**Question**What is the maximum edge density of a graph which has a good edge labeling?

We say that a graph is *good-edge-labeling critical*, if it has no good edge labeling, but every proper subgraph has a good edge labeling.

**Conjecture**For every , there is only a finite number of good-edge-labeling critical graphs with average degree less than .

Let be a finite undirected simple graph. A *good edge labeling* of is an assignment of distinct numbers to the edges such that every cycle has at least two local maxima. (The distinctness of the labels is required only to make the term `local maximum' unambiguous.)

Equivalently, a labeling of the edges is good, if for every pair of distinct vertices , there is at most one increasing path from to .

Having a good edge labeling is inherited by subgraphs.

It is easy to verify that the graphs and have no good edge labeling. In [ACGH2] an infinite class of graphs without good edge labelings is given, none of whom is a subgraph of the other. In [BFT] contains an example of a minimal graph without good edge labeling which as average degree < 3 (thus refuting an earlier conjecture saying that a good-edge-labeling critical graph with average degree less than three is either or ). In that same paper it is shown that every such graph must have girth at most 4.

Good edge labeling of graphs was introduced in [BCP] in the context of the so-called Routing and Wavelength Assignment (RWA) problem. The problems above are proposed in [ACGH1] and [ACGH2]. There the algorithmic problem of determining whether a graph has a good edge labeling is shown to be NP-hard. Moreover, the authors also prove that every planar graph with girth at least six has a good edge labeling.

## Bibliography

[BCP] J-C. Bermond, M. Cosnard, and S. Pérennes. Directed acyclic graphs with unique path property. Technical report 6932, INRIA, May 2009

[ACGH1] J. Araújo, N. Cohen, F. Giroire, F. Havet. Good edge-labelling of graphs. (English summary) LAGOS'09—V Latin-American Algorithms, Graphs and Optimization Symposium, 275–280, Electron. Notes Discrete Math., 35, Elsevier Sci. B. V., Amsterdam, 2009. MathSciNet

[ACGH2*] J. Araujo, N. Cohen, F. Giroire, and F. Havet. Good edge-labelling of graphs. Research Report 6934, INRIA, 2009.

[BFT] M. Bode, B. Farzad, D.O. Theis. Good edge-labelings and graphs of girth at least 5. (arXiv:1109.1125)

* indicates original appearance(s) of problem.