Pairs of Fan-type heavy subgraphs for pancyclicity of 2-connected graphs

الأصل · EN

A graph G on n vertices is Hamiltonian if it contains a spanning cycle, and pancyclic if it contains cycles of all lengths from 3 to n. In 1984, Fan presented a degree condition involving every pair of vertices at distance two for a 2-connected graph to be Hamiltonian. Motivated by Fan's result, we say that an induced subgraph H of G is f₁-heavy if for every pair of vertices u,v∈ V(H), dₕ(u,v)=2 implies {d(u),d(v)}≥ (n+1)/2. For a given graph R, G is called R-f₁-heavy if every induced subgraph of G isomorphic to R is f₁-heavy. In this paper we show that for a connected graph S with S≠ P₃ and a 2-connected claw-f₁-heavy graph G which is not a cycle, G being S-f₁-heavy implies G is pancyclic if S=P₄,Z₁ or Z₂, where claw is K₁,₃ and Zᵢ is the path a₁a₂a₃... aᵢ₊₂aᵢ₊₃ plus the edge a₁a₃. Our result partially improves a previous theorem due to Bedrossian on pancyclicity of 2-connected graphs.