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Degenerate colorings of planar graphs ★★★
Author(s): Borodin
A graph is -degenerate if every subgraph of has a vertex of degree .
Keywords: coloring; degenerate; planar
Partial List Coloring ★★★
Author(s): Iradmusa
Let be a simple graph, and for every list assignment let be the maximum number of vertices of which are colorable with respect to . Define , where the minimum is taken over all list assignments with for all .
Keywords: list assignment; list coloring
Cube-Simplex conjecture ★★★
Author(s): Kalai
Partial List Coloring ★★★
Author(s): Albertson; Grossman; Haas
Keywords: list assignment; list coloring
Combinatorial covering designs ★
Author(s): Gordon; Mills; Rödl; Schönheim
A covering design, or covering, is a family of -subsets, called blocks, chosen from a -set, such that each -subset is contained in at least one of the blocks. The number of blocks is the covering’s size, and the minimum size of such a covering is denoted by .
Keywords: recreational mathematics
Burnside problem ★★★★
Author(s): Burnside
Keywords:
Laplacian Degrees of a Graph ★★
Author(s): Guo
Keywords: degree sequence; Laplacian matrix
Random stable roommates ★★
Author(s): Mertens
Keywords: stable marriage; stable roommates
Chowla's cosine problem ★★★
Author(s): Chowla
Keywords: circle; cosine polynomial
End-Devouring Rays ★
Author(s): Georgakopoulos
Seagull problem ★★★
Author(s): Seymour
Keywords: coloring; complete graph; minor
$C^r$ Stability Conjecture ★★★★
Keywords: diffeomorphisms,; dynamical systems
Convex 'Fair' Partitions Of Convex Polygons ★★
Author(s): Nandakumar; Ramana
Basic Question: Given any positive integer n, can any convex polygon be partitioned into n convex pieces so that all pieces have the same area and same perimeter?
Definitions: Define a Fair Partition of a polygon as a partition of it into a finite number of pieces so that every piece has both the same area and the same perimeter. Further, if all the resulting pieces are convex, call it a Convex Fair Partition.
Questions: 1. (Rephrasing the above 'basic' question) Given any positive integer n, can any convex polygon be convex fair partitioned into n pieces?
2. If the answer to the above is "Not always'', how does one decide the possibility of such a partition for a given convex polygon and a given n? And if fair convex partition is allowed by a specific convex polygon for a give n, how does one find the optimal convex fair partition that minimizes the total length of the cut segments?
3. Finally, what could one say about higher dimensional analogs of this question?
Conjecture: The authors tend to believe that the answer to the above 'basic' question is "yes". In other words they guess: Every convex polygon allows a convex fair partition into n pieces for any n
Keywords: Convex Polygons; Partitioning
Growth of finitely presented groups ★★★
Author(s): Adyan
Keywords: finitely presented; growth
Ding's tau_r vs. tau conjecture ★★★
Author(s): Ding
Keywords: clutter; covering; MFMC property; packing
Equality in a matroidal circumference bound ★★
Keywords: circumference
Highly arc transitive two ended digraphs ★★
Author(s): Cameron; Praeger; Wormald
Keywords: arc transitive; digraph; infinite graph
Strong matchings and covers ★★★
Author(s): Aharoni
Let be a hypergraph. A strongly maximal matching is a matching so that for every matching . A strongly minimal cover is a (vertex) cover so that for every cover .
Keywords: cover; infinite graph; matching
Unfriendly partitions ★★★
If is a graph, we say that a partition of is unfriendly if every vertex has at least as many neighbors in the other classes as in its own.
Keywords: coloring; infinite graph; partition