(Sub) Pancyclic Conditions In Line Graph

Content : In this thesis, we mainly prove the following results about (sub)pancyclic condition in graph and line graph.Theorem 1 Let G be a simple connected graph with n vertices which satisfiescondition of p2(G) = min{d(u) + d(v) :uv∈ E(G), w,v∈F(G)}≥8. If n≥72 and girth g(G)≥5, δ2(G) = min{rf(w) + d(y): uv(?) E(G), u,v∈V(G)}>2n+1 , thenline graph L(G) is subpancyclic graph.Theorem2 Let G be a simple connected graph with n vertices which satisfies condition of p2(G)≥8. If n≥72 and girth g(G)≥4, δ22(G)- δ2(G) > 2n , thenline graph L(G) is subpancyclic graph.Theorem3 Let G be a simple connected graph with q edges and q ≥ 5. If itsline graph L(G) satisfies condition of δ(G) > , then line graph L(G) is subpancyclic graph.Theorem4 Let G be a simple connected graph. If its line graph L{G) satisfiescondition of δ2(L(G)) = min{dL(G)(e1) + dL(G)(e2): e1e2 (?) E(L(G)),e1,e2 ∈ V(L(G))} ≥ 9 and mcr(L(G)) ≤△ (G) +1 , then line graph L(G) is subpancyclic graph and thebound of A(G) +1 is the best possible.Theorem5 Let G be a simple connected graph with order n ≥ 51, which satisfies the girth g(G) ≥ 4 and δ2 (G) ≥ (2n - 9)/5. If G is Hamiltonian, then itsline graph L(G) is pancyclic graph.Theorem6 Let G be a claw-free graph with n vertices. If G satisfies condition of δ2 (G) > 9, p2 (G) ≥ n, then G is subpancyclic graph.