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Combinatorics

Sequence defined on multisets ★★

Author(s): Erickson

Conjecture   Define a $ 2 \times n $ array of positive integers where the first row consists of some distinct positive integers arranged in increasing order, and the second row consists of any positive integers in any order. Create a new array where the first row consists of all the integers that occur in the first array, arranged in increasing order, and the second row consists of their multiplicities. Repeat the process. For example, starting with the array $ [1; 1] $, the sequence is: $ [1; 1] $ -> $ [1; 2] $ -> $ [1, 2; 1, 1] $ -> $ [1, 2; 3, 1] $ -> $ [1, 2, 3; 2, 1, 1] $ -> $ [1, 2, 3; 3, 2, 1] $ -> $ [1, 2, 3; 2, 2, 2] $ -> $ [1, 2, 3; 1, 4, 1] $ -> $ [1, 2, 3, 4; 3, 1, 1, 1] $ -> $ [1, 2, 3, 4; 4, 1, 2, 1] $ -> $ [1, 2, 3, 4; 3, 2, 1, 2] $ -> $ [1, 2, 3, 4; 2, 3, 2, 1] $, and we now have a fixed point (loop of one array).

The process always results in a loop of 1, 2, or 3 arrays.

Keywords: multiset; sequence

Posted by Martin Erickson
updated July 18th, 2011
1 comment
Graph Theory

Vertex Coloring of graph fractional powers ★★★

Author(s): Iradmusa

Conjecture   Let $ G $ be a graph and $ k $ be a positive integer. The $ k- $power of $ G $, denoted by $ G^k $, is defined on the vertex set $ V(G) $, by connecting any two distinct vertices $ x $ and $ y $ with distance at most $ k $. In other words, $ E(G^k)=\{xy:1\leq d_G(x,y)\leq k\} $. Also $ k- $subdivision of $ G $, denoted by $ G^\frac{1}{k} $, is constructed by replacing each edge $ ij $ of $ G $ with a path of length $ k $. Note that for $ k=1 $, we have $ G^\frac{1}{1}=G^1=G $.
Now we can define the fractional power of a graph as follows:
Let $ G $ be a graph and $ m,n\in \mathbb{N} $. The graph $ G^{\frac{m}{n}} $ is defined by the $ m- $power of the $ n- $subdivision of $ G $. In other words $ G^{\frac{m}{n}}\isdef (G^{\frac{1}{n}})^m $.
Conjecture. Let $ G $ be a connected graph with $ \Delta(G)\geq3 $ and $ m $ be a positive integer greater than 1. Then for any positive integer $ n>m $, we have $ \chi(G^{\frac{m}{n}})=\omega(G^\frac{m}{n}) $.
In [1], it was shown that this conjecture is true in some special cases.

Keywords: chromatic number, fractional power of graph, clique number

Posted by Iradmusa
updated July 13th, 2011
1 comment
Graph Theory

Covering powers of cycles with equivalence subgraphs ★

Author(s):

Conjecture   Given $ k $ and $ n $, the graph $ C_{n}^k $ has equivalence covering number $ \Omega(k) $.

Keywords:

Posted by Andrew King
updated July 7th, 2011
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Theoretical Comp. Sci. » Complexity

Complexity of square-root sum ★★

Author(s): Goemans

Question   What is the complexity of the following problem?

Given $ a_1,\dots,a_n; k $, determine whether or not $ \sum_i \sqrt{a_i} \leq k. $

Keywords: semi-definite programming

Posted by abie
updated July 1st, 2011
5 comments
Number Theory » Combinatorial N.T.

Snevily's conjecture ★★★

Author(s): Snevily

Conjecture   Let $ G $ be an abelian group of odd order and let $ A,B \subseteq G $ satisfy $ |A| = |B| = k $. Then the elements of $ A $ and $ B $ may be ordered $ A = \{a_1,\ldots,a_k\} $ and $ B = \{b_1,\ldots,b_k\} $ so that the sums $ a_1+b_1, a_2+b_2 \ldots, a_k + b_k $ are pairwise distinct.

Keywords: addition table; latin square; transversal

Posted by mdevos
updated May 27th, 2011
1 comment
Graph Theory » Coloring » Nowhere-zero flows

3-flow conjecture ★★★

Author(s): Tutte

Conjecture   Every 4-edge-connected graph has a nowhere-zero 3-flow.

Keywords: nowhere-zero flow

Posted by mdevos
updated April 12th, 2011
1 comment
Analysis

Invariant subspace problem ★★★

Author(s):

Problem   Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial closed invariant subspace?

Keywords: subspace

Posted by tchow
updated April 11th, 2011
2 comments
Number Theory » Combinatorial N.T.

Sets with distinct subset sums ★★★

Author(s): Erdos

Say that a set $ S \subseteq {\mathbb Z} $ has distinct subset sums if distinct subsets of $ S $ have distinct sums.

Conjecture   There exists a fixed constant $ c $ so that $ |S| \le \log_2(n) + c $ whenever $ S \subseteq \{1,2,\ldots,n\} $ has distinct subset sums.

