Subpancyclicity of line graphs and degree sums along paths

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