Volume 118 No. 23 2018, 147-154 ISSN: 1314-3395 (on-line version) url: http://acadpubl.eu/hub ijpam.eu Equitable Colouring of Certain Double Vertex Graphs Venugopal P 1, Padmapriya N 2, Thilshath A 3 1,2,3 Department of Mathematics SSN College of Engineering Kalavakkam 603 110 1 venugopalp@ssn.edu.in 2 padmapriyan@ssn.edu.in 3 thilshathanwar@gmail.com Abstract A proper colouring of a graph is equitable if the sizes of colour classes differ by at most one. Let G = (V, E) be a graph of order n 2. The double vertex graph U 2 (G) is the graph whose vertex set consists of all nc 2 unordered pairs from V such that two vertices {x, y} and {u, v} are adjacent if and only if {x, y} {u, v} = 1 and if x = u then y and v are adjacent in G. In this paper the equitable colouring number of double vertex graphs for some simple graphs like path, cycle and star has been proposed. Keywords: Equitable colouring, equitable chromatic number, double vertex graphs. AMS Subject Classification: 05C15 1 Introduction Let G = (V, E) be a finite simple graph. If the vertices of a graph G can be partitioned into k classes V 1, V 2, V 3,... V k such that each V i is an independent set (none of its vertices are adjacent), then G is said to be k colourable and 147
the k classes are called as colour classes. A proper colouring of a graph is equitable if the sizes of colour classes differ by at most one. The graph G is said to be equitably k colourable if there is a k colouring whose colour classes satisfy the condition V i V j 1 for all i j (1) The smallest integer n for which G is equitably n colourable is defined to be the equitable chromatic number of G, denoted by χ e (G). Meyer [8] introduced the equitable chromatic number of a graph and Lih [7] have elaborately discussed about the equitable colouring of graphs, in particular bipartite graphs and trees. Equitable colouring generally arise in some scheduling, partitioning, and load balancing problems [3],[9],[10]. Dorothee [4] has disscussed about the equitable colouring of multipartite graphs. Let G = (V, E) be a graph of order n 2. Alavi et.al. [1], gave the following definition : The double vertex graph U 2 (G) is the graph whose vertex set consists of all nc 2 unordered pairs from V such that two vertices {x, y} and {u, v} are adjacent if and only if {x, y} {u, v} = 1 and if x = u then y and v are adjacent in G. Alavi et. al.[1], studied planarity of double vertex graphs. Survey of double vertex graph found in [2]. Jobby et. al.[5], introduced a natural generalization of this concept called the complete double vertex graph. The complete double vertex graph denoted by CU 2 (G), and is similar to U 2 (G), except that the vertex set is all (n + 1)C 2 unordered 2 multisets of elements of V. 2 Some Results of Double Vertex Graphs and Equitable Chromatic Number The notations used in this paper are (G) : Maximum vertex degree of G P n : Path of order n C n : Cycle of order n S 1,n : Star of order n+1 148
K m,n : Complete bipartite graph of order m+n The following results are used in the study Result 1 [2] The double vertex graph of a graph is a cycle if and only if G = K 3 or G = K 1, 3. Result 2 [2] If G is a connected graph, then the double vertex graph of G is a tree if and only if G = K 2 or G = P 3. Result 3 [2] If G is connected, then its double vertex graph is bipartite if and only if G is bipartite. Result 4 [2] cycle. The double vertex graph of path P n, n 4, contains a 4 Observation 1 [8] The equitable chromatic number of P n is χ e (P n ) = (P n ) = 2. (2) Observation 2 [8] The equitable chromatic number of C n is { 3 if n is odd χ e (C n ) = 2 if n is even. (3) Result 5 [6] Let G = G(X, Y ) be a connected bipartite graph with ϵ edges. Suppose X = m n = Y and ϵ < m/(n + 1) (m n) + 2m. Then χ e (G) m/(n + 1) + 1. Conjecture 1 [6] Let G be a connected bipartite graph, then 3 Main Results χ e (G) ( (G) + 3)/2. (4) In this section,equitable chromatic number for double vertex graphs of some simple graphs has been discussed. 149
