21. My Favorite Open Problems open problems in Discrete Math. Prizes. More online collections of open problemsin discrete math Dan Archdeacon's problems in topological graph theory; http://www.math.princeton.edu/~matdevos/open/open.html

Open Problems in Discrete Math Prizes Flows on Graphs Tutte's 3-flow, 4-flow, and 5-flow conjectures The 2+epsilon flow conjecture Jaeger's modular orientation conjecture Bouchet's 6-flow conjecture ... Unit vector flows Cycle Covers The cycle double cover conjecture The circular embedding conjecture Conjectures on (m,n)-cycle-covers Faithful circuit covers ... Circuit decompositions of Eulerian graphs Choosability for Ax=y A nowhere-zero point in a linear mapping The additive basis conjecture The permanent conjecture The Alon-Tarsi basis conjecture Edge Coloring The Berge-Fulkerson conjecture The Petersen coloring conjecture Packing T-joins Acyclic edge-coloring ... Conjectures on edge-connectivity Vertex Coloring Homomorphisms of planar graphs Homomorphisms to the circuit of length 5 Directed Graphs Seymour's second neighborhood conjecture Reversing edges in digraphs Packing dijoins Topological Graph Theory Grunbaum's conjecture Embedding digraphs Conway's thrackle conjecture Matroid Theory Bases of many weights Stable bases Additive Number Theory Kneser's critical problem Sets with distinct subset sums Miscellaneous Problems The lonely runner conjecture please feel free to email me if you know of unmentioned progress (possibly a solution) to any of these problems.

22. Open Problems In Combinatorics Part of the Combinatorics Net.Category Science Math Combinatorics open problems......open problems in Combinatorics. open problems of Paul Erdös, by Fan RK Chung; OpenProblems from the SIAM Activity Group Newsletter in Discrete Mathematics; http://www.combinatorics.net/problems/

Open Problems in Combinatorics Unsolved Combinatorial Problems by Zsolt Tuza Open Problems on Perfect Graphs , by Positivitiy Problems and Conjectures in Algebraic Combinatorics , by Richard Stanley Ten Problems in Combinatorics, Lecture given by Gian-Carlo Rota at the Conference on Combinatorics and Physics (CAP'98), held at the Los Alamos National Laboratory, August 10-12, 1998. To be announced. Problems in Signed, Gain, and Biased Graphs , compiled by Thomas Zaslavsky Some Problems in Matroid Theory , compiled by Thomas Zaslavsky Unsolved Problems in Graph Theory , compiled by Stephen C. Locke A problem on transitive permutation groups , by Peter J. Cameron A list of problems on permutation groups , by Peter J. Cameron Unsolved Mathematics Problems , compiled by Steven Finch. , by Fan R. K. Chung Open Problems from the SIAM Activity Group Newsletter in Discrete Mathematics Problems on the Enumeration of Matchings (ps file) , by James Propp

23. The Open Problems Project The open problems Project. This is the beginning of a project 1 to record open problemsof interest to researchers in computational geometry and related fields. http://cs.smith.edu/~orourke/TOPP/Welcome.html

Next: Numerical List of All The Open Problems Project edited by Erik D. Demaine Joseph S. B. Mitchell Joseph O'Rourke Introduction This is the beginning of a project to record open problems of interest to researchers in computational geometry and related fields. It commenced with the publication of thirty problems in Computational Geometry Column 42 [ ] (see Problems 1-30 ), but has grown much beyond that. We encourage correspondence to improve the entries; please send email to TOPP@cs.smith.edu . If you would like to submit a new problem, please fill out this template Each problem is assigned a unique number for citation purposes. Problem numbers also indicate the order in which the problems were entered. Each problem is classified as belonging to one or more categories. The problems are also available as a single Postscript or PDF file. To begin navigating through the open problems, you may select from a category of interest below, or view a list of all problems sorted numerically Categorized List of All Problems Below, each category lists the problems that are classified under that category. Note that each problem may be classified under several categories.

