 3, if n = 4 then e f () = n + n n + = and e f (0) = and if n = 5 then e f () = n + n n + = 3 and e f (0) = 3.

16 0 Seoud and Salman Case :If n 0(mod 4) then n = 4k for some positive integer number k and n = k, then E = (4k) + 3(4k) = 3k + k and f(v n ) = ( n ) (mod n + ) = ( k)(mod k + ) = 4k(mod k + ) = k while f(v n +) = n + = k +, thus the label of the edge v n v n + will be included in e f(0), therefor e f () = n + n n = 6k + 8k k = 6k + 6k and e f (0) = E e f () = 3k + k 6k 6k + = 6k + 6k we get e f (0) e f () =. Case 3 : If n (mod 4) then n = 4k + for some positive integer number k and n = k +, then E = (4k + ) + 3(4k + ) = 3k + 8k + 4 and f(v n ) = ( n ) (mod n + ) + n n + k + = (k + ) (mod k + ) + k + + = (4k + ) (mod k + ) + = k + while f(v n +) = n + = k +, thus the label of the edge v n v n + will be included in e f(), therefor e f () = n +n n + = (4k +) +(4k +) k + = 6k + 4k + and e f (0) = E e f () = 3k + 8k + 4 6k 4k = 6k + 4k + we get e f (0) e f () = 0 Case 4 : If n (mod 4) then n = 4k+ for some positive integer number k and n = k +, then E = (4k + ) + 3(4k + ) = 3k + 44k + 3 and f(v )(mod n ) = ( n n + ) + n n + k + = (k + ) (mod k + ) + k + + = (4k + ) (mod k + ) + = k +

17 On Dierence Cordial Graphs while f(v n +) = n + = k+, thus the label of the edge v n v n + will included in e f (), therefore e f () = n +n n + = (4k+) +(4k+) k + = 6k + k + 7 and e f (0) = E e f () = 3k + 44k + 3 6k k 7 = 6k + k + 6 we get e f (0) e f () = Case 5 : If n 3(mod 4) then n = 4k + 3 for some positive integer number k and n = k +, then E = (4k + 3) + 3(4k + 3) = 3k + 60k + 6 and f(v n ) = ( n ) (mod n + ) + n n + k + = ((k + ) )(mod k + 3) + k + + = (4k + 3)(mod k + 3) + = k +, while f(v n +) = n + = k+3, thus the label of the edge v n v n + will be included in e f (), therefor e f () = n +n n + = (4k +3) +(4k +3) k + = 6k + 30k + 3 and e f (0) = E e f () = 3k +60k +6 6k 30k 3 = 6k +30k +3 we get e f (0) e f () = 0. From the cases,,3,4and 5 we conclude that the double sided step ladder graph S(T n ) is a dierence cordial graph for all integer number n We discuss here some types of graphs not always dierence cordial such as diagonal ladder graph,diagonal grid graph and friendship graph. Diagonal ladder graph is a ladder with additional edges u i v i+ and u i+ v i, denoted by DL n, where n is half its vertices and the number of its edges is 5n 4. Corollary 3.4 The diagonal ladder graphs are dierence cordial graphs if n 3. Proof. Let the graph G be the diagonal ladder graph DL n with n vertices that means there are 5n 4 edges in G, G is a dierence cordial graph. Then we get by Proposition 5n 4 (n) n 3 then the diagonal ladder graph is dierence cordial when n = orn = 3 The following example shows that DL and DL 3 are dierence cordial

18 Seoud and Salman Example 3.5 The following are labeling for the diagonal ladder graphs DL, DL 3 Figure 6: The dierence cordial labelings for the diagonal ladder graphs DL & DL 3 The graph P m P n with diagonal edges is called diagonal grid graph and denoted by D( P m P n ). It has mn vertices and (mn + ) 3(m + n) edges. Remark Diagonal grid graph P m P n are not dierence cordial graphs for both m, n 3. Proof. Let G = D (P m P n ), then from Proposition if G is a dierence cordial graph then q p. Let m = n = 3, then q = (mn + ) 3(m + n) = ( ) 3(3 + 3) = 0 7 then D (P m P n ) cannot be a dierence cordial graph for bothm, n 3 This is consistent with corollary 3.5 since diagonal ladder graphs are diagonal grid graphs. Another type of graphs will be discussed here named one-point union fan graph, where a graph G in which a vertex distinguished from other vertices is called a rooted graph and the vertex is called the root of G. Let G be a rooted graph,the Graph G (n) obtained by identifying the roots of n copies of G is called a one-point union of the n copies of G. Proposition 3.6 The fan graph F n is dierence cordial for all n. citepon Proposition 3.7 The one-point union F n (m) dierence cordial for all n and for m 5. of m copies of a fan F n is Proof. Let G = F (m) n, then V (G) = mn + and E(G) = m(n ). These vertices are : v 00 is the central vertex and the other vertices are denoted by v ij, i nand j m

19 3 On Di erence Cordial Graphs Figure 7: The graph Fn(m) For each copy of a fan Fn there are n edges labelled, therefore there are m(n ) + edges labelled in Fn(m), where the central vertex is labelled (mod n) but is neither nor mn + then ef (0) = m(n ) m(n ) = mn Now ef (0) ef () = mn m(n ) = m 4 then ef (0) ef () for allm 6. We de ne the mapping f for m 5 and n N by f (v0 0 ) (mod n)and f (v0 0 ) 6=, mn + and f (vij ) = (j )n + i (j )n + i + if if (j )n + i < f (v0 0 ) (j )n + i > f (v0 0 ) for all i, j, i n, j m. As a special case,the friendship graph denotes by F(m) consists of one vertex union with m copies of paths P consisting of m + vertices and 3m edges as shown in Figure 8

20 4 Seoud and Salman Figure 8: The friendship graph F 5 Therefore the friendship graph F (m) is dierence cordial if and only if m 5. References [] I. Cahit, Cordial graph : A weaker version of graceful and harmonious graphs, Ars combinatoria 3(987),0-07. [] J. A. Gallian, Adynamic survey of graph labeling, The electronicjournal of combinatorics 8(03)#Ds6. [3] F. Harrary, Graph Theory, Addison-Wesely, Reading, Massachusetts,969 [4] R. Ponraj, S. Sathish Narayanan and R. Kala, Dierence cordial labeling of graphs, Global J. Math. Sciences: Theory and Practical, 3 (03) 9-0. [5] R. Ponraj and S. Sathish Narayanan, Dierence cordial labeling of subdivision of snake graphs, universal Journal of applied mathematics,() (04) [6] R. Ponraj, S. Sathish Narayanan and and R. Kala, A note on Dierence cordial graphs, Palestine Journal of mathematics, 4() (05) [7] Reinhard Diestel, Graph Theory, Electronic addition 005 Received: January, 05