Equitable Coloring On Mycielskian Of Wheels And Bigraphs

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1 Applied Mathematics E-Notes, 1(201), c ISSN Available free at mirror sites of amen/ Equitable Coloring On Mycielskian Of Wheels And Bigraphs Kaliraj Kalimuthu, Vernold Vivin Joseph, Akbar Ali Mohamed Moideen Received 6 August 201 Abstract In this paper we obtain interesting results regarding the equitable chromatic number χ = for the Mycielskian of wheel graph µ(w n) the Mycielskian of complete bipartite graph µ(k m,n). 1 Introduction The notion of equitable coloring was introduced by Meyer [14] in 197. This model of graph coloring has many applications. Everytime when we have to divide a system with binary conflicting relations into equal or almost equal conflict-free subsystems we can model this situation by means of equitable graph coloring. This subject is widely discussed in literature [1, 2,, 6, 7, 12, 1, 16, 17]. One motivation for equitable coloring suggested by Meyer [14] concerns scheduling problems. In this application, the vertices of a graph represent a collection of tasks to be performed, an edge connects two tasks that should not be performed at the same time. A coloring of this graph represents a partition of tasks into subsets that may be performed simultaneously. Due to load balancing considerations, it is desirable to perform equal or nearly-equal numbers of tasks in each time step, this balancing is exactly what an equitable coloring achieves. Furmańczyk [7] mentions a specific application of this type of scheduling problem, namely assigning university courses to time slots in a way that spreads the courses evenly among the available time slots avoids scheduling incompatible pairs of courses at the same time as each other, since then the usage of additional resources (e.g.rooms) is maximal. A straightforward reduction from graph coloring to equitable coloring by adding sufficiently many isolated vertices to a graph, proves that it is NP-complete to test whether a graph has an equitable coloring with a given number of colors (greater than two). Furmańczyk et al. [8] proved that the problem remains NP-complete for corona graphs. Bodlaender Fomin [1] showed that equitable coloring problem can Mathematics Subject Classifications: 05C15, 05C75. Department of Mathematics, R.V.S.College of Engineering Technology, Coimbatore , Tamil Nadu, India. Department of Mathematics, University College of Engineering Nagercoil, Anna University, Tirunelveli Region, Nagercoil , Tamil Nadu, India. School of Science Humanities, Department of Mathematics, Karunya University, Coimbatore Tamil Nadu, India. 174

2 Kaliraj. K et. al 175 be solved to optimality in polynomial time for trees (previously known due to Chen Lih []) outerplanar graphs. A polynomial time algorithm is also known for equitable coloring of split graphs [2], cubic graphs [4] some coronas [8]. In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph µ(g), we now call the Mycielskian of G, which has the same clique number as G whose chromatic number equals χ(g) + 1. Mycielski [15] showed that χ(µ(g)) = χ(g) + 1 for any graph G ω(µ(g)) = ω(g) for any graph G with at least one edge. Besides such interesting properties involving clique numbers chromatic numbers, Mycielski s graphs also have some other parameters that behave in a predictable way. For example, it was shown by Larsen et al. [11] that χ f (µ(g)) = χ f (G)+ 1 χ for any graph G, where χ f(g) f(g) is the fractional chromatic number of G. Mycielski s graphs were also used by Fisher [5] as examples of optimal fractional colorings that have large denominators. The problem of determining if χ = (G) = χ(g) or χ = (G) is close to χ(g) 1 is hard has been extensively studied for general graphs. This paper presents a polynomial time solution to the Mycielskian of wheel graphs complete bipartite graphs. 2 Preliminaries If the set of vertices of a graph G can be partitioned into k classes V 1, V 2,..., V k such that each V i is an independent set the condition V i V j 1 holds for every pair (i, j), then G is said to be equitably k-colorable. The smallest integer k for which G is equitably k-colorable is known as the equitable chromatic number of G is denoted by χ = (G). The open neighborhood of a vertex v in a graph G, denoted by N G (v), is the set of all vertices of G which are adjacent to v. Moreover, N G [v] = N G (v) {v} is called the closed neighborhood of v in the graph G. In this paper, by G we mean a connected graph. From a graph G, by Mycielski s construction [9, 10, 15], we get the Mycielskian µ(g) of G with V (µ (G)) = V U {z}, where V = V (G) = {x 1,..., x n }, U = {y 1,..., y n } E (µ (G)) = E (G) {y i x : x N G (x i ) {z}, i = 1,..., n}. We have the following example of a Mycielskian construction of a complete bigraph (Figure 1). For any integer n 4, the wheel graph W n is the n-vertex graph obtained by joining a vertex v 1 to each of the n 1 vertices {w 1, w 2,..., w n 1 } of the cycle graph C n 1. For any set S of vertices of G, the induced subgraph S is the maximal subgraph of G with vertex set S.

