Purpose

The web has become an excellent resource for locating and even learning established mathematics. Finding papers is easy with the ArXiv and MathSciNet, and (even rather technical) definitions can frequently be found on Mathworld and Wikipedia. On the other hand, the web still lacks good tools for exploration and discussion of the mathematical frontier. We hope that the Garden can help fill this void, playing a role complementary to that of these other sites. Thanks to a wonderful community of users, Wikipedia has quickly built an impressive library of established knowledge. We hope that the Garden will help our community to build an equally impressive online mapping of the frontier of our knowledge.

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Open problem structure

Associated with each open problem are the following fields.

Field Explanation
Title The title used by the Garden for reference purposes.
Statement A consise statment of the conjecture.
Importance
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Structure

Organization

Subject
Open problems are classified by Subject, a three level hierarchy in the form Area » Topic » Subtopic. This tree structure can be navigated using the links in the Navigate menu under "Subject".
Keywords/Authors
Open problems are also assigned both Authors and Keywords for reference purposes. Clicking on either "Author Index" or "Keyword Index" in the Navigate menu will bring you to a full list of these terms.
Recent Activity
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Graph Theory » Coloring » Edge coloring

List colorings of edge-critical graphs ★★

Author(s): Mohar

Conjecture   Suppose that $ G $ is a $ \Delta $-edge-critical graph. Suppose that for each edge $ e $ of $ G $, there is a list $ L(e) $ of $ \Delta $ colors. Then $ G $ is $ L $-edge-colorable unless all lists are equal to each other.

Keywords: edge-coloring; list coloring

Posted by Robert Samal
updated June 12th, 2007
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Graph Theory » Basic G.T. » Cycles

Barnette's Conjecture ★★★

Author(s): Barnette

Conjecture   Every 3-connected cubic planar bipartite graph is Hamiltonian.

Keywords: bipartite; cubic; hamiltonian

Posted by Robert Samal
updated August 9th, 2011
1 comment
Combinatorics » Matroid Theory

Aharoni-Berger conjecture ★★★

Author(s): Aharoni; Berger

Conjecture   If $ M_1,\ldots,M_k $ are matroids on $ E $ and $ \sum_{i=1}^k rk_{M_i}(X_i) \ge \ell (k-1) $ for every partition $ \{X_1,\ldots,X_k\} $ of $ E $, then there exists $ X \subseteq E $ with $ |X| = \ell $ which is independent in every $ M_i $.

Keywords: independent set; matroid; partition

Posted by mdevos
updated June 11th, 2007
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Outstanding ✭✭✭✭ One of the top problems in the area.
High ✭✭✭ A problem which has received attention in the literature.