Search

A connected edge-colored graph G is rainbow-connected if any two distinct vertices of G are connected by a path whose edges have pairwise distinct colors; the rainbow connection number rc ( G ) of G is the minimum number of colors such that G is rainbow-connected. We consider families F of connected graphs for which there is a constant k F such that, for every connected F -free graph G , rc ( G ) ź diam ( G ) + k F , where diam ( G ) is the diameter of G . In the paper, we finalize our previous considerations and give a complete answer for any finite family F .

The smoothed aggregation method has became a widely used tool for solving the linear systems arising by the discretization of elliptic partial differential equations and their singular perturbations. The smoothed aggregation method is an algebraic multigrid technique where the prolongators are constructed in two steps. First, the tentative prolongator is constructed by the aggregation (or, the generalized aggregation) method. Then, the range of the tentative prolongator is smoothed by a sparse linear prolongator smoother. The tentative prolongator is responsible for the approximation, while the prolongator smoother enforces the smoothness of the coarse-level basis functions.

We study the stability of some classes of claw-free graphs defined in terms of forbidden subgraphs under the closure operation defined in [10]. We characterize all connected graphs A such that the class of all CA -free graphs (where C denotes the claw) is stable. Using this result, we prove that every 2-connected and CHP 8 -free, CHZ 5 -free or CHN 1, 1, 4 -free graph is either hamiltonian or belongs to some classes of exceptional graphs (all of them having connectivity 2).

We study the stability of some classes of graphs defined in terms of forbidden subgraphs under the closure operation introduced by the second author. Using these results, we prove that every 2-connected claw-free and P7-free, or claw-free and Z4-free, or claw-free and eiffel-free graph is either hamiltonian or belongs to a certain class of exceptions (all of them having connectivity 2).