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Graph Theory » Coloring » Vertex coloring

Reed's omega, delta, and chi conjecture ★★★

Author(s): Reed

For a graph $ G $, we define $ \Delta(G) $ to be the maximum degree, $ \omega(G) $ to be the size of the largest clique subgraph, and $ \chi(G) $ to be the chromatic number of $ G $.

Conjecture   $ \chi(G) \le \ceil{\frac{1}{2}(\Delta(G)+1) + \frac{1}{2}\omega(G)} $ for every graph $ G $.

Keywords: coloring

Posted by mdevos
updated September 24th, 2007
2 comments
Graph Theory

Pebbling a cartesian product ★★★

Author(s): Graham

We let $ p(G) $ denote the pebbling number of a graph $ G $.

Conjecture   $ p(G_1 \Box G_2) \le p(G_1) p(G_2) $.

Keywords: pebbling; zero sum

Posted by mdevos
updated September 24th, 2007
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Unsorted

Rendezvous on a line ★★★

Author(s): Alpern

Problem   Two players start at a distance of 2 on an (undirected) line (so, neither player knows the direction of the other) and both move at a maximum speed of 1. What is the infimum expected meeting time $ R $ (first time when the players occupy the same point) which can be achieved assuming the two players must adopt the same strategy?

Keywords: game theory; optimization; rendezvous

Posted by mdevos
updated September 21st, 2007
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Graph Theory » Coloring

Linial-Berge path partition duality ★★★

Author(s): