Chapter 5. E-cordial Labeling of Graphs

Transcription

1 Chapter 5 E-cordial Labeling of Graphs 65

2 Chapter 5. E-cordial Labeling of Graphs Introduction In 987 the concept of cordial labeling was introduced by Cahit [6] as a weaker version of graceful and harmonious labeling. After that some more labelings are also introduced having cordial theme. Some of them are A-cordial labeling, product cordial labeling, total product cordial labeling and E-cordial labeling. This chapter is focused on E-cordial labeling. 5. E-cordial labeling 5.. Binary vertex labeling Let G be a graph. A mapping f : V (G) {,} is called a 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 = uv) = f (u) f (v). Let v f (), v f () be the number of vertices of G having labels and respectively under f and let e f (), e f () be the number of edges of G having labels and respectively under f. 5.. Cordial labeling A binary vertex labeling of a graph G is cordial labeling if v f () v f () and e f () e f (). A graph is cordial if it admits cordial labeling Edge graceful labeling A graph G = (V (G),E(G)) with p vertices and q edges is said to be edge graceful if there exists a bijection f : E(G) {,,...,q} such that the induced mapping f :

3 Chapter 5. E-cordial Labeling of Graphs 67 V (G) {,,,..., p } given by f (x) = ( f (xy))(mod p) taken over all edges xy is a bijection. In 985 Lo[3] has introduced the notion of edge-graceful graphs E-cordial labeling-a weaker version of edge graceful labeling In 997 Yilmaz and Cahit [49] introduced E-cordial labeling as a weaker version of edge graceful labeling having blend of cordial labeling and harmonious labeling. A function f : E(G) {,} is called an E-cordial labeling of a graph G if the induced function f : V (G) {,} defined by f = { f (uv)/uv E(G)}(mod) and the number of vertices labeled and the number of vertices labeled differ by at most and the number of edges labeled and the number of edges labeled differ by at most. A graph that admits E-cordial labeling is called E-cordial Illustration In Figure 5. K 4 and its E-cordial labeling is shown. v v v4 v3 FIGURE 5.: K 4 and its E-cordial labeling

4 Chapter 5. E-cordial Labeling of Graphs Some existing results Yilmaz and Cahit [49] have proved that The trees with n vertices is E-cordial if and only if n (mod4). C n is E-cordial if and only if n (mod4). K n is E-cordial if and only if n (mod4). Regular graphs of degree on n vertices is E-cordial if and only if n is even. Friendship graph C (n) 3 is E-cordial n. Fan F n is E-cordial if and only if n (mod4). Wheel W n is E-cordial if and only if n (mod4). K m,n admits E-cordial labeling if and only if m + n (mod4). The graphs with p (mod4) can not be E-cordial. Devaraj [] has proved that M m,n which is the the mirror graph of K m,n is E-cordial when m + n is even. The general Petersen graph P n,k is E-cordial when n is even. Vyas [47] has discussed the E-cordial labeling of some mirror graphs.

5 Chapter 5. E-cordial Labeling of Graphs E-cordial labeling of some graphs Theorem M(P n ) admits E-cordial labeling. Proof. Let v,v,..., v n be the vertices and let e,e,..., e n be the edges of path P n. Let M(P n ) be the middle graph of path P n. According to the definition of the middle graph V (M(P n )) = V (P n ) E(P n ) and E(M(P n )) = {v i e i ; i n,v i e i ; i n,e i e i+ ; i n }. To define f : E(M(P n )) {,} two cases are to be considered. Case : n is even. f (v e ) = f (v i e i ) = ; for i n f (v i e i ) = ; for i n f (v n e n ) =. For i n f (e i e i+ ) = ; if i is odd. = ; if i is even. Case : n is odd. f (v e ) = f (v i e i ) = ; for i n f (v i e i ) = ; for i n For i n f (e i e i+ ) = ; if i is odd. =; if i is even. In view of the above defined labeling pattern f satisfies the conditions for E-cordial labeling as shown in Table 5.. That is, M(P n ) admits E-cordial labeling. n Vertex Condition Edge Condition Even v f () = v f () + = n e f () = e f () = 3n 4 Odd v f () = v f () + = n e f () = v f () + = 3n 3 Table 5. Illustration In Figure 5. M(P 6 ) and its E-cordial labeling is shown.

