Cordial Labelling Of K-Regular Bipartite Graphs for K = 1, 2, N, N-1 Where K Is Cardinality of Each Bipartition

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1 IOSR Journal of Mathematics (IOSR-JM) e-issn: p-ISSN: X Volume 6 Issue 4 (May. - Jun. 3) PP 35-4 Cordial Labelling Of K-Regular Bipartite Graphs for K = N N- Where K Is Cardinality of Each Bipartition Pranali Sapre JJT University Jhunjhunu Rajasthan India Assistant ProfessorVidyavardhini s College of Engineering & Technology Vasai Road Dist. Thane. Maharashtra Abstract: In the labelling of graphs one of the types is cordial labelling. In this we label the vertices or and then every edge will have a label or if the end vertices of the edge have same or different labellings respectively. Here we are going find whether a k-regular bipartite graph can be cordial for different values of k. I. Introduction : Regular graph: A graph G is said to be a regular graph if degree of each vertex a same. It is called k-regular if degree of each vertex is k. Bipartite graph: Let G be a graph. If the vertex of G is divided into two subsets A and B such that there is no edge ab with a b A or a b B then G is said to be bipartite that is Every edge of G joins a vertex in A to a vertex in B. The sets A and B are called partite sets of G. Complete graph: A graph in which every vertex is adjacent to every other vertex is a complete graph. For a complete graph on n vertices degree of each vertex is n-. Complete bipartite graph: A bipartite graph G with bipartition ( AB ) is said to be complete bipartite if every vertex in A is adjacent to every vertex in B and vice versa. Cycle: A closed path is called a cycle. Labelling of a graph: Vertex labelling :It is a mapping from set of vertices to set of natural numbers. Edge labelling :It is a mapping from set of edges to set of natural numbers. Cordial labelling: For a given graph G label the vertices of G or. And every edge 'ab' of G will be labeled as if the labeling of the vertices a and b are same and will be labeled as if the labeling of the vertices a and b are different. Then this labeling is called a cordial labeling or the graph G is called a cordial graph iff number of vertices labeled '' number of vertives labeled ' ' number of edges labeled '' number of edges labeled Theorem. If G is a - regular bipartite graph with partite sets A and B with A = B = n Then G is cordial iff n = 3 mod 4. ' ' like Case I. n = 4m Let m = i.e. n = 4 then G looks 35 Page

2 Cordial labeling of k-regular bipartite graphs for k = n n- where k is cardinality of This can be labeled as shown is figure. This is CORDIAL. Now for n = 4m for m > G can be considered as combination of graph shown above and hence can be labeled repeatedly as above which is CORDIAL. Hence G IS CORDIAL FOR n = 4m Case II. n = 4m + Consider n 4 m for mz u A v B Consider any edge uv E ( G ) for Then G {uv} is a bipartite graph with n = 4m. Hence it has a cordial labeling as given in case ( i ) which gives # of edges labeled - # of edges labeled = hence along with the same labeling if we label u as and v as we get # of vertices labeled - # of vertices labeled = and # of edges labeled - # of edges labeled = Hence it is cordial. G IS CORDIAL FOR n = 4m+ Case III. Let n = 4m + Consider n 4 m for mz A = B = 4m+ Assume G is cordial. Let be the number of vertices in A labeled Let be the number of vertices in A labeled Let be the number of vertices in B labeled Let be the number of vertices in B labeled Total number of edges ab where a is labeled and b B = Total number of edges ab where b is labeled and a A= * Total number of edges ab where a is labeled and b B = Total number of edges ab where b is labeled and a A= Let number of edges of type ( i.e. edge ab where a and b both are labeled ) be x And Let number of edges of type be y From * Number of edges of type and are x Number of edges of type and are x = y But By assumption G is cordial x = y y But total number of edges = 4m+ Number of edges labeled = Number of edges labeled G is cordial x + y = m+ 36 Page

