On Harmonious Colouring of Line Graph. of Central Graph of Paths

Transcription

1 Applied Mathematical Scieces, Vol. 3, 009, o. 5, O armoious Colourig of Lie Graph of Cetral Graph of Paths Verold Vivi J * Ad K. Thilagavathi # * Departmet of Mathematics Sri Shakthi Istitute of Egieerig ad Techology Coimbatore , Idia # Departmet of Mathematics, Koguadu Arts ad Sciece College Coimbatore , Idia verold_vivi@yahoo.com Abstract I this paper, we preset some properties of the cetral graph P ) of a path, ad its lie graph L P )]. We maily have our discussio o the harmoious chromatic umber of P ) ad the lie graph of cetral graph of P, deoted by L P )]. Mathematics Subject Classificatio: 05C75, 05C15 Keywords: Cetral graph, paths, lie graph ad harmoious Colourig INTRODUCTION All graphs cosidered here are udirected. All the otios that are ot defied i this paper ca be foud i 3,14]. The cetral graph1,31,33,34] of G, deoted by G) is obtaied by subdividig each edge of G exactly oce ad joiig all the o-adjacet vertices of G. By defiitio p G) = p+q, where p ad q deotes the umber of vertices ad edges i G. The lie graph 3,14] L(G) of a graph G=(V, E) is a graph with vertex set E(G) i which two vertices are adjacet if ad oly if the correspodig edges i G are adjacet.

2 06 V. Vivi ad K. Thilagavathi A harmoious colourig,4,6,7,8,9,10,11,1,15,16,18,0,1,,3,4]is a proper vertex colourig i which every pair of colours appears o at most oe pair of adjacet vertices. The harmoious chromatic umber χ (G) of a graph G is the miimum umber of colours eeded for ay harmoious colourig of G; The above defiitio implies that harmoious colourig is defied for simple graphs rather tha multigraphs 1. A BRIEF REVIEW OF ARMONIOUS COLOURING The first paper o harmoious graph colourig was published i 198 by Frak arary ad M. J. Platholt 15]. owever, the proper defiitio of this otio is due to J. E. opcroft ad M. S. Krishamoorthy 18] i1983. S.Lee ad Joh Mitchum,3] published a paper cosistig of the upper boud for the harmoious chromatic umber of graphs i I 1988, Z. Miller ad D. Pritiki, worked o harmoious colourig ad gave the harmoious chromatic umber of graphs, i the Proceedigs of 50th Aiversary Coferece o Graph Theory (Fort Waye, Idiaa, 1986) (eds. K. S. Bagga et al.), Cogressus Numeratium. D.G. Beae, N.L.Biggs ad B.J. Wilso, studied the growth rate of harmoious chromatic umber i Agai Z. Miller ad D. Pritiki 7] published a paper o the topic the harmoious colourig umber of a graph i I 1991 C. J.. McDiarmid ad Luo Xihua 6] gave the Upper bouds for harmoious colourigs. Zhikag Lu 38] gave a published work o the harmoious chromatic umber of a complete biary ad triary tree, i e also published a paper o Estimates of the harmoious chromatic umbers of some classes of graphs (Chiese), Joural of Systems Sciece ad Mathematical Scieces, 13 (1993). A combied work by L. R. Casse, C. M. O Keefe ad B. J. Wilso 5] gave us the Miimal harmoiously colourable desigs i I the same year, I. Krasikov ad Y. Roditty 1] gave a paper o bouds for the harmoious chromatic umber of a graph. Zhikag Lu,38] i 1995, published a paper o the harmoious chromatic umber of a complete 4-ary tree. Also K. J. Edwards 6] worked ad gave results o the harmoious chromatic umber of almost all trees. I the ext year (1996) he ivestigated o the harmoious chromatic umber of bouded degree trees 7]. Joh P. Georges 0] published a paper o the harmoious Colourigs o collectio of graphs i I 1996, a paper o the harmoious chromatic umber of quasistars, was give by I. avel ad J.M. Laborde Mauscript, Prague ad Greoble, I 1997, K. J. Edwards,8] cotiued his work o the harmoious chromatic umber of bouded degree graphs, ad also published papers relatig harmoious colourig ad achromatic umber. Zhikag Lu 41,4] published a paper o the exact value of the harmoious chromatic umber of a complete biary tree i 1997 ad triary tree i I 1998, K. J. Edwards 9] published a work emphasizig a ew upper boud for the harmoious chromatic umber, ad i 1999 o the harmoious chromatic umber of complete r-ary trees.