24. Open Problems In Visib. + Illumn. open problems in the Combinatorics of Visibility and Illumination. I willpost notices of solutions to the open problems listed in the paper. http://cs.smith.edu/~orourke/Open.html

Open Problems in the Combinatorics of Visibility and Illumination Last update to this page: I will post notices of solutions to the open problems listed in the paper. Please keep me apprised of progress! orourke@cs.smith.edu Illuminating Rectangles. I quoted a lower bound of n-1 and an upper bound of ~(4/3)n in the paper. Jorge Urrutia shows in his chapter, "Art gallery and illumination problems" (to appear in the North-Holland Handbook of Computational Geometry ) that the Hoffman-Kaufman-Kriegel [n/4] result on orthogonal art galleries with holes establishes immediately that n+1 lights suffice. So the lower and upper bounds are nearly identical now: [n-1,n+1]. Stage Illumination. Proven NP-complete, even with these restrictions: the floodlights are all affixed to just two points, and those points have the same x coordinate, or the same y coordinate. See: H. Ito, H. Uehara, M. Yokayama, "NP-completeness of stage illumination problem", Proc. Japan Conf. Discrete Comput. Geom. '98 , Tokyo, 1998, pp. 88-92. ito@tutics.tut.ac.jp

25. Open Problems Originally from the Katsiveli 2000 open problems Session, now maintained by Sergiy Kolyada. PDF/PS.Category Science Math Differential Equations Dynamical Systems......open problems in Dynamical Systems Ergodic Theory. Welcome! 9, Polygonal billiardssome open problems Submitted by Pascal Hubert and Serge Troubetzkoy. http://www.math.iupui.edu/~mmisiure/open/

Other sites with this page O pen P roblems in D ynamical S ystems E rgodic T heory Welcome! Katsiveli - 2000 Open Problems Session. New problems are being added to it. If you would like to submit some open problems to this page, please send them to Sergiy Kolyada If you have any remarks about this page, please write to Sergiy Kolyada or Michal Misiurewicz Geometric models of Pisot substitutions and non-commutative arithmetic Submitted by Pierre Arnoux (corrections - November 29, 2001) Ergodic Ramsey Theory - an update Submitted by Vitaly Bergelson (see also here Dense periodic points in cellular automata Submitted by Francois Blanchard Non-discrete locally compact second countable groups Submitted by Sergey Gefter Martingale convergence and ergodic theorems Submitted by Alexander Kachurovskii The problem is closed (October 21, 2002) Entropy, periodic points and transitivity of maps Submitted by Sergiy Kolyada and Lubomir Snoha (corrections - October 26, 2002) Natural spectral isomorphisms Submitted by Jan Kwiatkowski Density of periodic orbit measures for piecewise monotonic interval maps Submitted by Peter Raith Polygonal billiards: some open problems Submitted by Pascal Hubert and Serge Troubetzkoy Is any kind of mixing possible in "ToP" N-actions?

26. Survey Of Venn Diagrams Open Problems In the Electronic Journal of Combinatorics.Category Science Math Combinatorics open problems......Venn Diagram References combinatorial facts, beautiful figures, n=3and beyond, some open problems. Venn Diagram Survey open problems. http://sue.csc.uvic.ca/~cos/venn/VennOpenEJC.html

27. L-functions And Random Matrix Theory Conjectures and open problems concerning Lfunctions, focussing on the areas in which there has been recent progress using results from Random Matrix Theory. Maintained at AIM. http://www.aimath.org/WWN/lrmt/

L-functions and Random Matrix Theory This web page highlights some of the conjectures and open problems concerning L-functions, focussing on the areas in which there has been recent progress using results from Random Matrix Theory. Click on the subject to see a short article on that topic. If you would like to print a hard copy of the entire web page, you can download a postscript or pdf version. Distribution of zeros of L-functions The GUE hypothesis Correlations of zeros Neighbor spacing ... GOE and Graphs

28. Mesh Generation: Open Problems ICS Theory Group ICS 280G, Spring 1997 Mesh Generation for Graphicsand Scientific Computation. open problems. We proved (4/3/97) the http://www.ics.uci.edu/~eppstein/280g/open.html

ICS 280G, Spring 1997: Mesh Generation for Graphics and Scientific Computation Open Problems We proved ( ) the existence of triangulations of any polygon or straight line graph, and of convex quadrilateralizations of any orthogonal polygon. What about curved objects? Do spline-polygons have spline-triangulations? An example formed by connecting four quarter-ellipses shows Steiner points may be needed, even for quadratic splines, but maybe they only need to be added in the interior of the splinegon. On we went over dynamic programming techniques for optimal triangulation (e.g. minimum total edge length) of simple polygons, in O(n ) time or O(E ) if the visibility graph has E edges. So the slowest case is seemingly the most simple, when the polygon is convex. Can we find the minimum length triangulation of convex polygons in o(n ) time? Steve S. suggested Frances Yao's generalization of Knuth's speedup to optimal binary search tree construction (which has the same general dynamic programming form) but it doesn't seem to work. The same dynamic programming methods also work for optimal quadrilateralization, in time O(n