Theorem 3.1 If H is the subgraph of a connected graph G, then χ e [U 2 (G)] χ e [U 2 (H)] (5) Proof : Let G be a connected graph and H is its subgraph. Then the number of vertices and edges of the graph H is less than the number of vertices and edges of graph G. Thus the maximum vertex degree of the double vertex graph H is less than or equal to the maximum vertex degree of the double vertex graph G. This implies the vertex colouring of U 2 (H) is less than or equal to the graph U 2 (G). Therefore, the equitable chromatic number of U 2 (H) cannot exceed the equitable chromatic number of the graph U 2 (G). Theorem 3.2 The equitable chromatic number of U 2 (P n ) is χ e (U 2 (P n )) = 3, n 4. (6) Proof : The double vertex graph U 2 (P n ) of path P n is a bipartite graph (by Result 3) with ϵ edges. Let the vertex set of U 2 (P n ) be partitioned into two vertex sets X and Y. Suppose X = m k = Y and ϵ < m/(k + 1) (m k) + 2m. Then by Result 5, χ e (U 2 (P n )) m/(k + 1) + 1 = 3. Hence χ e (U 2 (P n )) 3. (a) (b) Figure 1: (a) Path P 5 (b) Double vertex graph of path U 2 (P 5 ) Suppose, we assume that χ e (U 2 (P n )) = 2. By Result 4, the double vertex graph of path P n, n 4, contains a 4 cycle. For colouring a 4 cycle, atleast two colours are needed. Hence all the four cycles in U 2 (P n ) can be coloured with 2 colours. By the structure of U 2 (P n ) (Figure1(b)), there are 150
2 pendant vertices, both of them have to be coloured with the same colour. This makes the difference of colour classes as two, which contradicts the equitable colouring. This implies χ e (U 2 (P n )) = 3, n 4. Result 3.1 χ e [U 2 (P 3 )] = χ e (P 3 ) = 2. (7) Proof : The double vertex graph of a path P 3 is also a path P 3 by Result 2. So equitable chromatic numbers of P 3 and U 2 (P 3 ) graphs are equal. Hence by Observation.1, χ e [U 2 (P 3 )] = χ e (P 3 ) = 2. (8) Theorem 3.3 The equitable chromatic number of U 2 (C n ) is Proof : The double vertex graph U 2 (C 3 ) is C 3 [2]. By Observation 2, χ e (U 2 (C 3 )) = 3. χ e (U 2 (C n )) = 3, n 3. (9) (a) (b) Figure 2: (a)cycle C 5 (b) Double vertex graph of cycle U 2 (C 5 ) It is observed that for each C n, n 4, U 2 (C n ) has n 2 more edges than U 2 (P n ). These n 2 edges does not affect the equitable coloring of U 2 (C n ) (Figure 2(b)). Hence, the equitable chromatic number of U 2 (C n ) is 3, for n 3. Theorem 3.4 The equitable chromatic number of U 2 (S 1,n ) is χ e (U 2 (S 1,n )) n/2 + 1, n > 2. (10) 151
Proof : For n > 2, the double vertex graph of star is a bipartite graph (by Result 3). By Conjecture 1, χ e (U 2 (S 1,n )) ([ (U 2 (S 1,n ))] + 3)/2. Clearly, (U 2 (S 1,n )) = n 1 Hence χ e (U 2 (S 1,n )) n/2 + 1, n > 2. Theorem 3.5 The equitable chromatic number of U 2 (K m,n ) is χ e (U 2 (K m,n )) ( (K m,n ) + 3)/2. (11) Proof : The double vertex graph U 2 (K m,n ) of a complete bipartite graph K m,n is also a bipartite graph for any m, n and m n [1]. Hence by Conjecture 1, χ e (U 2 (K m,n )) ( (K m,n ) + 3)/2. Result 3.2 χ e [U 2 (K 1,3 )] = χ e (C 6 ) = 2. (12) Proof : The double vertex graph of a complete bipartite graph K 1,3 is a cycle C 6 by Result 1. By Observation 2, if n is even, χ e (C n ) = 2. Hence χ e [U 2 (K 1,3 )] = χ e (C 6 ) = 2. 4 CONCLUSION In this paper, equitable chromatic number for double vertex graphs of some simple graphs were discussed. This work can be extended to study the double vertex graph of other graphical structures. Acknowlegement The authors wish to thank the SSN management for its continuous support and encouragement. References [1] Y. Alavi, M. Behzad, and J. E. Simpson, Planarity of double vertex graphs, Graph Theory Combinatorics, algorithms and applications, SIAM, pp. 472 485, 1991. 152
[2] Y. Alavi, Don R. Lick, and Jiuqiang Liu. Survey of double vertex graphs. Graphs and Combin., Vol.18, Issue4, pp.709 715, 2002. [3] J. Blazwicz, K. Ecker, G. Schmidt and J. Weglarz: Scheduling computer and manufacturing processes, 2nd ed., Springer, Berlin, pp.485, 2001. [4] Dorothee Blum, Jorrey.D and Hammack R, Equitable chromatic number of complete multipartite graphs, Missouri Journal of Math. Sci.,Vol.15, Issue2, pp.75 81, 2003. [5] Jobby Jacob, Wayne Goddard and Renu Laskar, Double vertex graphs and complete double vertex graphs,congressus Numerantium, Vol.184 185, 2007. [6] Ko-Wei Lih, Pou-Lin Wu, On equitable colouring of bipartite graphs, Discrete Mathematics, Vol.151 Issues 1 3, pp.155 160, 1996. [7] K.W.Lih, The equitable coloring of graphs, Handbook of Combinatorial Optimization, Vol.3, pp.543 566, 2013. [8] W.Meyer, Equitable coloring, American Mathematical Monthly, Vol.80, pp.920 922, 1973. [9] B. F. Smith, P. E. Bjorstad and W. D. Gropp: Domain decomposition; Parallel multilevel methods for elliptic partial differential equations; Cambridge University Press, Cambridge, pp.224, 1996. [10] A. Tucker: Perfect graphs and an application to optimizing municipal services, SIAM Review 15,pp. 585 590, 1973. 153
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