3 176 Equitable coloring on Mycielskian of wheels bigraphs Figure 1: The Mycielskian graph of complete bipartite graph K, Equitable Coloring of Mycielskian of Wheel Graph We have the following Theorem 1. THEOREM 1. Let µ(w n ) be the Mycielskian of wheel graph. Then { 5, for n = 4, χ = (µ(w n )) =, for n 5. PROOF. Let V (W n ) = {x i : 1 i n} E(W n ) = {x 1 x i : 2 i n} {x i x i+1 : 2 i n 1} {x n x 2 }. By Mycielski s construction, V (µ (W n )) = V (W n ) {y i : 1 i n} {z}. In µ (W n ), y 1 is adjacent to each vertex of {x i : 2 i n}, each y i (2 i n) is adjacent to the set of vertices of N (x i ). z is adjacent to each vertex of {y i : 1 i n}. 1. n = 4

4 Kaliraj. K et. al 177 Since {x 1, x 2, x, x 4 } is K 4, each of these vertices receives distinct colors c 1, c 2, c, c 4 respectively. The same colors can be used to color the vertices y 1, y 2, y y 4 in the same order. Since z is adjacent to each of y 1, y 2, y y 4 there exists a new color to color z. Hence χ = (µ(w n )) = 5 for n = n 5 By the definition of Mycielskian, x 1 is adjacent to each of {x i : 2 i n} {y i : 2 i n}. There exists a color (say c 1 ), which is assigned to x 1, so that it cannot be assigned to any vertex of {x i : 2 i n} {y i : 2 i n}. The color c 1 can be assigned at most two times in µ (W n ). By the definition of equitable coloring, the number of vertices of µ (W n ), receiving the same color, is either 2 or. Therefore χ = (µ (W n )). We use the following partition to color the vertices of µ(w n ) equitably in the following subcases: (a) n = 5 (b) n = 6 (c) n = 7 V 1 = {x 1, y 1 }, V 2 = {x 2, x 4, z}, V = {x, y, x 5 }, V 4 = {y 2, y 4, y 5 }. V 1 = {x 1, y 1 }, V 2 = {x 2, x 5, y 2 }, V = {x, y, y 5 }, V 4 = {x 4, y 4, y 6 }, V 5 = {x 6, z}. V 1 = {x 1, y 1 }, V 2 = {x 2, y 2, y 6 }, V = {x, y, y 7 }, V 4 = {x 4, y 4 }, V 5 = {x 5, y 5, x 7 }, V 6 = {x 6, z}. (d) n 8 V (µ (W n )) = {x i : 1 i n} {y i : 1 i n} {z}. We first partition this vertex set as follows: V (µ (W n )) = V V

