A total coloring of a graph G in an assignment of colors to the vertices and edges of G such that no two adjacent or incident

Transcription

1 International Journal of Scientific and Research Publications, Volume 7, Issue 7, July Total Coloring of Closed Helm, Flower and Bistar Graph Family R. ARUNDHADHI *, V. ILAYARANI ** * Department Mathematics, D.G.Vaisnav College, Chennai-106. ** Department Mathematics Mother Teresa University Saidapet Chennai Abstract- A coloring of vertices and edges of a graph G is said to be total coloring if no two adjacent or incident elements of have the same color. The minimum number of colors required for a total coloring is said to be the total chromatic number of G and is denoted by χ T (G). In this paper, we have discussed the total chromatic number of closed Helm, Flower and Bistar Graph Families.. Key words- Total coloring, total chromatic number, closed Helm, Flower Graph, Bistar Graph. 1.INTRODUCTION A total coloring of a graph G in an assignment of colors to the vertices and edges of G such that no two adjacent or incident elements have the same color.the minimum number of colors required for a total coloring of a graph G is known as total chromatic number and is denoted by χ T (G).The total coloring was introduced independently by Behzad[1] and vizing[10] as Total coloring conjecture (TCC),which states that χ T (G) (GG)+2 for any graph G with maximum degree (GG). Hilton et al[5] has verified this conjecture for graphs with (GG) 3 V(w).TCC has also been verified by Kostochka[7] for graphs with (GG)+5. Molloy and Reed [9] 4 has shown that there is a constant C such that χ T (G) (GG)+C.The research work of finding total chromatic number for paths cycles, complete and complete bipartite graphs[2], complete multipartite graphs of odd order[4,6], planar graphs G with (GG 11[3], and outer planar graphs [12] has also been done in the past.sudha et al [8] has found the lower and upper bound for the total chromatic number of a new graph called S(n,m)-graph for even n 2m+2 and for odd m >1.They has also found the total chromatic number of S(n,2) for all n 6 and S(n,3) for odd n 7. Vaidya et al [11] have investigated the total chromatic number of some cycle related graphs. In this paper, we investigate the total chromatic number of closed Helm, flower graph and bistar graphs family. II. Total coloring of closed Helm Graph family. In this section, we have obtained the total chromatic number of Closed Helm Graph Family 2.1 Definition (Closed Helm) A closed helm CH n is the graph obtained by taking a helm H n and adding edges between the pendant vertices. Fig.1.Closed Helm CH Coloring Algorithm: Input ; Closed Helm CH n V v, v 1, v 2,..., v n, w 1, w 2,..., w n E x k v v k (k = 1 to n) ;y k v k v k+1 (k = 1 to n 1), y n v n v 1 ; e k v k w k (k = 1to n) z k w k w k+1 (k= 1 to n-1) ; Z n w n w 1 v 1 ;

2 International Journal of Scientific and Research Publications, Volume 7, Issue 7, July For k = 1o n x k k+1 ; r k + 2 ; If r n+1, v k r ; v k r n ; For k= 1 to n s k + 4 ; if s n+1, z k, y k s, z k, y k s n For k= 1 to n e k 1 ; end procedure Output : vertex, edge colored closed Helm. 2.3 Theorem: The Total chromatic number of closed Helm is n+1. (i.e) χ T (CH n ) = n+1, n 4. Proof : Since the maximum degree is n, χ T (CH n ) n+1.the color class of 1 isv, e k ; k = 1 to n.the color class of k (2 k n + 1) is x k 1, v t, y s, z s, w k 1 ; t = k 2 if k > 2 and t = n if k = 2, s = k 4 if k > 4 and s = n + k 4, 2 k 4. Clearly, the elements in each color classes are neither adjacent nor incident. Therefore, the coloring given in the algorithm 2.2 is a total coloring of closed Helm. Hence, χ T (CH n ) = n + 1, n 4. Fig.2.χ T (CH n )=5

3 International Journal of Scientific and Research Publications, Volume 7, Issue 7, July III.Total coloring of Flower Graph family In this section, we have obtained the total chromatic number of Flower Graph Family. 3.1 Definition (Flower Graph) A flower graph Fn is the graph obtained from a helm by joining each pendant vertex to the central vertex of the helm. Fig.3.Flower graph F Coloring Algorithm: Input : Fn (n 4) V v, v 1, v 2,..., v n, w 1, w 2,..., w n E x k v v k (k = 1 to n);y k v k v k+1 (k = 1 to n 1), y n v n v 1 ; z k v k w k (k = 1to n); e k vw k (k = 1to n) For k = 1to n v, z k 1 ; x k k+ 1 ; For k= 1 to n e k n k ; r k+4 ; if r n + 1, y k r ; y k r n ; s k +2 ; if s n+1, v k s ; v k s n ;