3 O harmoious colourig of lie graph 07 J. Mitchem ad E. Schmeichel, published a paper The armoious Chromatic Number of Deep ad Wide Complete -ary Trees, i The Nith Quadreial Iteratioal Coferece o Graph Theory, Combiatorics, Algorithms ad Applicatios (Kalamazoo, Michiga, 000) (eds. Y. Alavi, D. Joes, D. R. Lick ad Jiuqiag Liu), Electroic Notes i Discrete Mathematics, 11 (00). A work o the harmoious chromatic umber of P(α,K ), P(α,K 1, ) ad P(α,K m, ),was published by M. F.Mammaa 5] i 003. D. Campbell ad K. J. Edwards 4] agai gave a ew lower boud for the harmoious chromatic umber i 004. I 006, K. Thilagavathi ad J. V. Vivi,30] published a paper armoious colourig of graphs. We had a detailed study o the harmoious chromatic umber of lie graph of cetral graph of C, K, K 1,, K, ad its lie graphs deoted by L C )] L K )] L K 1, )] ad L K, )] respectively i 31,33,34] Also we have discussed the harmoious chromatic umber of middle graph of cetral graph of C, K, K 1, ad P deoted by M C )], M K )], M )] ad M P )] respectively i 1,35]. K 1, The problem of harmoious colorig ca be applied i such diverse areas as radio avigatio, data acquisitio ad image compressio. Ufortuately, the problem is NPhard.. STRUCTURAL PROPERTIES OF P ) Throughout this paper P deotes a path o edges. The cetral graph of P, P ) has (i) (+1) vertices of degree. (ii) vertices of degree. (iii) The umber of vertices i p = ( 1). C ( P ) + (iv) The umber of edges i q P ) + 3 =.

4 08 V. Vivi ad K. Thilagavathi 3. STRUCTURAL PROPERTIES OF L P )] (i) The umber of vertices havig degree maximum degree Δ deoted ( 1) by ( pδ ) =, similarly ( pδ ) =. (ii) + 3 The umber of vertices i L P )], pl C ( P )] =. (iii) 3 + The umber of edges i L P )], ql P )] =. (iv) I L P )], miimum degree δ = ad maximum degree Δ = ( 1). ece L P )] is ( δ =, Δ 0(mod )) graphs. (v) L P )] is always hamiltoia; if is eve the L P )] is both Euleria ad amiltoia 17]. Example 3.1 L P )] 4 L P4 )] is both Euleria ad amiltoia. Figure 1 4. ARMONIOUS COLOURING ON P ) AND L P )] Observatio 4.1 χ ( ) = P { }

5 O harmoious colourig of lie graph 09 Example 4.1 χ { 1+ (4) + 1} 4 ( P 4 ) = = Figure Theorem 4. χ P )] = + 3, ( >1). Proof Figure 3 Figure 4 Let v 1, v,.v +1 be the vertices of the P. Let u i be the vertex of subdivisio of the edge v i v i+1 of P. Let S = { ui /1 i }. Assig the colour c i to the vertex v i ( 1 i + 1) assig c + to the vertex u i if i is odd ad assig c +3 to the vertex u i if i is eve. The clearly the above said colourig is a harmoious colourig with miimum umber of colours. Therefore χ P )] = + 3.