5 178 Equitable coloring on Mycielskian of wheels bigraphs where V = { x i : 1 i 2n 2 } { y i : 1 i 2n 2 } { } x, x n, z V = { x i : 2n + 4 } { i n 1 y i : 2n + 1 } i n. Clearly V V = V V V = with V = 2 2n V = 2n Now we partition V into independent sets each containing two vertices of V as follows: ( V i = {x i, y i } 1 i ) 2n 2 { } V = x, x n, z. Clearly i=1 V i = V V i V j = for i j V i = 2 for each i. Now we add the vertices of V into the partition of V without violating the independence of each V i s. Since each x i (2 i n) is adjacent to y 1 each y i (2 i n) is adjacent to x 1, V 1 cannot be altered. Similarly V cannot be altered. Since z is adjacent with each y i (1 i n). Hence V = V 2 V V 2n 2 V = ( 2n 2 2n + 1 ) + 1 2n Also, V 2, V,..., V 2n 2 can be altered by adding at most one vertex from V in such a way that the resultant V i s contain at most three vertices. Hence V i V j 1 for i j each V i is an independent set. This partition implies V (µ (W n )) can be colored equitably with at most colors. Hence χ = (µ(w n )) = for n 5. This completes the proof of theorem 1. 4 Equitable Coloring of Mycielskian of Bipartite Graph We have the following Theorem 2

6 Kaliraj. K et. al 179 THEOREM 2. Let µ(k m,n ) be the Mycielskian of bipartite graph. Then, if m = n 4, 4, if m = n 5, χ = (µ(k m,n )) =, if 2m = n,, otherwise. PROOF. Let By Mycielski s construction, V (µ(k m,n )) = {x i : 1 i m} {y i : 1 j n} E(K m,n ) = m i=1 {e ij = x i y j : 1 i n}. V (µ (K m,n )) = V (K m,n ) {x i : 1 i m} { y j : 1 j n } {z}. 1. m = n 4 Since {x i : 1 i 4}, {y i : 1 i 4} {y i : 1 i 4} are totally disconnected, each of these subgraphs can be assigned with single color. Assign the color c 1 to each of {x i : 1 i 4}, c 2 to each of {y i : 1 i 4} c to each of {y i : 1 i 4}. Assign c 1 to x 1, c 2 to z c to x 2. By this process of assigning color, each color is assigned exactly 4 times. To color the remaining vertices x x 4, we can use the colors c 1 c without affecting the equitable condition. Hence χ = (µ(k m,n )) = for m = n m = n 5 By applying the above said procedure to color equitably, we color each of {y i : 1 i n} {z} by c 2. Hence the remaining number of vertices to be colored is n. These n vertices should be divided into two sets, each containing n 2 ( n 2 + 1) vertices. Since n 2 n+1 ( n 2 + 1) n+1for n 5, we conclude that the partition of V (µ(k m,n ))into three sets satisfying V i V j 1for i jis not possible hence χ = (µ (K m,n )) 4. We use the following partition to color the vertices of µ(k m,n )for m = n 5equitable in the following: V 1 = {x i : 1 i n} {z}, V 2 = {y i : 1 i n}, V = {x i : 1 i n}, V 4 = {y i : 1 i n}. Clearly V i V j 1 for i j. Hence χ = (µ(k m,n )) 4 so χ = (µ (K m,n )) = 4.

