Hamiltonian Laceability in Line Graphs

Abstract

A Connected graph G is a Hamiltonian laceable if there exists in G a Hamiltonian path between every pair of vertices in G at an odd distance. G is a Hamiltonian-t-Laceable (Hamiltoniant*-Laceable) if there exists in G a Hamiltonian path between every pair (at least one pair) of vertices at distance't' in G. 1≤ t ≤ diamG. In this paper we explore the Hamiltonian-t*-laceability numberof graph L (G) i.e., Line Graph of G and also explore Hamiltonian-t*-Laceable of Line Graphs of Sunlet graph, Helm graph and Gear graph for t=1,2 and 3. KeywordsConnected graph, Line graph, Sun let graph, Helm graph, Wheel graph, Gear graph and Hamiltonian-t-laceable graph. INTRODUCTIONAll graphs considered here are finite, simple, connected and undirected graph. Let [7] the authors have studied Hamiltonian-t-laceability and Hamiltonian-t*laceability of various graph structures. In this paper we explore the Hamiltonian-t*-laceability number of Line graph L(G) and also Hamiltonian-t*-laceability of Line graph L(G) of the sun let graph, Helm graph and Gear graph. DEFINITION 1The Line graph L(G) of G is the graph of E in which E y x  , are adjacent as vertices if and only if they are adjacent as edges in G. In Figure 1, we display the graph G and its Line graph L (G). DEFINITION 2The Sun let graph S n is a graph with cycle where by each vertex of the cycle is attached to one pendent vertex. Each sun let graph contains r-vertices with r-edges.In Figure 2, we display the Sun let graph S n DEFINITION 3The wheel graph with n spokes, W n is the graph that consists of an n-cycle and one additional vertex, say u, which is adjacent to all the vertices of the cycle. In Figure 3, we display the Wheel graph W 6 . International Journal of Computer Applications (0975 -8887)Volume 98-No.12, July 2014 17 Hamiltonian Laceability in Line GraphsManjunath.G KeywordsConnected graph, Line graph, Sun let graph, Helm graph, Wheel graph, Gear graph and Hamiltonian-t-laceable graph. INTRODUCTIONAll graphs considered here are finite, simple, connected and undirected graph. Let [7] the authors have studied Hamiltonian-t-laceability and Hamiltonian-t*laceability of various graph structures. In this paper we explore the Hamiltonian-t*-laceability number of Line graph L(G) and also Hamiltonian-t*-laceability of Line graph L(G) of the sun let graph, Helm graph and Gear graph. DEFINITION 1The Line graph L(G) of G is the graph of E in which E y x  , are adjacent as vertices if and only if they are adjacent as edges in G. In Figure 1, we display the graph G and its Line graph L (G). DEFINITION 2The Sun let graph S n is a graph with cycle where by each vertex of the cycle is attached to one pendent vertex. Each sun let graph contains r-vertices with r-edges.In Figure 2, we display the Sun let graph S n DEFINITION 3The wheel graph with n spokes, W n is the graph that consists of an n-cycle and one additional vertex, say u, which is adjacent to all the vertices of the cycle. In Figure 3, we display the Wheel graph W 6 . International Journal of Compute...