Hamiltonian Decomposition of Wheel Related Graphs

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1 Hamiltonian Decomposition of Wheel Related Graphs L.T.Cherin Monish Femila 1 and S.Asha 2 1 Research Scholar ( Reg.No ), Department of Mathematics, Nesamony Memorial Christian College, Marthandam ,Tamil Nadu, INDIA. cherinmonishfemila@gmail.com 2 Assistant Professor, Department of Mathematics, Nesamony Memorial Christian College, Marthandam , Tamil Nadu, INDIA. ashanugraha@yahoo.co.in Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli , Tamil Nadu, INDIA. Abstract A path in a graph G that contains every vertex of G is called Hamiltonian path. A cycle in a graph G that contains every vertex of G is called a Hamiltonian cycle of G. A Hamiltonian graph is a graph that contains a Hamiltonian cycle. A graph is uniquely Hamiltonian if it contains exactly one Hamiltonian cycle. A decomposition of a graph G into Hamiltonian cycles will be called Hamiltonian decomposition of G. In this paper, we decompose wheel related graphs such as wheel graph, line graph of wheel graph, and closed helm graph using the concept of Hamiltonian decomposition. Keywords Hamiltonian cycle, Hamiltonian graph, uniquely Hamiltonian, Hamiltonian decomposition, wheel graph. I. INTRODUCTION Graph theory is proved to be tremendously useful in modeling the essential features of systems with finite components. Graphical models are used to represent telephone network, railway network, communication problems, traffic network etc. A graph is a convenient way of representing information involving relationship between objects. The objects are represented by vertices and the relations by edges. The concept of decomposition of graphs in Hamiltonian cycles, Hamiltonian path decomposition of regular graphs was introduced by Klas Markstrom[4]. Decomposition of complete graphs into Hamilton cycles was discussed in [5] and J.C. Bermond [1] deals about Hamiltonian Decompositions of Graphs, Directed Graphs and Hypergraphs. A path in a graph G that contains every vertex of G is called Hamiltonian path. A cycle in a graph G that contains every vertex of G is called a Hamiltonian cycle of G. A Hamiltonian graph is a graph that contains a Hamiltonian cycle. A graph is uniquely Hamiltonian if it contains exactly one Hamiltonian cycle. A decomposition of a graph G into Hamiltonian cycles will be called Hamiltonian decomposition of G. For n 3, the wheel Wn is defined as C n + K 1, the join of C n and K 1, constructed by joining a new vertex to every vertex of C n. The line graph of a graph G, denoted by L(G), is a graph whose vertex set V(L(G)) is E(G), where two vertices in L(G) are adjacent iff the corresponding edges are adjacent in G. Page No: 338

2 The Helm Hn is the graph obtained from wheel W n by attaching a pendant edge to each of its rim vertices. The Closed Helm CHn is the graph obtained from a helm by joining each pendent vertex to form a cycle. II. HAMILTONIAN DECOMPOSITION OF WHEEL GRAPH W N+1. Theorem: 2.1 The Wheel graph W n+1, n 3 can be decomposed into n Hamiltonian cycles and the rest of the edges can be decomposed into n copies of star K 1,n-2 and n copies of K 2. Proof: Let W n+1, n 3 is the Wheel graph with vertex set V:{v 1,v 2,...,v n,v n+1}. We construct n cyclic permutations, σ 1: ( (n+1) 2 3 (n-2) (n-1) n) σ 2: ( (n+1) (n-1) n 1) σ 3: ( (n+1) n 1 2) σ n: ( (n+1) n 1 2 (n-3) (n-2) (n-1) ), n 3. Now, we form n spanning cycles using these permutations C 1: ( v n+1 v 1 v 2 v 3 v n-2 v n-1 v n ) C 2: ( v n+1 v 2 v 3 v 4 v n-1 v n v 1 ) C 3: ( v n+1 v 3 v 4 v 5 v n v 1 v 2 ) C n : ( v n+1 v n v 1 v 2 v n-3 v n-2 v n-1 ) The remaining edges joining the vertices of the cycle C n : { v 1 v 2 v 3 v n-2 v n-1 v n } and the vertex v n+1 makes n copies of star K 1,n-2, n 3. Also, the remaining edges of the cycle C : { v 1 v 2 v 3 v n-2 v n-1 v n } makes n copies of K 2. Example: 2.2 Page No: 339