7 180 Equitable coloring on Mycielskian of wheels bigraphs. 2m = n Since there exists an odd cycle x 1 y 1zx 1y 1 x 1 in µ (K m,n ), χ = (µ(k m,n )) for 2m = n, V 1 = {x i, x i : 1 i m}, V 2 = {y i : 1 i n} {z}, V = {y i : 1 i n}. Clearly V i V j 1 for i j. Hence χ = (µ(k m,n )) so χ = (µ (K m,n )) =. 4. V (µ (K m,n )) = 2m+ without loss of generality assume that m < n. Then by the construction of equitable coloring, all the vertices of X = {x i : 1 i m} {x i : 1 i m} receive the colors according to any one of the following cases: (a) If X receives the same color 1. In this case the color 1, appears at the maximum of 2m times. The other colors 2,,... are used at the maximum of 2m+1 times to color the vertices of {y j : 1 j n} { y j : 1 j n} {z}. Therefore each color appear at most 2m + 1 times at least 2m times. (b) If X receives the same color 1 with an another color, then the color 1, appears at the maximum of 2m 1 times other colors 2,,... are used at the maximum of 2m times to color the vertices of {y j : 1 j n} { y j : 1 j n } {z}. Therefore in this case each color appear at most 2m times at least 2m 1 times. Therefore we conclude from the subcases (a) (b) that the minimum number of colors to be used to get equitable coloring of µ (K m,n ) is either [ ] 2m+ [ ] 2m or 2m+. (i.e.) 2m+1 m. In general. In such a coloring, out of colors, ( ) 2m+ (2m + 2n + 1) colors appear 2m+ + 1 times 2m + 2n m + n (2m + 2n + 1) min (m, n) colors appear the proof of theorem 2. 2m+ times. Hence χ = (µ(k m,n )) =. This completes

8 Kaliraj. K et. al Summary This paper establishes an optimal polynomial time solution to the equitable chromatic number of Mycielskian of wheels bigraphs. A further study on equitable coloring Mycielskian graph families would result in some nice contributions to the literature applications in this area. Acknowledgment. The authors sincerely thank the anonymous reviewer for his/her careful reading, constructive comments fruitful suggestions that have resulted in the improvement of the quality of this manuscript. References [1] H. L. Bodleer F. V. Fomin, Equitable colorings of bounded treewidth graphs, Theoret. Comput. Sci., 49(1)(2005), [2] B. L. Chen, M. T. Ko K. W. Lih, Equitable m-bounded coloring of split graphs, in: Combinatorics Computer Science (Brest, 1995) LCNS 1120, Springer (1996), 1 5. [] B. L. Chen K.W. Lih, Equitable coloring of trees, J. Combin. Theory Ser. B, 61(1994), [4] B. L. Chen, K.W. Lih P.L. Wu, Equitable coloring the maximum degree, European J. Combin., 15(5)(1994), [5] D. C. Fisher, Fractional colorings with large denominators, J. Graph Theory, 20(1995), [6] H. Furmańczyk, Equitable coloring of graphs, in: Graph Colorings, (M. Kubale, ed.) American Mathematical Society Providence, Rhode Isl (2004). [7] H. Furmańczyk, Equitable coloring of graph products, Opuscula Math., 26(1)(2006), [8] H. Furmańczyk, K. Kaliraj, M. Kubale V. J. Vernold, Equitable coloring of corona products of graphs, Adv. Appl. Discrete Math., 11(2)(201), [9] G. J. Chang, L. Huang X. Zhu, Circular chromatic numbers of Mycielski s graphs, Discrete Math., 205(1 )(1999), 2 7. [10] J. Miškuf, R. Škrekovski, M. Tancer, Backbone Colorings generalized Mycielski Graphs, SIAM J. Discrete Math., 2(2)(2009), [11] M. Larsen, J. Propp D. Ullman, The fractional chromatic number of Mycielski s graphs, J. Graph Theory, 19(1995), [12] K. W. Lih P. L. Wu, On equitable coloring of bipartite graphs, Discrete Math., 151(1 )(1996),

9 182 Equitable coloring on Mycielskian of wheels bigraphs [1] W. H. Lin G. J. Chang, Equitable colorings of Cartesian products of graphs, Discrete Appl. Math., 160(2012), [14] W. Meyer, Equitable coloring, Amer. Math. Monthly, 80(197), [15] J. Mycielski, Sur le coloriage des graphes, Colloq. Math., (1955), [16] W. Wang K. Zhang, Equitable colorings of line graphs complete r-partite graphs, Systems Sci. Math. Sci., 1(2)(2000), [17] H. P. Yap Y. Zhang, Equitable colourings of planar graphs, J. Combin. Math. Combin. Comput., 27(1998),