29. Home Page For Constructive Approximation Open Problem Section Research problems section of the journal, edited by Peter Borwein, Albert Cohen, Ingrid Daubechies and Vilmos Totik. Includes problem statement (PostScript) and discussion. http://www.cecm.sfu.ca/personal/pborwein/CA_MOSAIC/CA_problems.html

Research Problems Section Edited by: Peter Borwein, Albert Cohen, Ingrid Daubechies and Vilmos Totik About the open problems section Problems available online Return to the Constructive Approximation Homepage The Brachistochrone Challenge "I, Johann Bernoulli, greet the most clever mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to earn the gratitude of the entire scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall then publicly declare him worthy of praise." Groningen, 1 January 1697 http://www.cecm.sfu.ca/personal/pborwein/CA_MOSAIC/CA_problems.html Modified: 06/13/1995 by pborwein@cecm.sfu.ca (Peter Borwein).

30. Open Problems In Graph Theory Involving Steiner Distance open problems involving Steiner distance. http://www.uwinnipeg.ca/~ooellerm/open_problems/index.html

Some Open Problems in Graph Theory It has been shown by Chartrand, Oellermann, Tian, and Zou that, for a tree T: diam n n This inequality does not hold for graphs in general as was shown by Henning, Oellermann, and Swart . It was shown in the same paper that for a graph G and n=3 and 4: diam n n G. It was shown by Oellermann and Tian that for a tree T: C n-1 (T) is contained in C n It remains an open problem to determine whether this containment holds for general graphs. In other words, it is not known if the Steiner (n-1)-center of a graph is contained in its Steiner n-center. It was shown by Beineke, Oellermann and Pippert that if T is a tree, then M n-1 (T) is contained in M n It remains an open problem to determine whether this containment holds for general graphs. In other words, it is not known if the Steiner (n-1)-median of a graph is contained in its Steiner n-median. Oellermann and Tian ). It is known that every graph is the 2-median of some graph (see Holbert ,and Hendry ). Steiner n-medians of trees have been completely characterized by

31. Problems In Analysis Of Algorithms A list of open problems with updates and solutions. http://pauillac.inria.fr/algo/AofA/Problems/

PROBLEMS in ANALYSIS of ALGORITHMS PAGES Home Page Research Problems Bulletin board ... Resources Return to Analysis of Algorithms Home Page This page contains a list of interesting problems that we are aware of. You are encouraged to submit new ones by posting directly on the Bulletin Board . Typically 5 to 10 lines of TeX should be best for further editing. A digest of the main problems will be compiled here periodically by the Problem Editor Summer 97 Problem 1. Problem Solution ] By Conrado Martinez , 11-Jul-97. The depth of the j-th element in a random binary search tree of size n. Problem 2. Problem ] By Conrado Martinez , 11-Jul-97. Quicksort with median-of-three partitioning and halting on small subfiles. Problem 3. Problem Comments ] By Hsien-Kuei Hwang , 23-Jul-97. A limit distribution and zeros of a polynomial. Problem 4. Problem ] By Wojtek Szpankowski , 25-Jul-97. What is the distribution of node levels and height in digital search trees built on Bernoulli sources? Problem 5. Problem ] By Ed Coffman , 01-Aug-97. Analyse the waste in First-Fit bin-packing.

32. Open Problems On Perfect Graphs Unsolved problems on perfect graphs.Category Science Math Combinatorics Graph Theory open problems...... In February of that year, Bruce and I prepared a list of open problems,which was then sent to all the invited participants. The http://www.cs.rutgers.edu/~chvatal/perfect/problems.html

