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Ramsey properties of Cayley graphs ★★★

Author(s): Alon

Conjecture   There exists a fixed constant $ c $ so that every abelian group $ G $ has a subset $ S \subseteq G $ with $ -S = S $ so that the Cayley graph $ {\mathit Cayley}(G,S) $ has no clique or independent set of size $ > c \log |G| $.

Keywords: Cayley graph; Ramsey number

Bases of many weights ★★★

Author(s): Schrijver; Seymour

Let $ G $ be an (additive) abelian group, and for every $ S \subseteq G $ let $ {\mathit stab}(S) = \{ g \in G : g + S = S \} $.

Conjecture   Let $ M $ be a matroid on $ E $, let $ w : E \rightarrow G $ be a map, put $ S = \{ \sum_{b \in B} w(b) : B \mbox{ is a base} \} $ and $ H = {\mathit stab}(S) $. Then $$|S| \ge |H| \left( 1 - rk(M) + \sum_{Q \in G/H} rk(w^{-1}(Q)) \right).$$

Keywords: matroid; sumset; zero sum

The Erdos-Turan conjecture on additive bases ★★★★

Author(s): Erdos; Turan

Let $ B \subseteq {\mathbb N} $. The representation function $ r_B : {\mathbb N} \rightarrow {\mathbb N} $ for $ B $ is given by the rule $ r_B(k) = \#\{ (i,j) \in B \times B : i + j = k \} $. We call $ B $ an additive basis if $ r_B $ is never $ 0 $.

Conjecture   If $ B $ is an additive basis, then $ r_B $ is unbounded.

Keywords: additive basis; representation function

Rota's unimodal conjecture ★★★

Author(s): Rota

Let $ M $ be a matroid of rank $ r $, and for $ 0 \le i \le r $ let $ w_i $ be the number of closed sets of rank $ i $.

Conjecture   $ w_0,w_1,\ldots,w_r $ is unimodal.
Conjecture   $ w_0,w_1,\ldots,w_r $ is log-concave.

Keywords: flat; log-concave; matroid

A conjecture on iterated circumcentres ★★

Author(s): Goddyn

Conjecture   Let $ p_1,p_2,p_3,\ldots $ be a sequence of points in $ {\mathbb R}^d $ with the property that for every $ i \ge d+2 $, the points $ p_{i-1}, p_{i-2}, \ldots p_{i-d-1} $ are distinct, lie on a unique sphere, and further, $ p_i $ is the center of this sphere. If this sequence is periodic, must its period be $ 2d+4 $?

Keywords: periodic; plane geometry; sequence

Unions of triangle free graphs ★★★

Author(s): Erdos; Hajnal

Problem   Does there exist a graph with no subgraph isomorphic to $ K_4 $ which cannot be expressed as a union of $ \aleph_0 $ triangle free graphs?

Keywords: forbidden subgraph; infinite graph; triangle free

The Two Color Conjecture ★★

Author(s): Neumann-Lara

Conjecture   If $ G $ is an orientation of a simple planar graph, then there is a partition of $ V(G) $ into $ \{X_1,X_2\} $ so that the graph induced by $ X_i $ is acyclic for $ i=1,2 $.

Keywords: acyclic; digraph; planar

Half-integral flow polynomial values ★★

Author(s): Mohar

Let $ \Phi(G,x) $ be the flow polynomial of a graph $ G $. So for every positive integer $ k $, the value $ \Phi(G,k) $ equals the number of nowhere-zero $ k $-flows in $ G $.

Conjecture   $ \Phi(G,5.5) > 0 $ for every 2-edge-connected graph $ G $.

Keywords: nowhere-zero flow

Gao's theorem for nonabelian groups ★★

Author(s): DeVos

For every finite multiplicative group $ G $, let $ s(G) $ ($ s'(G) $) denote the smallest integer $ m $ so that every sequence of $ m $ elements of $ G $ has a subsequence of length $ >0 $ (length $ |G| $) which has product equal to 1 in some order.

Conjecture   $ s'(G) = s(G) + |G| - 1 $ for every finite group $ G $.