3 Cordial labeling of k-regular bipartite graphs for k = n n- where k is cardinality of x m Contradiction. and y m Hence G is not cordial for n=4m+ G IS NOT CORDIAL FOR n = 4m+ Case IV Let n = 4m + 3 Consider n 4 m 3 for mz A = B = 4m+3 Let G be a graph n = 4m+3 Consider any three edges u v )( u v )( u ) with u Aand v B for i =3 ( 3 v3 v u v u3 v Then G - u 3 is a graph with n = 4m By case i it has a cordial labeling with # of vertices labeled - # of vertices labeled = and # of edges labeled - # of edges labeled = Along with the same labeling label u v u v u3 v3 as follows u as o v as o u as o v as u3 as v3 as Then number of vertices labeled = Then number of vertices labeled And labeling of ( u v ) is ( u v) is ( u3 v3) is Then number of vertices labeled = Then number of vertices labeled + # of vertices labeled - # of vertices labeled = and # of edges labeled - # of edges labeled = G is cordial G IS CORDIAL FOR n = 4m+3 Hence the result. i i Theorem. If G is a - regular bipartite graph with partite sets A and B with Then G is cordial iff every component of G can be written as cycle of length 4m A = B = n Let G be a bipartite regular graph of degree. We know that If G is a regular bipartite graph of degree then it can always be written as disjoint union of even cycles. Let G be the graph which is the cycle of length n Claim : Cycle of length n is cordial iff n is even. Part (a) : To prove: n is even G is cordial. Part (b) : To prove: G is cordial n is even. i.e. To prove: n is odd G is not cordial. Proof of (a): Consider a cycle of length m = n where n is even. Let n = p m = 4p Let a a a3 a4 a5... a4 p a be the given cycle where ai is adjacent to a i For i 3...4p and a4 p is adjacent to a 37 Page

4 Cordial labeling of k-regular bipartite graphs for k = n n- where k is cardinality of Label a a a3 a4 a5... a4 p as As m = 4p we can have repeatedly p times the labeling of vertices as Which gives edges labeling as [ last edge will be labeled as a4 p and a both labeled. We can see that the labeling is cordial. Hence cycle of length n where n is even is always cordial. Proof of (b): Consider a cycle of length m = n where n is odd. Let n p m 4p We can write this graph as a bipartite graph with partite sets A and B where A = B = p+ Let number vertices of A labeled be Let number vertices of A labeled be Let number vertices of B labeled be Let number vertices of B labeled be Notation: i j --- number of edges ab where a is a vertex in A labeled i and b is a vertex in B labeled j With this notation we have a a a a a a a a Assuming G is cordial we get a a a * a But a a a a 4p a a a 4p a ** From *and * a a p a a p and 3 a a a p a Which is a contradiction as L.H.S. is mod and R.H.S. is mod Hence this cycle cannot be cordial. Hence a cycle of length n is not cordial if n is odd. Thus we have proved: Cycle of length n is cordial iff n is even. A regular bipartite graph of degree is cordial iff its every component can be written as a cycle of length 4n. Hence the proof. 38 Page

5 Cordial labeling of k-regular bipartite graphs for k = n n- where k is cardinality of Theorem 3. If G is a n- regular bipartite graph with partite sets A and B with Then G is cordial A = B = n A { a a... an} Let B { b b... bn} Case I : n is even Let n = m Label a a... an as and am am... am as. For set B label any m vertices as any m as W.l.g. let b b... bn are and bm bm... bm are For the vertex a :The edges incident on a are a b ab... abm abm abm... ab m out of which a b ab... ab m are labeled and a bm abm... ab m are labeled. Which gives equal number of edges labeled and each equal to m Similarly for remaining vertices. Hence we have in all m edges labeled o and m edges labeled. # of edges labeled - # of edges labeled = G is cordial Case II : n is odd Let n = m+ Consider A { u } and B { v } for some uv V (G) Then by case ( i ) it has a cordial labeling with m edges labeled and m edges labeled Now label u as and v as As it is complete bipartite we have edges ub ub... ub ub ub... ub uv m m m m which are labeled as. respectively giving m edges labeled and m+ edges labeled. Also a v av... amv am v amv... amv which are labeled as. respectively giving m edges labeled and m edges labeled. m m and In all we get Number of edges labeled = Number of edges labeled = m m # of edges labeled - # of edges labeled = G is cordial Hence the proof. Theorem 4. If G is a n- regular bipartite graph with partite sets A and B with Then G is cordial iff n = mod 4. A = B = n Case ( i ) : n = 4m Let A = { a a... an} and B ={ b... b bn} And G is a (k-) bipartite graph with partite sets AB. It can be obtained by removing one and only one edge of every vertex of a complete graph. Label a a... am as and a m am... a4m as Then we can have G is a graph obtained by removing bi abi... a mbi where bi bi for i j ik from the complete bipartite graph. a 4 4 m We will rename the vertices in the set B such that the edges removes are a b a b a b... 4m 4m j k 39 Page

