
Problem If
is a
-connected finite graph, is there an assignment of lengths
to the edges of
, such that every
-geodesic cycle is peripheral?





A cycle is
-geodesic if for every two vertices
on
there is no
-
~path in
shorter, with respect to
, than both
-
~arcs on
.
It is not hard to prove [GS] that for every finite graph and every assignment of edge lengths
the
-geodesic cycles of
generate its cycle space. Thus, a positive answer to the problem would imply a new proof of Tutte's classical theorem [T] that the peripheral cycles of a
-connected finite graph generate its cycle space.
Bibliography
*[GS] Angelos Georgakopoulos, Philipp Sprüssel: Geodesic topological cycles in locally finite graphs. Preprint 2007.
[T] W.T. Tutte, How to draw a graph. Proc. London Math. Soc. 13 (1963), 743–768.
* indicates original appearance(s) of problem.