Keywords: subsequence sum; zero sum

Universal point sets for planar graphs ★★★

Author(s): Mohar

We say that a set $ P \subseteq {\mathbb R}^2 $ is $ n $-universal if every $ n $ vertex planar graph can be drawn in the plane so that each vertex maps to a distinct point in $ P $, and all edges are (non-intersecting) straight line segments.

Question   Does there exist an $ n $-universal set of size $ O(n) $?

Keywords: geometric graph; planar graph; universal set

Antichains in the cycle continuous order ★★

Author(s): DeVos

If $ G $,$ H $ are graphs, a function $ f : E(G) \rightarrow E(H) $ is called cycle-continuous if the pre-image of every element of the (binary) cycle space of $ H $ is a member of the cycle space of $ G $.

Problem   Does there exist an infinite set of graphs $ \{G_1,G_2,\ldots \} $ so that there is no cycle continuous mapping between $ G_i $ and $ G_j $ whenever $ i \neq j $ ?

Keywords: antichain; cycle; poset

Fat 4-polytopes ★★★

Author(s): Eppstein; Kuperberg; Ziegler

The fatness of a 4-polytope $ P $ is defined to be $ (f_1 + f_2)/(f_0 + f_3) $ where $ f_i $ is the number of faces of $ P $ of dimension $ i $.

Question   Does there exist a fixed constant $ c $ so that every convex 4-polytope has fatness at most $ c $?

Keywords: f-vector; polytope

The Crossing Number of the Complete Bipartite Graph ★★★

Author(s): Turan

The crossing number $ cr(G) $ of $ G $ is the minimum number of crossings in all drawings of $ G $ in the plane.

Conjecture   $ \displaystyle   cr(K_{m,n}) = \floor{\frac m2} \floor{\frac {m-1}2}                      \floor{\frac n2} \floor{\frac {n-1}2}  $

Keywords: complete bipartite graph; crossing number

Woodall's Conjecture ★★★

Author(s): Woodall

Conjecture   If $ G $ is a directed graph with smallest directed cut of size $ k $, then $ G $ has $ k $ disjoint dijoins.

Keywords: digraph; packing

Pentagon problem ★★★

Author(s): Nesetril

Question   Let $ G $ be a 3-regular graph that contains no cycle of length shorter than $ g $. Is it true that for large enough~$ g $ there is a homomorphism $ G \to C_5 $?

Keywords: cubic; homomorphism

Ryser's conjecture ★★★

Author(s): Ryser

Conjecture   Let $ H $ be an $ r $-uniform $ r $-partite hypergraph. If $ \nu $ is the maximum number of pairwise disjoint edges in $ H $, and $ \tau $ is the size of the smallest set of vertices which meets every edge, then $ \tau \le (r-1) \nu $.

Keywords: hypergraph; matching; packing

The Erdös-Hajnal Conjecture ★★★

Author(s): Erdos; Hajnal

Conjecture   For every fixed graph $ H $, there exists a constant $ \delta(H) $, so that every graph $ G $ without an induced subgraph isomorphic to $ H $ contains either a clique or an independent set of size $ |V(G)|^{\delta(H)} $.

Keywords: induced subgraph

Hamiltonian paths and cycles in vertex transitive graphs ★★★

Author(s): Lovasz

Problem   Does every connected vertex-transitive graph have a Hamiltonian path?

Keywords: cycle; hamiltonian; path; vertex-transitive

57-regular Moore graph? ★★★

Author(s): Hoffman; Singleton

Question   Does there exist a 57-regular graph with diameter 2 and girth 5?

Keywords: cage; Moore graph

Few subsequence sums in Z_n x Z_n ★★

Author(s): Bollobas; Leader

Conjecture   For every $ 0 \le t \le n-1 $, the sequence in $ {\mathbb Z}_n^2 $ consisting of $ n-1 $ copes of $ (1,0) $ and $ t $ copies of $ (0,1) $ has the fewest number of distinct subsequence sums over all zero-free sequences from $ {\mathbb Z}_n^2 $ of length $ n-1+t $.

Keywords: subsequence sum; zero sum