![](/files/happy5.png)
Problem If
is a
-connected finite graph, is there an assignment of lengths
to the edges of
, such that every
-geodesic cycle is peripheral?
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ 3 $](/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png)
![$ \ell: E(G) \to \mathb R^+ $](/files/tex/aad2d29bf9a23e904e02b7e5f616c403cb1b95df.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ \ell $](/files/tex/d2c5960dd9795a1b000a5843d282c97268e303c4.png)
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.