PERFECT PROBLEMS Created on 22 August, 2000 Last updated on 11 February, 2003 The Strong Perfect Graph Conjecture has become the Strong Perfect Graph Theorem Details are here. As a part of the 1992 1993 Special Year on Combinatorial Optimization at DIMACS ftp://dimacs.rutgers.edu/pub/perfect/problems.tex If you have information on progress towards solving these problems or complaints in case I did not give credit where credit was due or suggestions for problems to add, please, send them to me Related pages: Problems from the AIM Workshop on Perfect Graphs (compiled by Maria Chudnovsky) A bibliography on perfect graphs Home pages of people interested in perfect graphs This collection is written for people with at least a basic knowledge of perfect graphs. Uninformed neophytes may look up the missing definitions on the web in Alexander Schrijver's lecture notes or in Jerry Spinrad's draft of a book on efficient graph representations etc. or in Eric Weisstein's World of Mathematics . Books on perfect graphs include M. C. Golumbic

33. Steiner Trees: Open Problems open problems with Steiner Trees, maintained by Joe Ganley.Category Science Math Geometry open problems......ganley.org The Steiner Tree Page - open problems. open problems.Of course, there are probably about a zillion open problems related http://ganley.org/steiner/open.html

ganley.org The Steiner Tree Page Open Problems Of course, there are probably about a zillion open problems related to Steiner trees, but here are a few I've thought about. Please email me others if you like, and I'll include them here. Full trees Hwang's theorem allows us to construct an optimal rectilinear Steiner tree of a full set in linear time. I know of no other metric or type of graph in which computing the optimal Steiner tree of a full set is polynomial-time solvable but computing a general Steiner tree is NP-hard. Note that there isn't even a sufficiently strong analogue of Hwang's theorem for rectilinear Steiner trees in three dimensions. Multidimensional rectilinear Steiner ratio . What is the rectilinear Steiner ratio in arbitrary dimension d ? It is at least 2-1/ d , as the d -dimensional analogue of the "cross" has this ratio. It is obviously at most 2. It is generally believed that the lower bound is correct, but this hasn't been proven. Even an upper bound lower than 2 would be interesting. Rectilinear Steiner arborescence . These are Steiner-like trees on points in the (first quadrant of the) plane, in which every segment in the tree is directed left to right or bottom to top. It is unknown whether computing an RSA is NP-complete. (A good reference to start with is

34. Unsolved Problems Including the list of 50 problems of Bondy and Murty with current status. Compiled by Stephen C. Locke.Category Science Math Combinatorics Graph Theory open problems...... A good paper on open and solved cageproblems is PK Wong, Cages-A Survey,, JGT,Vol.6 (1982) 1-22 Newer results can be found on Gordon Royle's www-Page http http://www.math.fau.edu/locke/unsolved.htm

Unsolved Problems You can contact Stephen C. Locke at LockeS@fau.edu Several people have asked me about unsolved problems. I will take the easy way out: see the list of 50 problems in Bondy and Murty . I hope it will not annoy the authors of that text if I will reproduce that list here, and perhaps (eventually) add to it. Problems number above 50 are from other sources. Some of these problems have been solved (and thus the title is slightly incorrect) and I won't claim to be familiar with all current results. If you find that one of them has been solved (or even that some reasonable progress has been made), please e-mail me . Also, I'm not giving you all of the references in Bondy and Murty . You should get yourself a copy of that book. Problems 26-56 Problems 57-61 The reconstruction conjecture . (S.M. Ulam, 1960) 2. A graph G is embeddable in a graph H if G is isomorphic to a subgraph of H . Characterise the graphs embeddable in the k -cube. (V.V. Firsov, 1965) 3. Prove: Every 4-regular simple graph contains a 3-regular subgraph. (N. Sauer, 1973) Conjecture 3 was proved in 1985 by L. Zhang: Every 4-regular simple graph contains a 3-regular subgraph, J. of Changsha Railway Institute 1 (1985), 130-154.

35. Open Problems Columns - Douglas B. West open problems Columns Douglas B. West. Thispage has moved. Please update your bookmark. http://www.math.uiuc.edu/~west/pcolink.html

Open Problems Columns - Douglas B. West This page has moved. Please update your bookmark.