6 Chapter 5. E-cordial Labeling of Graphs 7 v v v3 v4 v5 v6 e e e3 e4 e5 FIGURE 5.: M(P 6 ) and its E-cordial labeling Theorem T (P n ) is E- cordial. Proof. Let v,v,..., v n be the vertices and e,e,..., e n be the edges of path P n. Let T (P n ) be the total graph of path P n with V (T (P n )) = V (G) E(G) and E(T (P n )) = {v i v i+ ; i n,v i e i ; i n,e i e i+ ; i n,v i e i ; i n}. Define f : E(T (P n )) {,} as follows. For i n f (v i v i+ ) = ; if i is odd. =; if i is even. For i n f (v i e i ) = ; if i is odd. =; if i is even. For i n f (e i e i+ ) = ; if i is odd. =; if i is even. For i n f (v i e i ) = ; if i is odd. =; if i is even. Thus above defined function f satisfies the conditions of E-cordial labeling as shown in Table 5.. That is, f provides E-cordial labeling for the graph T (P n ).

7 Chapter 5. E-cordial Labeling of Graphs 7 n Vertex Condition Edge Condition Even v f () = v f () + = n e f () = e f () + = n Odd v f () = v f () + = n e f () = e f () + = n Table 5. Illustration In Figure 5.3 T (P 7 ) and its E-cordial labeling is shown. v v v3 v4 v5 v6 v7 e e e3 e4 e5 e6 FIGURE 5.3: T (P 7 ) and its E-cordial labeling Theorem spl(p n ) is E-cordial for even n. Proof. Let v,v,..., v n be the vertices and e,e,..., e n be the edges of path P n. Let v,v,..., v n be the newly added vertices to form the split graph of path P n. Let spl(p n ) be the split graph of path P n. V (spl(p n )) = {v i } {v i }, i n and E(spl(P n)) = {v i v i+; i n,v i v i ; i n,v i v i+ ; i n }. Define f : E(spl(P n )) {,} as follows. For i n f (v i v i+) = ; if i is odd. =; if i is even. For i n f (v i v i ) = ; if i is odd. =; if i is even. For i n f (v i v i+ ) = ; if i is odd. =; if i is even.

8 Chapter 5. E-cordial Labeling of Graphs 7 The above defined function f satisfies the conditions of E-cordial labeling as shown in Table 5.3. That is, spl(p n ) is an E-cordial graph. Vertex Condition v f () = v f () = n Edge Condition e f () = e f () + = 3n Table 5.3 Illustration spl(p 6 ) and its E-cordial labeling is shown in Figure 5.4. v' v' v3' v4' v5' v6' v v v3 v4 v5 v6 FIGURE 5.4: spl(p 6 ) and its E-cordial labeling Theorem spl(c n ) is E-cordial for even n. Proof. Let v,v,...v n be the vertices of cycle C n and v,v,...v n be the newly added vertices where n is even. Let spl(c n ) be the split graph of cycle C n with V (spl(c n )) = {v i,v i ; i n} and E(spl(C n)) = {v i v i+ ; i n,v n v,v i v i+; i n,v nv,v i v i+ ; i n,v nv }. To define f : E(spl(C n)) {,} two cases are to be considered. Case : n (mod4) For i n f (v i v i+ ) = ; if i is odd. =; if i is even. f (v n v ) =

9 Chapter 5. E-cordial Labeling of Graphs 73 For i n f (v i v i+ ) = ; f (v n v ) = For i n f (v i v i+) = ; f (v nv ) =. if i,(mod4) =; if i, 3(mod4) if i,(mod4) =; if i, 3(mod4) Case : n (mod4) For i n f (v i v i+ ) = ; f (v n v ) = For i n f (v i v i+ ) = ; f (v n v ) = For i n f (v i v i+) = ; f (v nv ) =. if i is odd. =; if i is even. if i,(mod4) =; if i, 3(mod4) if i,(mod4) =; if i, 3(mod4) In view of the above defined labeling pattern f satisfies the conditions for E-cordial labeling as shown in Table 5.4. That is, the spl(c n ) for n is even is E-cordial. n Vertex Condition Edge Condition n,(mod4) v f () = v f () = n e f () = e f () = 3n Table 5.4 Illustration Figure 5.5 shows the E-cordial labeling of spl(c 8 ).

