Subpancyclicity of line graphs and degree sums along paths ?
A graph is called subpancyclic  if it contains a cycle of length ? ? for each ? ? between 3 and the circumference of the graph. We show that if G G is a connected graph on n?146 n ? 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,y u , v , x , y of any path P=uvxy P = uvxy in G G , then the line graph L(G) L ( G ) is subpancyclic, unless G G 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.
05c45 05c35