Recomm. for undergrads: yes
Posted by: Agelos
on: August 4th, 2007
Problem   If $ G $ is a $ 3 $-connected finite graph, is there an assignment of lengths $ \ell: E(G) \to \mathb R^+ $ to the edges of $ G $, such that every $ \ell $-geodesic cycle is peripheral?

A cycle $ C $ is $ \ell $-geodesic if for every two vertices $ x,y $ on $ C $ there is no $ x $-$ y $~path in $ G $ shorter, with respect to $ \ell $, than both $ x $-$ y $~arcs on $ C $.

It is not hard to prove [GS] that for every finite graph $ G $ and every assignment of edge lengths $ \ell: E(G) \to \mathb R^+ $ the $ \ell $-geodesic cycles of $ G $ 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 $ 3 $-connected finite graph generate its cycle space.


*[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.


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