10 Chapter 5. E-cordial Labeling of Graphs 74 v7' v8' v6' v' v8 v v3 v7 v6 v4 v5 v5' v' v v4' v3' FIGURE 5.5: spl(c 8 ) and its E-cordial labeling Theorem P n [P ] is E-cordial when n is even. Proof. Consider two copies of P n. Let v,v,..., v n be the vertices of first copy of P n and u,u,..., u n be the vertices of the second copy of P n. Let G be P n [P ]. V (G)=T T where T = {(v i,u ), i n} and T = {(v i,u ), i n}, E(G)={(v i,u )(v i+,u ); i n,(v i,u )(v i+,u ); i n,(v i,u )(v i+,u ); i n,(v i,u )(v i+,u ); i n,(v i,u )(v i,u ); i n}. Define f : E(G) {,} as follows. For i n f ((v i,u )(v i+,u )) = ; if i is odd. = ; if i is even. For i n f ((v i,u )(v i+,u )) = ; if i is odd. = ; if i is even. For i n f ((v i,u )(v i+,u )) = ; if i is odd. = ; if i is even.

11 Chapter 5. E-cordial Labeling of Graphs 75 For i n f ((v i,u )(v i+,u )) = ; if i is odd. = ; if i is even. For i n f ((v i,u )(v i,u ))= ; if i is odd. =; if i is even. f ((v n,u )(v n,u ))=. In view of the above defined labeling pattern f satisfies the conditions for E-cordial labeling as shown in Table 5.5. That is, P n [P ] is E-cordial when n is even. Vertex Condition v f () = v f () = n Edge Condition e f () = e f () = 5n 4 Table 5.5 Illustration In Figure 5.6 the graph P 6 [P ] and its E-cordial labeling is shown. (v,u) (v,u) (v3,u) (v4,u) (v5,u) (v6,u) (v,u) (v,u) (v3,u) (v4,u) (v5,u) (v6,u) FIGURE 5.6: P 6 [P ] and its E-cordial labeling Theorem The graph obtained by duplication of an arbitrary vertex of C n admits E-cordial labeling. Proof. Let v,v,..., v n be the vertices of the cycle C n. Let G be the graph obtained by duplicating an arbitrary vertex of C n. Without loss of generality let this vertex be v and

12 Chapter 5. E-cordial Labeling of Graphs 76 the newly added vertex be v. E(G)={E(C n),e,e } where e = v v and e = v n v.to define f : E(G)) {,} two cases are to be considered. Case : n,(mod4) f (e ) = f (e ) = For i n f (e i ) = ; for i,(mod4), =; for i, 3(mod4) Case : n 3(mod4) f (e ) = f (e ) = For i n f (e i ) = ; for i,(mod4), =; for i, 3(mod4) In view of the above defined labeling pattern f satisfies the conditions for E-cordial labeling as shown in Table 5.6. That is, the graph obtained by the duplication of an arbitrary vertex in cycle C n admits E-cordial labeling. n vertex condition edge condition n (mod4) v f () = v f () + = n+ e f () = e f () = n+ n (mod4) v f () = v f () + = n+ e f () = e f () = n+ n 3(mod4) v f () = v f () = n+ e f () = e f () + = n+3 Table 5.6 Illustration Figure 5.7 shows the E-cordial labeling of the graph obtained by the duplication of an arbitrary vertex in cycle C.

13 Chapter 5. E-cordial Labeling of Graphs 77 e'' v' e' e e9 e e v v v v v v9 v8 v6 e8 v7 e7 e e6 v3 v5 e v4 e5 e3 e4 FIGURE 5.7: The graph obtained by the duplication of an arbitrary vertex in cycle C and its E-cordial labeling Theorem The graph obtained by duplication of an arbitrary edge in C n admits E-cordial labeling. Proof. Let v,v,..., v n be the vertices of the cycle C n. Let G be the graph obtained by duplicating an arbitrary edge of C n. Without loss of generality let this edge be e = v v and the newly added edge be e = v v. E(G)={E(C n),e,e,e } where e = v v 3 and e = v n v. Define f : E(G) {,} as follows. Case : n,(mod4) f (e ) = f (e ) = f (e ) = For i n f (e i ) = ; for i,(mod4), =; for i, 3(mod4)

