Radio coloring for families of Bistar graphs

Transcription

1 Inter national Journal of Pure and Applied Mathematics Volume 113 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu Radio coloring for families of Bistar graphs February 13, 2017 A.Vimala Rani 1 and N.Parvathi 2 Department of Mathematics, SRM University Kattangulathur, India vimalarani92a@gmail.com Department of Mathematics, SRM University Kattangulathur, India parvathi.n@ktr.srmuniv.ac.in Abstract An assignment of colors to the vertices of a graph G so that adjacent vertices have different colors is a coloring of G. Concepts and questions arise naturally from practical problems and have found applications in many areas, including Information Theory and most notably Theoritical computer science. Radio Coloring is one of the coloring concepts in Graph Coloring. It is due to Harary. A k-radio coloring of a graph is a function f from the vertex set V(G) to the set of all non negative integers 1, 2, k such that i f(x) f(y) 1 if d(x, y) = 2 and ii f(x) f(y) 2 if d(x, y) = 1 The radio number of G is the smallest k for which G has radio coloring and is denoted by rn(g) of G. In this paper we find the radio number of Square, Shadow, Splitting and degree splitting graphs of Bistar Graph. ijpam.eu

2 AMS Subject Classification: 05CO7, 05C12,05C15, 05C76. Keywords: Radio coloring, Radio number, Bistar Graph, Square of a graph, Shadow graph, Spiltting graph and degree Spiltting graph. 1 INTRODUCTION All graphs considered in this paper are finite, nontrivial, simple, connected and undirected. A k-coloring of a graph G is an assignment of k-colors 1, 2, k to the vertices of G. The coloring is proper if no two adjacent vertices have the same color. The minimum number of colors used for proper coloring is called chromatic number of G and is denoted by χ(g). A graph G is radio colored if the colors f(v i ) assigned to every vertex v i V verify the following two conditions. i f(x) f(y) 1 if d(x, y) = 2 and ii f(x) f(y) 2 if d(x, y) = 1 The minimum number of colors used in a radio coloring is the radio chromatic number rn(g) of G. The bistar B n,n is graph obtained by joining the center (apex) vertices of two copies of K 1,n by an edge. For a simple connected graph G the square of graph G is denoted by G 2 and defined as the graph with the same vertex set as of G and two vertices are adjacent in G 2 if they are at a distance 1 or 2 apart in G. The shadow graph D 2 (G) of a connected graph G is constructed by taking two copies of G say G and G. Join each vertex u in G to the neighbours of the corresponding vertex v in G. For a graph G the splitting graph S (G) of a graph G is obtained by adding a new vertex v corresponding to each vertex v of G such that N(v) = N(v ). ijpam.eu

3 Let G = (V (G), E(G)) be a graph with V = S 1 S 2 S 3 S i T where each S i is a set of vertices having at least two vertices of the same degree and T = V \ S i. The degree splitting graph of G denoted by DS(G) is obtained from G by adding vertices w 1, w 2, w 3, w t and joining to each vertex of S i for 1 i t. In this article we find the radio chromatic number for Square, Shadow, Splitting and Degree Splitting graphs of Bistar graph. Analysis: By the analysis of various graph families for finding radio number, it is found that on radio coloring of a graph using positive integers, the radio number is obtained by using either only odd integers or even integers. The usage of both odd and even integers alternatively for coloring the graphs leads to large number of colors compared to using only odd or even. 2 Results Result: 1 The radio chromatic number of path P n is three for atleast three vertices. Result: 2 The radio chromatic number of comb graph G is four for n 3. Theorem 2.1. is n + 1 The radio chromatic number of star graph S n Theorem 2.2. The radio chromatic number of cycle + 1 n = 3i, i = 1, 2,... rn(c n ) = + 2 n = 3i + 1, i = 1, 2, n = 3i + 2, i = 1, 2,... (1) Theorem 2.3. The radio chromatic number of sunlet graph S n ijpam.eu