4 International Journal of Scientific and Research Publications, Volume 7, Issue 7, July w k k + 1 ; end procedure Output : vertex, edge colored F n. 3.3 Theorem The total chromatic number of Flower Graph is 2n + 1. (i.e) χ T (F n ) = 2n + 1, n 4 Proof : Since (F n ) = 2n χ T (Fn) 2n+1.The color class of 1 isv, z k ; k = 1 to n.the color class of k (2 k n + 1) is x k 1, v t, y s, w k 1 ; t = k 2 if k > 2. t = n if k = 2, s = k 4 if k > 4.s = n + k 4, 2 k 4.The color class of k (k= n + 2 to 2n+1) is e k-n-1. Each color class is independent and hence the coloring given in the algorithm 3.2 is a total coloring of F n. χ T (F n ) = 2n+1, n 4. Fig.4. χ T (F n )=11 IV. Total coloring of Bistar graph family. This section deals with the total chromatic number of a special type of tree, called as Bistar Graph. 4.1 Definition (Bistar) Bistar graph B m, n is the graph obtained from K 2 by joining m pendant edges to one end and n pendent edges to the other end of K Coloring Algorithm Input : B m,n (m, n 2) Fig.5.Bistar graph B m,n V u, v, u 1...u m, v 1, v 2,... v n E e uv, x k uu k (k = 1 to m) ;y k vv k (k = 1to n) v 1 ; e 2 ; u 3 ; For k = 1 to m

5 International Journal of Scientific and Research Publications, Volume 7, Issue 7, July x k k+ 3 ; y k k + 3 ; u k 3 ; v k 1 ; end procedure. Output : Vertex and edge colored B m, n. 4.3 Theorem: The total chromatic number of Bistar graph family is max (m, n) + 3. (i-e χ T (B m,n ) = max (m, n) + 3, m, n 2. Proof : Consider the Bistar graph B m,n whose vertices and edges are colored as in algorithm 4.2 Since (B m, n ) is either m(m > n) or n(n>m) χ T (B m,n ) max (m,n)+1.the color class of 1 is u, v k ; k = 1 to n.the color class of 2 is e.the color class of 3 is v u k ; k = 1 to n.the color class of k is (4 k max(m,n)+3) is x k-3, y k-3.clearly, the elements of each color classes are neither adjacent nor incident. Therefore, the coloring given in the algorithm 4.2 is a total coloring of B m,n. Hence, χ T (B m,n ) = max(m,n) + 3 m,n 2. Fig.6. χ T (B 4,5 )=8 In this paper, we have obtained the following results. (i). χ T (CH n ) = n + 1, n 4. (ii). χ T (F n ) = 2n + 1, n 4. (iii).χ T (B m, n ) = max (m, n) + 3, m, n 2 V.Conclusion VI.References [1]. M. Behzad, Graphs and their chromatic numbers, Ph.D Thesis, Michigan State University, 1965 [2]. M.Behzad, G.Chartrand, and J.K.Cooper Jr.,The Colour number of complete graphs. J.LOndan Math.Sec.42(1967), [3]. O.V.Boroding, A.V.Kostochka, and D.R.woodall, Total colourings of planar graphs with large maximum degree. J.Graph Theory 26(1997), [4]. K.H.Chew and H.P.yap. Total Chromatic number of complete r-partite graphs. J.Graph Theory 16(1992), [5]. A.J.W.Hilton and H.R.Hind, Toal Chromatic number of graphs having large maximum degree, Discrete Math, 117 (1993), [6]. D.G.Hoffman and C.A.Rodger, The Chromatic index of complete multipartite graphs.j.graph theory 16(1992), [7]. A.V.Kostochka, The total coloring of a multigraph with maximal degree 4, Discrete Math, 17, 1989, [8]. S.Sudha and K. Manikandan, Total coloring of s(m,n) graph, International Journal of scientific and Innovative Mathematical Research (IJSIMR) Volume2,Issue I, January-2014,PP ISSN X(Print)& ISSN (Online). [9]. M.Molloy and B.Reed, A bound on the total chromatic number, combinatorica 18 (1998) PP

6 International Journal of Scientific and Research Publications, Volume 7, Issue 7, July [10]. M.V.G.Vizing, Some unsolved problems in graph theory, Uspekhi Mat. Nauk (in Russian) 23(6), 1968, (in Russian) and in Russian Mathematical Surveys, 23(6), 1968, [11]. S.K. Vaidya,total coloring of cycle related graphs,iosr Journal of Mathematics (IOSR-JM) e-issn: , p-issn: X. Volume 11, Issue 3 Ver. V (May - Jun. 2015), PP [12]. Z.Zhang, J.Zhand,and J.Wang, The total chromatic number of some graphs, Scientia Sinica A 31(1988), AUTHORS First Author Dr R. ARUNDHADHI.M.Sc.,M.Phil.,P.hd, Assistant Professor, D.G.Vaisnav College, Arumbakkam, Chennai, Tamilnadu, India arundhadhinatarajan@gmail.com. Second Author V.ILAYARANI.M.Sc,B.Ed, Mother Teresa University, Saidapet, Chennai, Tamilnadu, India ilayamohan29@gmail.com Correspondence Author Dr R. ARUNDHADHI M.Sc., M.Phil., P.hd, Assistant Professor, D.G.Vaisnav College, Arumbakkam, Chennai,Tamilnadu, India arundhadhinatarajan@gmail.com, Contact No:

7 International Journal of Scientific and Research Publications, Volume 7, Issue 7, July