36. Interval Computations Presents a guide to Interval Arithmetic research featuring open problems, applications, generalizations, bibliography, researchers' sites, ftp site, mailing list, etc. http://www.cs.utep.edu/interval-comp/

Interval Computations: Introduction and Technical issues Interval Computations: Introduction ( PostScript dvi see also the following page Frequently Asked Questions (FAQ) ... Interval Notations (original suggestion, reaction) Open Problems Software and Languages Languages for Interval Computations Interval and Related Software Interval Arithmetic in the Forte Fortran 95 Compiler More Information about Interval Computations Books and Courses on Interval Computations Bibliographies on Interval and Related Methods Interval Ftp Site and Interval Mailing Lists The Interval Computations Community Homepages of Interval Computations Research Centers and Special Interest Groups Personalia: Homepages of Interval Computations Researchers Reliable Computing (formerly ... an International Journal, and other journals of interest to the interval computations community Conferences: Forthcoming and Previous Honors Received by Interval Researchers Important Web Sites of Interest to the Interval Computations Community Applications Applications of Interval Computations (see also GlobSol - software for global solutions Generalizations of Interval Arithmetic and their Applications Targeted funding opportunities Interval-Related Grants: Calls for Participation European Mirror Website,

37. Ideas, Concepts, And Definitions open problems. open problems are not a source of frustration, they are asource of delight. open problems are the lifeblood of mathematics. http://www.c3.lanl.gov/mega-math/gloss/math/openpr.html

Open Problems In a community of mathematicians, an open problem is a question that no one has found the answer to. Open problems are not a source of frustration, they are a source of delight. Open problems are the lifeblood of mathematics. For you, as an individual mathematician, your own open problems are the questions you raise that you cannot answer. Anyone who does mathematics for very long soon discovers that open problems are abundant, and even more of them are generated as mathematicians think about something and ask themselves questions in an effort to understand it. One of a mathematician's hardest choices is deciding which open problems to give focused attention to and try to solve. When you raise a question for yoursef, and you cannot answer it, it becomes an open problem for you. It is a good idea to share this problem with other mathematicians friends, classmates, teachers, etc. to see if they know of a solution or have ideas about solving it. Share your open problems with MegaMath! Don't abandon your open problems just because they remain unsolved for a long time. Set them aside and think about them gently . A solution might surprise you and arrive when you least expect it. You might even dream it!

38. Open Problems In Graph Theory Involving Steiner Distance open problems involving Steiner distance.Category Science Math Combinatorics Graph Theory open problems......Some open problems in Graph Theory. It has been shown by Chartrand,Oellermann, Tian, and Zou that, for a tree T diam n T = n/(n http://www.uwinnipeg.ca/~ooellerm/open_problems/

Some Open Problems in Graph Theory It has been shown by Chartrand, Oellermann, Tian, and Zou that, for a tree T: diam n n This inequality does not hold for graphs in general as was shown by Henning, Oellermann, and Swart . It was shown in the same paper that for a graph G and n=3 and 4: diam n n G. It was shown by Oellermann and Tian that for a tree T: C n-1 (T) is contained in C n It remains an open problem to determine whether this containment holds for general graphs. In other words, it is not known if the Steiner (n-1)-center of a graph is contained in its Steiner n-center. It was shown by Beineke, Oellermann and Pippert that if T is a tree, then M n-1 (T) is contained in M n It remains an open problem to determine whether this containment holds for general graphs. In other words, it is not known if the Steiner (n-1)-median of a graph is contained in its Steiner n-median. Oellermann and Tian ). It is known that every graph is the 2-median of some graph (see Holbert ,and Hendry ). Steiner n-medians of trees have been completely characterized by

39. Open Problems On Graph Minors By Nathaniel Dean. http://dimacs.rutgers.edu/~nate/publics/open.ps

40. The RTA List Of Open Problems The RTA list of open problems. The RTA 98. We are currently workingto transform the list of open problems into a WWW service. This http://www.lsv.ens-cachan.fr/~treinen/rtaloop/

The RTA list of open problems The RTA list of open problems summarizes open problems in the field of the International Conference on Rewriting Techniques and Applications (RTA). For the RTA 2002 conference, the topics of RTA were given as Applications: case studies; rule-based programming; symbolic and algebraic computation; theorem proving; functional and logic programming; proof checking. Foundations: matching and unification; completion techniques; strategies; constraint solving; explicit substitutions. Frameworks: string, term, and graph rewriting; lambda-calculus and higher-order rewriting; conditional rewriting; proof nets; constrained rewriting and deduction; categorical and infinitary rewriting. Implementation: compilation techniques; parallel execution; rewriting tools. Semantics: equational logic; rewriting logic. The RTA list of open problems was created in 1991 by Nachum Dershowitz Jean-Pierre Jouannaud and Jan Willem Klop on occasion of the RTA'91 conference. Updated lists have since been published at RTA'93, RTA'95 and RTA'98. We are currently working to transform the list of open problems into a WWW service. This effort is, at the moment, led by

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