4 rn(s n ) = { + 2 n 1 mod Otherwise (2) 3 The radio chromatic number for Square, Shadow, Splitting and Degree Splitting graphs of Bistar graph Theorem 3.1. Let G be a bistar graph of order 2n+2. Then where (G) is the maximum degree of G. rn(b n,n ) = n + 2 (3) Proof. The Star K 1,n has exactly n + 1 vertices while the Star K 1,n has exactly n+1 vertices. Hence, the bistar has n + n + 2 ie., 2n + 2 vertices. On the other hand K 1,n has n edges and K 1,n has n edges. Since the bistar graph is obtained by adding the edge uv, the size of the bistar B n,n is 2n + 1. The rn of k 1,n is n+1. Hence both K 1,n and K 1,n can have the same n+1 colors. Since B n,n is obtained by joining two Star graphs by an edge the two centre(apex) vertices becomes adjacent and they require different color other than n+1. Hence the minimum number of colors required is n+1+1. Thus radio number of (B n,n ) is n+2. Theorem 3.2. For all n, rn(b 2 n,n) = 2n + 2 Proof. The order of Square graph of Bistar graph is same as that of bistar graph. For the size of square graph of Bistar graph, each edge otherthan the edge uv is doubled since the centre vertex of K 1,n is at distance two from the pendant vertices of K 1,n. Simillarly for K 1,n. Hence the size of the graph is 4n + 1. Let G = B 2 n,n. Let f : v {Set of positive odd integers} be radio coloring of G. In G every vertex V i can be reached from all the other vertices by distance two and the adjacency of B n,n is also ijpam.eu

5 preserved. By the definition of radio coloring if d(x, y) = 2 then f(x) f(y) 1. Therefore each vertex of G receives a unique color. Hence the radio number of Bn,n 2 = 2n + 2 which is the order of G. Theorem 3.3. For all n, rn(d 2 (B n,n )) = 2n + 4 Proof. By the definition of shadow graph the order of D 2 (B n,n ) is twice that of bistar graph since it has two copies of bistar graph G 1, G 2. Hence the order of shadow graph of bistar graph is 4n + 4. The size of D 2 (B n,n ) is sum of the size of G 1, G 2 and sum of the degree of vertices of a copy of bistar graph since by definition each vertex v i in G 1 should be adjacent to the neighbours of the corresponding v i in G 2. Therefore the size of D 2 (B n,n ) is 8n+4. Let f : v {Set of positive odd integers} be radio coloring of D 2 (B n,n ). The structure of shadow graph of bistar graph gives that each center vertex v i is at distance one or two from the other vertices. So, the four centre vertices are assigned unquie colors and each vertex v i other than the center vertices receives 2n colors since the n vertices in each copy G 1 receives the same color. Adding all the colors gives the desired result. Theorem 3.4. For all n > 0, rn(s (B n,n )) = 2n + 3 Proof. The Spiltting graph of a graph has order 4n + 4 since a vertex is added corresponding to each vertex of (B n,n ). The size of S (B n,n ) is 6n+3 since N(v) = N(v ). Let f : v {Set of positive odd integers} be radio coloring of S (B n,n ). The structure of Spiltting graph of bistar graph gives that each center vertex v i is at distance one or two from the other vertices. So, the two centre vertices are assigned unquie colors and each vertex v i other than the center vertices receives 2n + 1 colors since the n vertices in (B n,n ) receives same colors and the newly added vertices corresponding to vertices u and v receives the same ijpam.eu