3 Theorem: 2.3 The Wheel graph W n+1 e, n 3 is uniquely Hamiltonian and the rest of the edges form a star K 1,n-2. Proof : Wheel graph W n+1 e, n 3 is obtained from a path graph P n by adding a complete graph K 1. Let W n+1 e, n 3 is a graph with vertex set V:{v 1, v 2, v 3,, v n,v n+1}. Clearly, it has a path P n: v 1 v 2 v 3 v n. Form a spanning cycle by connecting the end vertices of P n to K 1. This is the only Hamiltonian cycle, for, the internal vertices of P n joining K 1 forms a cycle but it is not a spanning cycle. Therefore, W n+1 e is uniquely Hamiltonian. The rest of the edges joining K 1 and internal vertices of P n forms star K 1, n-2. Example: 2.4 Theorem: 2.5 The wheel graph W n+2 can be decomposed into 2n 2 + 6n + 4 Hamiltonian paths. Proof: Let the wheel graph W n+3 with vertex set V:{v 0, v 2, v 3, v n, v n+1, v n+2}. Consider the vertex v 0, we construct n+2 paths excluding the vertex v 0. Q 1: v 1 v 2 v 3 v n v n+1 v n+2 Q 2: v 2 v 3 v 4 v n+1 v n+2 v 1 Q 3: v 3 v 4 v 5 v n+2 v 1 v 2 Q n+2 : v n+2 v 1 v 2 v n-1 v n v n+1 Now connect the vertex v 0 to the starting vertex of each paths Q i, 1 i n+2, we get n+2 spanning paths. Also, connect the vertex v 0 to the end vertex of each paths Q i, 1 i n+2, we get n+2 spanning paths. Therefore, by considering the vertex v 0, we get 2n+4 Hamiltonian paths. Next, by considering the vertex v 1, construct 2n+1 paths except the vertex v 1. R 1: v 2 v 3 v 4 v n+1v n+2 v 0 Page No: 340

4 R 2: v 3 v 4 v 5 v n+2 v 0 v 2 R 3: v 4 v 5 v 6 v 0 v 2 v 3 R 2n+1: v n+2 v 0 v 2 v n-1 v n v n+1 Now, connect the vertex v 1 to the starting vertex of each paths R i, 1 i 2n+1, we get 2n+1 spanning paths. Also, connect the vertex v 1 to the end vertex of each paths R i, 1 i 2n+1, we get 2n+1 spanning paths. Therefore, considering the vertex v 1, we get 4n+2 spanning paths. Each W n+3 contains 2 repeated spanning paths, so we remove this two spanning paths. Hence, by considering the vertex v 1, we get 4n Hamiltonian paths. Proceeding like this we get a sequence of Hamiltonian paths 2n+4, 4n, 3n-1, 3n-3,, n+3, n+1. The number of Hamiltonian paths in W n+3 = (2n+4+4n)+(3n-1+3n-3+ +n+3+n+1) = (2n+4+4n) + 2n 2 = 2n 2 +6n+4. Hence the wheel graph W n+3, can be decomposed into 2n 2 +6n+4 Hamiltonian paths. Example: 2.6 III.HAMILTONIAN DECOMPOSITION OF LINE GRAPH OF WHEEL GRAPH L(W n+3) Theorem: 3.1 The Line graph of Wheel graph L(W n+3) can be decomposed into 2n+4 Hamiltonian cycles. Proof: Let L(W n+3) is a Line graph of Wheel graph with vertex set V:{u 1, u 2,..., u n+1, u n+2, v 1, v 2,, v n+1, v n+2}. We construct n+2 paths in the inner cycle Page No: 341

5 Q 1 : u 1 u 2 u 3 u n u n+1 u n+2 Q 2 : u 2 u 3 u 4. u n+1 u n+2 u 1 Q n+2 : u n+2 u 1 u 2 u n-1 u n u n+1 Also we construct n+2 paths in the outer cycle R 1 : v 1 v 2 v 3 v n v n+1 v n+2 R 2 : v 2 v 3 v 4 v n+1 v n+2 v 1 R n+2 : v n+2 v 1 v 2 v n-1 v n v n+1 Now, connect the starting and ending vertices of each paths Q i and R i, 1 i n+2, we get n+2 spanning cycles. Also, the remaining edges joining the vertices of the inner cycle and outer cycle makes n+2 spanning cycles. Therefore, we get 2n+4 Hamiltonian cycles. Case 1: n is odd In L(W n+3), the vertices of inner cycle joining internally makes copies of C n+2, n=1, 3, 5, Case 2: n is even In L(W n+3), the non-consecutive vertices u i, u i+2, i = 1, 2, 3, of inner cycle joining internally makes copies of K 2, n = 2, 4, 6, The remaining vertices of inner cycle joining internally makes copies of C n+2, n = 2, 4, 6, Example: 3.2 Page No: 342

