ON THE ADJACENT VERTEX-DISTINGUISHING EDGE COLORING OF C F

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1 JP Joral of Matheatical Scieces Vole 16, Isse 2, 2016, Pages Ishaa Pblishig Hose This paper is available olie at ON THE ADJACENT VERTEX-DISTINGUISHING EDGE COLORING OF C F Lazho City Uiversity Lazho , P. R. Chia Abstract Spposig C = 1 2Lv1, V ( C F ) = { i = 1, 2, K, } U{ v i = 1, i ij j = 1, }, E( C F ) = E( C ) U { v i = 1, j = 1, 2, K } i ij, U { vijvi( j+ 1) i = 1, j = 1, 1}. I this paper, we preset Adjacet Vertex-distigishig Edge Chroatic Nber of C F ( 2). 1. Itrodctio We discssed adjacet vertex-distigishig edge colorig of graph i [1]-[3], it is very difficlt qestio. We itrodce the cocept of adjacet vertex-distigishig edge colorig, thogh it is easier tha vertex-distigishig edge colorig, it is very difficlt too Matheatics Sbject Classificatio: 05C15. Keywords ad phrases: graph, cycle, fa, adjacet vertex-distigishig edge colorig. This stdy is spported by Lazho City Uiversity Ph. D. Research Fd (LZCU-BS ad LZCU-BS ). Received Je 3, 2016

2 48 Defiitio 1 [1]. G is a siple graph ad k is a positive iteger, if it exists a f appig f, E( G) { 1, k}, ad satisfied with f ( e) f ( e ) for adjacet edge e, e E( G), the f is called a Proper Edge Colorig of G, is abbreviated k -PEC of G, ad χ ( G) = i{ k k-pec of G} is called the Edge Chroatic Nber of G. Defiitio 2 [2-5]. For the proper edge colorig f of siple graph, if it is satisfied with C( ) C( v) for V ( G) ( v), where C( ) = { f ( v) v E( G) }, the f is called the Vertex-distigishig Edge Colorig, is abbreviated G, ad χ vd ( G) = i{ k k-vdec of G} is called the Vertex-distigishig Edge Chroatic Nber of G. the k -VDEC of Cojectre. G is a coected graph where V ( G) 3, if G C5 (5-cycle), χ ( G) + 2. I which ( G) is axi degree of G. as Defiitio 3. For a graph G, i is the vertex ber which degree is i, sig δ, deoted the ii, axi degree of G, it is called Cobiatorial Degree of G. λ µ ( G) = ax{ i{ λ i }, σ i }, i Cojectre. For coected graph G ad V ( G) 3, the the left of the cojectre is obviosly tre. µ ( G) χ ( G) vd µ ( G ) + 1

3 ON THE ADJACENT VERTEX-DISTINGUISHING EDGE COLORING 49 Defiitio 4 [6]. Spposig G ad H are two siple graphs which are vertex disjoited ad edge disjoited, the V ( G H ) = V ( G) U V ( H ), E( G H ) = E( G) U E( H ) U { v V ( G), v V ( H )}, G H is called Joi-graph of G ad H. I this paper, we discss Adjacet Vertex-distigishig Edge Chroatic Nber of C F. The ters ad sigs we se i this paper bt ot deoted ca be fod i [5] ad [6]. 2. Mai Reslts Lea 1 [4]. G is a coected graph where V ( G) 3, if there are vertices of axi degree which is adjacet, the χ ( G) + 1. Theore 1. If 2, χ as ( C F ) = + 3. as Proof. Clearly, ( C F ) = + 2, we obtai χ ( C F ) + 3 Lea 1. I order to prove the reslt is tre, we eed prove ( + 3) -AVDEC oly. Case 1. If = 0( od 3). Sppose f is 12, 23, K, 1, we ca color the edges with colors 1, 2, 3, repeatedly. Case 1.1. If = 2. For i 1( od 3), f ( v i1 ) = 2, f ( v i v i2 ) = 4, f ( v i1 v i2 ) = 1. as by C F is