6 10 V. Vivi ad K. Thilagavathi Example 4. χ P4 )] = = 7 Figure 5 Theorem χ { L P )]} =, if >. Proof Let v 1, v,.v +1 be the vertices of the P. By the defiitio of P ), each edge v i v i+1 of P is subdivided by the vertex u i. Clearly the umber of edges i P ) is ( +1) i.e., q P ) =. Each vertex v i is o adjacet with exactly (-1) vertices i P. i.e., Each vertex v i is adjacet with u i ad hece deg C ( P ) v i = 1+ 1 = for each i = 1,,.,. v i ad v i+1 are o adjacet vertices i P ). Let E i = {e i1, e i,..,e i } be the edges i P ) icidet with v i for i = 1,,.,. Clearly the edges e i1, e i,..,e i are mutually adjacet for each i. Clearly E i ad E i+1 are mutually disjoit. The edges of E i form a clique K 1 of vertices i L P )] similarly the edges of E i+1 form a clique K of vertices i L P )]. The edges of K 1 ad K are disjoit. Clearly there exist a vertex v of E{ L P )]} ( K 1 K ) such that v is either adjacet with K 1 or K but ot with both. I ay harmoious colourig colours are assiged to the vertices of K 1 ad K ad 1 colours are assiged to the vertices of E L P )]} ( K 1 K ). Also it is miimal harmoious colourig. { Therefore χ { L P )]} = + 1 ( 1) = =

7 O harmoious colourig of lie graph 11 = + 3. Ackowledgemets. This work was doe whe the first author was a research scholar i Koguadu Arts ad Sciece College, Coimbatore, Idia. REFERENCES 1] Akbar Ali.M.M ad Verold Vivi.J, armoious Chromatic Number of Cetral graph of Complete Graph Families, Joural of Combiatorics, Iformatio ad System Scieces. Vol. 3 (007) No.1-4 (combied), pp ] D.G. Beae, N.L. Biggs ad B.J. Wilso, The growth rate of harmoious chromatic umber, Joural of Graph Theory, Vol.13 (1989) pp ] J. A. Body ad U.S.R. Murty, Graph theory with Applicatios. Lodo: MacMilla (1976). 4] D.Campbell ad K. J. Edwards, A ew lower boud for the harmoious chromatic umber, Australasia Joural of Combiatorics, 9 (004), pp ] L. R. Casse, C. M.O Keefe ad B. J. Wilso, Miimal harmoiously colourable desigs, Joural of Combiatorial Desigs, (1994), pp ] K. J. Edwards, The harmoious chromatic umber of almost all trees, Combiatorics, Probability ad Computig, 4 (1995), pp ] K. J. Edwards, The harmoious chromatic umber of bouded degree trees, Combiatorics, Probability ad Computig, 5 (1996), pp ] K. J. Edwards, The harmoious chromatic umber of bouded degree graphs, Joural of the Lodo Mathematical Society (Series ), 55 (1997), pp ] K. J. Edwards, A ew upper boud for the harmoious chromatic umber, Joural of Graph Theory, 9 (1998), pp ] K. J. Edwards, The harmoious chromatic umber of complete r-ary trees, Discrete Mathematics, 03 (1999), pp

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10 14 V. Vivi ad K. Thilagavathi 37] Zhikag Lu, O a upper boud for the harmoious chromatic umber of a graph, Joural of Graph Theory, 15 (1991), pp ] Zhikag Lu, The harmoious chromatic umber of a complete biary ad triary tree, Discrete Mathematics, 118 (1993), pp ] Zhikag Lu, Estimates of the harmoious chromatic umbers of some classes of graphs, Joural of Systems Sciece ad Mathematical Scieces, 13 (1993), pp ] Zhikag Lu, The harmoious chromatic umber of a complete 4-ary tree, Joural of Mathematical Research ad Expositio, 15 (1995), pp ] Zhikag Lu, The exact value of the harmoious chromatic umber of a complete biary tree, Discrete Mathematics, 17 (1997), pp ] Zhikag Lu, Exact value of the harmoious chromatic umber of a complete triary tree, Systems Sciece ad Mathematical Scieces, 11 (1998), pp Received: July, 008