6 Now label Cordial labeling of k-regular bipartite graphs for k = n n- where k is cardinality of b b... b4m as b m bm bm... b b b... as m as b m bm... b3m as b3 m b3m... b4m as Then the edges removed from the complete graph are labeled as a b a b a b m m total m edges am bm ambm... ambm total m edges a m bm ambm... a3mb3m total m edges a3 m b3m a3mb3m... a4mb4m total m edges For G Number of vertices labeled = number of vertices labeled = 4m And Number of edges labeled = number of edges labeled = ( from the case k = n ) # of vertices labeled - # of vertices labeled = and # of edges labeled - # of edges labeled = G is cordial Case ( ii ) : n = 4m+ 8m m As G is a (n-)-regular bipartite graph with the partitions A and B Where A = { a a... an} and B ={ b... b bn} By case ( i ) with the similar arguments let G is obtained from the complete bipartite graph by deleting the edges a b a b... a b a b Where 4m 4m 4m 4m a... am am a are labeled as and a m am... a4m as. Now label the vertices of B as follows. b b... b m as b m bm... bm bm as b m bm... b3m as b 3 m b3m... b4m as The edges removed from the complete graph and there labelings are as follows : a b a b... a b m m total m edges a total m+ edges m bm ambm... ambm am bm mbm am3bm3.. a3m b3m 3 mb3m a3m3b3m3... a4m b4m a total m edges a total m edges Number of edges labeled are 8m 8m 4m m m m Number of edges labeled are 8m 8m 4m m m m Number of edges labeled = Number of edges labeled Hence G is cordial. (n-) regular graph is cordial for n = 4m+ Case ( iii ) : n = 4m+ As G is a (n-)-regular bipartite graph with the partitions A and B Where A = { a a... an} and B ={ b... b bn} 4 Page

7 Cordial labeling of k-regular bipartite graphs for k = n n- where k is cardinality of Number of edges in G is 4m 4m 6m m Let be number of vertices in A labeled be number of vertices in A labeled be number of vertices in B labeled be number of vertices in B labeled Total number of edges ab where a is labeled and b B = (4m ) Total number of edges ab where b is labeled and a A= (4m ) Total number of edges ab where a is labeled and b B = (4m ) Total number of edges ab where b is labeled and a A= (4m ) Let Total number of edges ab where a and b both are labeled =x Total number of edges ab where a and b both are labeled =y (i.e. - and - type) The number of edged ab of labeling & And & are 4m (4m ) x Also The number of edged ab of labeling & And &are 4m (4m ) y m (4m ) x m (4m ) = y 4 4 (4 ) (4m ) x = m y But = as G is cordial x = y Now as total number of edges are 6m m number of edges labeled = 8m 6m x+y = 8m 6m x = 8m 6m x y asgis cordial Contradiction L.H.S. is even & R.H.S. is odd Hence G is Not cordial. (n-) regular graph is not cordial for n = 4m+ Case ( iii ) : n = 4m+3 Total number of edges = (4m 3)(4m ) 6m m 6 Using the same notations and the same arguments as in case ( iii ) we get m (4m ) x = m (4m ) y 4 4 (4 ) (4m ) x = m y But = as G is cordial x = y Now as total number of edges are 6m m 6 number of edges labeled = 8m m 3 4 Page

8 Cordial labeling of k-regular bipartite graphs for k = n n- where k is cardinality of x+y = 8m m 3 x = 8m m 3 x y asgis cordial Contradiction L.H.S. is even & R.H.S. is odd Hence G is NOT cordial (n-) regular graph is not cordial for n = 4m+3 Hence the proof. II. Conclusion If G is a -regular bipartite graph with partite sets A and B such that A = B = n then G is cordial iff n = 3 mod 4 If G is a -regular bipartite graph with partite sets A and B such that A = B = n then G is cordial iff its every component is a cycle of length 4m. If G is a n-regular n 3 bipartite graph with partite sets A and B such that A = B = n then G is cordial. If G is a n -regular bipartite graph with partite sets A and B such that A = B = n then G is cordial iff n mod 4. References [] L.CAIX.ZHU () Gaming coloring index of graphs Journal of graph theory [] XUDING ZHU (999) The game coloring number of planar graphs Journal of Combinatorial Theory Series B [3] H. YEH XUDING ZHUING (3) 4-colorable 6-regular toroidal graphs Discrete Mathematics [4] X.ZHU (6) Recent development in circular coloring of graphs Topics in discrete mathematics spinger [5] H.CHANG X.ZHU (8) Coloring games on outerplanar graphs & trees Discrete mathematicsdoi.6/j.disc 9-5 [6] H. HAJIABLOHASSAN X. ZHU (3): The circular chromatic number and mycielski construction Journal of graph theory Page