6 IV. HAMILTONIAN DECOMPOSITION OF CLOSED HELM GRAPH CH N+2 Theorem: 4.1 The Closed Helm graph CH n+2 can be decomposed into n 2 +3n+2 Hamiltonian cycles. Proof: Let CH n+2 be the Closed Helm graph with vertex set V:{u 0, u 1, u 2,..., u n+1, u n+2, v 1, v 2,, v n+1, v n+2}, where {u 0, u 1, u 2,..., u n+1, u n+2} be the vertex set of wheel W n+3 and {v 1, v 2,, v n+1, v n+2} be the vertex set of cycle C n+2. Consider the cycle, we construct n+2 paths of length n+2 Q 1: v 1 v 2 v 3 v n v n+1 v n+2 Q 2: v 2 v 3 v 4 v n+1 v n+2 v 1 Page No: 343

7 Q 3: v 3 v 4 v 5 v n+2 v 1 v 2 Q n+2 : v n+2 v 1 v 2 v n-1 v n v n+1 In wheel graph W n+3, consider the vertices of wheel W n+3 with the condition that u 0 is not the end vertices of paths and also the end vertices of paths are consecutive, we get n 2 +3n+2 paths. Considering the vertex u 1, we construct n+1 paths R 1:u 1 u 0 u 2 u 3 u 4 u n u n+1 u n+2 R 2:u 1 u 2 u 0 u 3 u 4 u n u n+1 u n+2 R 3:u 1 u 2 u 3 u 0 u 4 u n u n+1 u n+2 R n+1:u 1 u 2 u 3 u 4 u n+1 u 0 u n+2 Also, we construct n+1 paths S 1:u 1 u 0 u n+2 u n+1 u n u 4 u 3 u 2 S 2:u 1 u n+2 u 0 u n+1 u n u 4 u 3 u 2 S 3:u 1 u n+2 u n+1 u 0 u n u 4 u 3 u 2 S n+1:u 1 u n+2 u n+1 u n u 4 u 3 u 0 u 2 Now, connect the end vertices of path Q 1 (which starts with the vertex v 1) to each paths R i, 1 i n+1, we get n+1 spanning cycles. Also, connect the end vertices of path Q 2 (which ends with vertex v 1) to each paths S i, 1 i n+1, we get n+1 spanning cycles. Therefore, considering the vertex u 1, we get 2n+2 Hamiltonian cycles. Considering the vertex u 2, we construct n+1 paths X 1: u 2 u 0 u 1 u n+2 u n+1 u 5 u 4 u 3 X 2: u 2 u 1 u 0 u n+2 u n+1 u 5 u 4 u 3 X 3: u 2 u 1 u n+2 u 0 u n+1 u 5u 4u 3 X n+1: u 2 u 1 u n+2 u n+1 u 5 u 4 u 0 u 3 Also, we construct n+1 paths Y 1: u 2 u 0 u 3 u 4 u 5 u n+1 u n+2 u 1 Y 2: u 2 u 3 u 0 u 4 u 5 u n+1 u n+2 u 1 Page No: 344

8 Y 3: u 2 u 3 u 4 u 0 u 5 u n+1 u n+2 u 1 Y n+1: u 2 u 3 u 4 u 5 u n+1 u n+2 u 0 u 1 In Wheel graph W n+3, considering any vertex u j, 2 j n+1, the n+1 paths X j, 1 j n+1 (which starts or ends with the vertex u j-1) are repeated. Therefore, considering the vertex u 2, we get n+1 paths. Now, connect the end vertices of path Q 3 (which starts with vertex v 2) to each paths X i, 1 i n+1, we get n+1 spanning cycles. Therefore, considering the vertex u 2, we get n+1 Hamiltonian cycles. Proceeding like this we get a sequence of Hamiltonian cycles 2n+2, n+1, n+1,, n+1. The number of Hamiltonian cycles in CH n+2 = (2n+2)+(n+1+ n+1+ +n+1+n+1) = (2n+2) + n(n+1) = n 2 +3n+2. Hence the closed helm graph CH n+2, can be decomposed into n 2 +3n+2 Hamiltonian cycles. Example:4.2 REFERENCES [1] Bermond J.C., Hamiltonian Decompositions of Graphs, Directed Graphs and Hypergraphs, Advances in Graph Theory, Annals of Discrete Mathematics 3 (1978) [2] Bondy J.A., Murty U.S.R., Graph Theory with Applications, 1976, Elsevier, New York. [3] Bosak J., Decompositions of Graphs, 1990, Kluwer Academic Press, Dordrecht. [4] Klas Markstrom, Even cycle decomposition of 4-regular graphs and line graphs. [5] Murugesan N., Vanadhi T., Hamiltonian decomposition of complete graphs and complete digraphs, Global Journal of theoretical and applied Mathematics Sciences, Vol 1, No 1, (2011), ISSN [6] Vaidya S.K., Pandit R.M., Independent Domination in Some Wheel Related Graphs, Applications and Applied Mathematics: An International Journal, Vol 11, Issue 1 (June 2016), pp , ISSN: Page No: 345