Subpancyclicity of line graphs and degree sums along paths

A graph is called subpancyclic if it contains a cycle of length ℓ for each ℓ between 3 and the circumference of the graph. We show that if G is a connected graph on n {greater than or slanted equal to} 146 vertices such that d ( u ) + d ( v ) + d ( x ) + d ( y ) > ( n + 10 / 2 ) for all four vertices u, v, x, y of any path P = uvxy in G, then the line graph L ( G ) is subpancyclic, unless G is isomorphic to an exceptional graph. Moreover, we show that this result is best possible, even under the assumption that L ( G ) is hamiltonian. This improves earlier sufficient conditions by a multiplicative factor rather than an additive constant.