14 Chapter 5. E-cordial Labeling of Graphs 78 Case : n 3(mod4) f (e ) = f (e ) = f (e ) = For i n f (e i ) = ; for i,(mod4), =; for i, 3(mod4) f (e n ) =. In view of the above defined labeling pattern f satisfies the conditions for E-cordial labeling as shown in Table 5.7. That is, the graph obtained by the duplication of an arbitrary edge in cycle C n is E-cordial. n vertex condition edge condition n (mod4) v f () = v f () + = n+3 e f () = e f () = n+3 n (mod4) v f () = v f () = n+ e f () = e f () + = n+4 n 3(mod4) v f () = v f () + = n+3 e f () = e f () = n+3 Table 5.7 Illustration The E-cordial labeling of the graph obtained by the duplication of an arbitrary edge in cycle C is as shown in Figure 5.8. e e9 e' v' v' e'' e e v v v v3 v v4 v9 v5 e8 v8 e7 v7 e6 e' v6 e e5 e3 e4 FIGURE 5.8: The graph obtained by the duplication of an arbitrary edge in cycle C and its E-cordial labeling

15 Chapter 5. E-cordial Labeling of Graphs 79 Theorem Joint sum of two copies of C n of even order produces an E-cordial graph. Proof. We denote the vertices of first copy of C n by v,v,...v n and second copy by v, v,v 3,...v n, e i and e i where i n be the corresponding edges. Join the two copies of C n with a new edge and let G be the resultant graph. Without loss of generality we assume that the new edge be e=v v. To define f : E(G) {,} two cases are to be considered. Case : n (mod4) f (e) = For i n f (e i ) = ; for i,(mod4), =; for i, 3(mod4) For i n f (e i ) = ; for i,(mod4), =; for i, 3(mod4) Case : n (mod4) f (e) = For i n f (e i ) = ; for i,(mod4), =; for i, 3(mod4) For i n f (e i ) = ; for i,(mod4), =; for i, 3(mod4) In view of the above defined labeling pattern f satisfies the conditions for E-cordial labeling as shown in Table 5.8. That is, the joint sum of two copies of even cycle C n is E-cordial. n Vertex Condition Edge Condition n (mod4) v f () = v f () = n e f () = e f () + = n + n (mod4) v f () = v f () = n e f () = e f () + = n + Table 5.8

16 Chapter 5. E-cordial Labeling of Graphs 8 Illustration E-cordial labeling of joint sum of two copies of cycle C is shown in Figure 5.9. e9 e v e v e e' v' v v' e e9' e' v' e' v9 v3 v3' v9' e3 e3' e8 e8' v4' v8 v4 v8' e4' e4 e7' e7 v5' v7 v5 v7' v6 e5 v6' e5' e6 e6' FIGURE 5.9: The joint sum of two copies of cycle C and its E-cordial labeling Theorem D (P n ) is E-cordial when n is even. Proof. Let P n P n be two copies of path P n. We denote the vertices of first copy of P n by v,v,...v n and second copy by v, v,...v n. Let G be D (P n ) with v(g) = n and E(G) = 4n 4. To define f : E(G) {, } two cases are to be considered. Case : n (mod4) For i n f (v i v i+ ) = ; For i n if i 3 j =; if i = 3 j, j =,,... n 3 f (v i v i+ ) = ; if i 3 j For i n f (v i v i+ ) = ; f (v n v n) = =; if i = 3 j, j =,,... n 3 if i is odd. =; if i is even.

17 Chapter 5. E-cordial Labeling of Graphs 8 For i n f (v i v i+ ) = ; if i is odd. =; if i is even. Case : n (mod4) For i n f (v i v i+ ) = ; For i n if i,(mod4) =; if i, 3(mod4) f (v i v i+ ) = ; if i,(mod4) For i n =; if i, 3(mod4) f (v i v i+ ) = ; if i is odd. f (v n v n) = For i n f (v i v i+ ) = ; =; if i is even. if i is odd. =; if i is even. In view of the above defined labeling pattern f satisfies the conditions for E-cordial labeling as shown in Table 5.9. That is, D (P n ) where n is even is E-cordial. n Vertex Condition Edge Condition n,(mod4) v f () = v f () = n e f () = e f () = n Table 5.9 Illustration Figure 5. shows the E-cordial labeling of D (P 6 ).

18 Chapter 5. E-cordial Labeling of Graphs 8 v'' v'' v3'' v4'' v5'' v6'' v' v' v3' v4' v5' v6' FIGURE 5.: D (P 6 ) and its E-cordial labeling 5.4 Concluding Remarks and Further Scope of Research Nine new results have been investigated on E-cordial labeling. To derive similar results for various labeling problems and for different graph families is an open area of research. The next chapter is targeted to discuss odd sequential labeling of graphs.