6 color. Adding all the colors gives the desired result. Theorem 3.5. For all n > 0, rn(ds(b n,n )) = 2n + 3 Proof. The Degree Spiltting graph of a graph has order 2n + 4 and size 4n + 3. Let f : v {Set of positive odd integers} be radio coloring of DS(B n,n ). The structure of Degree Spiltting graph of bistar graph shows that each vertex other than the new vertices receives the different colors and the newly added two vertices receives unique colors. Adding all the colors gives the desired result. 4 Conclusion In this paper we obtain the radio chromatic number for Square, Shadow, Splitting and Degree Splitting graphs of Bistar graph. Despite of the few results in radio coloring we still experiment on radio number of various families of graphs. 5 Application The problem of reducing interference in wireless networks is becoming increasingly important with the continuous deployment of larger and more sophisticated wireless networks. The interference reduction problem is modeled as a graph coloring problem. However, additional constraints to graph coloring scenarios that account for various networking conditions result in additional complexity to standard graph coloring. Signal interference is a major drawback of wireless networks. The aim of reducing interference is to prevent adjacent or connected nodes, which are linked by radio signals, from receiving and transmitting signals which conflict or blend together. Thus, interference occurs when conflicting transmissions over one radio frequency are received by one or more nodes in a wireless network. This inhibits ijpam.eu

7 the ability of the receiver to decipher incoming signals. Applying the graph coloring problem to reducing interference in wireless networks is practical compared to investing large amounts of money in radio transmitter technology or simply hard-wiring a temporary fix. The problem of reducing interference in arbitrary networks turns out to be very difficult, and for this reason, simpler network layouts have been investigated.numerous methods for reducing interference exist, such as topology control, power control, and channel assignment. This paper will focus exclusively on the latter method, which seeks to assign channels of different frequencies to interfering nodes or edges. Through careful assignment of communication channels to nodes in a network, interference could be greatly reduced. It is important to note, however, that the number of radio frequencies is finite, and therefore, the problem of minimizing the number of channels allocated to a specific network is worthy of thorough investigation as well. In some instances, channel overlap is necessary if the number of assigned channels for a network is inadequate to connect all nodes. References [1] Graph Radiocoloring Concepts,R. Kalfakakou, G. Nikolakopoulou, E. Savvidou, M. Tsouros saristotle university of Thessaloniki, Faculty of engineering, Thessaloniki, Greece. [2] A.Vimala Rani and Dr.N.Parvathi, Radio Number of Star Graph and Sunlet Graph, Global Journal of Pure and Applied Mathematics (GJPAM) Volume 12,Number 1 (2016). [3] Vaidya, S.K., Shah, N.H., Cordial labeling for some Bistar Related Graphs, International Journal of Mathematics and Soft Computing, (4), (2) (2014). [4] Harary, F., Private communication. [5] Aardal, K., Hurkens, A., Lenstra, J.K., and Tourine, S.R., Algorithms for frequency assignment problems, CWI Quarterly, 9 (1996) 1-9. ijpam.eu

8 [6] Cameron, and Wu, Y., A frequency assignment algorithm based on a minimum residual difficulty heuristic, Proc. IEEE Int.Symp. EMC 79 (CH EMC), 1979, [7] Hale, W.K., Frequency assignment: Theory and applications, Proceedings of the IEEE, 69 (12) (1980). [8] Harary, F., Graph Theory, Addison-Wesley, Reading MA, [9] Michaels, J.G., and Rosen, K.H., Applications of Discrete Mathematics, McGraw-Hill Inc, [10] Murphey, R.A., Pardalos, P.M., and Resende, M.G.C., Frequency assignment problems, in: Handbook of Combinatorial Optimization, Kluwer Academic Press, [11] R.K.Yeh, Labelling graphs with a condition ar distance two. Ph.D Thesis, University of South Cardino(1990). [12] Savvidou, E., Algorithm for graph coloring, radiocoloring and predefined coloring separation, Dpt of Appl. Informatic, Univ. of Macedonia,Thessaloniki, [13] Christofides, N., Graph Theory, an Algorithmic Approach, Academic Press, New York, [14] Garey, M.R., and Johnson, D.S., Computers and Intractability : A Guide to the Theory of NPCompleteness, Freeman, San Francisco, ijpam.eu

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