Heavy fans, cycles and paths in weighted graphs of large connectivity

A set of paths joining a vertex yy and a vertex set LL is called (y,L)(y,L)-fan if any two of the paths have only yy in common, and its width is the number of paths forming it. In weighted graphs, it is known that the existence of heavy fan is useful to find a heavy cycle containing some specified vertices.In this paper, we show the existence of heavy fans with large width containing some specified vertices in weighted graphs of large connectivity, which is a weighted analogue of Perfect's theorem. Using this, in 33-connected weighted graphs, we can find heavy cycles containing three specified vertices, and also heavy paths joining two specified vertices containing two more specified vertices. These results extend the previous results in 22-connected weighted graphs to 33-connected weighted graphs.