Keywords: subset sum

Posted by mdevos
updated April 7th, 2011
1 comment
Graph Theory » Directed Graphs

Seymour's Second Neighbourhood Conjecture ★★★

Author(s): Seymour

Conjecture   Any oriented graph has a vertex whose outdegree is at most its second outdegree.

Keywords: Caccetta-Häggkvist; neighbourhood; second; Seymour

Posted by nkorppi
updated March 18th, 2011
5 comments
Group Theory

Which lattices occur as intervals in subgroup lattices of finite groups? ★★★★

Author(s):

Conjecture  

There exists a finite lattice that is not an interval in the subgroup lattice of a finite group.

Keywords: congruence lattice; finite groups

Posted by williamdemeo
updated February 22nd, 2011
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Number Theory

Quartic rationally derived polynomials ★★★

Author(s): Buchholz; MacDougall

Call a polynomial $ p \in {\mathbb Q}[x] $ rationally derived if all roots of $ p $ and the nonzero derivatives of $ p $ are rational.

Conjecture   There does not exist a quartic rationally derived polynomial $ p \in {\mathbb Q}[x] $ with four distinct roots.

Keywords: derivative; diophantine; elliptic; polynomial

Posted by mdevos
updated February 16th, 2011
1 comment
Topology

Nonseparating planar continuum ★★

Author(s):

Conjecture   Does any path-connected, compact set in the plane which does not separate the plane have the fixed point property?

A set has the fixed point property if every continuous map from it into itself has a fixed point.

Keywords: fixed point

Posted by porton
updated February 16th, 2011
1 comment
Topology

Hilbert-Smith conjecture ★★

Author(s): David Hilbert; Paul A. Smith

Conjecture   Let $ G $ be a locally compact topological group. If $ G $ has a continuous faithful group action on an $ n $-manifold, then $ G $ is a Lie group.

Keywords:

Posted by porton
updated February 6th, 2011
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Algebra

trace inequality ★★

Author(s):

Let $ A,B $ be positive semidefinite, by Jensen's inequality, it is easy to see $ [tr(A^s+B^s)]^{\frac{1}{s}}\leq [tr(A^r+B^r)]^{\frac{1}{r}} $, whenever $ s>r>0 $.

What about the $ tr(A^s+B^s)^{\frac{1}{s}}\leq tr(A^r+B^r)^{\frac{1}{r}} $, is it still valid?

Keywords:

Posted by Miwa Lin
updated December 20th, 2010
3 comments
Graph Theory » Coloring » Nowhere-zero flows

Real roots of the flow polynomial ★★

Author(s): Welsh

Conjecture   All real roots of nonzero flow polynomials are at most 4.

Keywords: flow polynomial; nowhere-zero flow

Posted by mdevos
updated December 17th, 2010
2 comments
Graph Theory » Basic G.T. » Cycles

Hamiltonicity of Cayley graphs ★★★

Author(s): Rapaport-Strasser

Question   Is every Cayley graph Hamiltonian?

Keywords:

Posted by tchow
updated December 15th, 2010
4 comments
Algebra

Finite Lattice Representation Problem ★★★★

Author(s):

Conjecture  

There exists a finite lattice which is not the congruence lattice of a finite algebra.

Keywords: congruence lattice; finite algebra

Posted by williamdemeo
updated December 10th, 2010
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Topology

Outer reloid of restricted funcoid ★★

Author(s): Porton

Question   $ ( \mathsf{RLD})_{\mathrm{out}} (f \cap^{\mathsf{FCD}} ( \mathcal{A} \times^{\mathsf{FCD}} \mathcal{B})) = (( \mathsf{RLD})_{\mathrm{out}} f) \cap^{\mathsf{RLD}} ( \mathcal{A} \times^{\mathsf{RLD}} \mathcal{B}) $ for every filter objects $ \mathcal{A} $ and $ \mathcal{B} $ and a funcoid $ f\in\mathsf{FCD}(\mathrm{Src}\,f; \mathrm{Dst}\,f) $?

Keywords: direct product of filters; outer reloid

Posted by porton
updated December 3rd, 2010
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Graph Theory

Star chromatic index of complete graphs ★★

Author(s): Dvorak; Mohar; Samal

Conjecture   Is it possible to color edges of the complete graph $ K_n $ using $ O(n) $ colors, so that the coloring is proper and no 4-cycle and no 4-edge path is using only two colors?

Equivalently: is the star chromatic index of $ K_n $ linear in $ n $?

Keywords: complete graph; edge coloring; star coloring

Posted by Robert Samal
updated November 16th, 2010
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Graph Theory

Star chromatic index of cubic graphs ★★

Author(s): Dvorak; Mohar; Samal

The star chromatic index $ \chi_s'(G) $ of a graph $ G $ is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi-colored.

Question   Is it true that for every (sub)cubic graph $ G $, we have $ \chi_s'(G) \le 6 $?

Keywords: edge coloring; star coloring

Posted by Robert Samal
updated November 16th, 2010
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