4 50 For i 2( od 3), f ( v i1 ) = 3, f ( v i2 ) = 5, f ( v i1 v i2 ) = 2. For i 0( od 3), f ( v i1 ) = 4, f ( v i2 ) = 5, f ( v i1 v i2 ) = 3. For f, we have C( ) C( ) ( i = 1, 2, K, j = 1, 2), i v ij C( v ) C( v ) ( i = 1, ). i1 i2 Sppose C ( ) = {, 2, 3, 4, 5} \ C( ). The i 1 i C ( ) = { 5}, i 1( od 3); C ( ) = { 4}, i 2( od 3); C ( ) = { 1}, i 0( od 3). So f is a appig abot 5 -AVDEC of C F 2, this proves that the reslt is tre. Case 1.2. If = 3. For i = 1( od 3), f ( v i1 ) = 2, f ( v i2 ) = 4, f ( v i3 ) = 5, f ( v i v i ) 1, f ( v i v i ) =

5 ON THE ADJACENT VERTEX-DISTINGUISHING EDGE COLORING 51 For i 2( od 3), For i 0( od 3), f ( v i1 ) = 3, f ( v i2 ) = 5, f ( v i3 ) = 6, f ( v i v i ) 1, f ( v i v i ) = f ( v i1 ) = 4, f ( v i2 ) = 5, f ( v i3 ) = 6, f ( v i v i ) 2, f ( v i v i ) = For f, sae as Case 1.1, we eed check C ( i ) C( i+1) ( i = 1, 2, K, 1) ad C( 1 ) C( ) oly. The C ( ) = { 6}, i 1( od 3); C ( ) = { 4}, i 2( od 3); C ( ) = { 1}, i 0( od 3). Hece f is a appig abot 6 -AVDEC of C F 3, this proves that the reslt is tre. Case 1.3. If 3. For i 1( od 3), f ( v i1 ) = 2, f ( v ij ) = j + 2 ( j = 2, 3, K, ); f ( v i v i ) 1, f ( vij vi( j+ 1 ) ) = j + 1 ( j = 2, 3, K, 1).

6 52 For i 2( od 3), f ( i vij ) = j + 3 ( j = 1, ); f ( vij vi( j ) ) = 1 ( j = 1, 2, K, 1). + 1 j + For f, sae as Case 1.1, we eed check C ( i ) C( i+1) ( i = 1, 2, K, 1) ad C( 1 ) C( ) oly. The C ( ) = { + 3}, i 1( od 3); C ( ) = { 4}, i 2( od 3); C ( ) = { 1}, i 0( od 3). Hece f is a appig abot ( + 3) -AVDEC of C F ( 3), this proves that the reslt is tre. Case 2. If 1( od 3), sppose f is 12, 23, K, 1. First we ca color the edges with colors 1, 2, 3, 4, the color the edges with colors 1, 2, 3, repeatedly. For ( i 1, 2, K, j = 1, ) ivij = ad v ijv i ( j+1) ( i = 1, j = 1, 2, K, 1), we ca color the edges like Case 1, we ca obtai C F is ( + 3) -AVDEC. This proves that the reslt is tre. Case 3. If 2( od 3), sppose f is 12, 23, K, 1. First we ca color the edges with colors 1, 2, 3, 4, 5, the color the edges with colors 1, 2, 3, repeatedly. The rest edges ca be colored like Case 1, we ca obtai ( + 3) -AVDEC. This proves that the reslt is tre. C F is All i all, the theore is tre.

7 ON THE ADJACENT VERTEX-DISTINGUISHING EDGE COLORING 53 Refereces [1] A. C. Brris ad R. H. Schelp, Vertex-distigishig proper edge-colorigs, J. Graph Theo. 26 (1997), [2] C. Bazga, A. Harkat-Behadie, Li Hao ad M. Woźiak, O the vertexdistigishig proper edge-colorig of graphs, J. Cobi. Theo. Ser. B, 75 (1999), [3] P. N. Balister, B. Bollobás ad R. H. Schelp, Vertex-distigishig colorigs of graphs with G = 2, Discr. Math. 252 (2002), [4] Zhogf Zhag, etc., Adjacet strog edge colorig of graphs, Appl. Math. Lett. 15 (2002), [5] J. A. Body ad U. S. R. Marty, Graph Theory with Applicatios, The Macilla Press, Ltd., New York, [6] P. Hase ad O. Marcotte, Editors, Graph Colorig ad Applicatio, AMS Providece, Rhode, Islad USA, 1999.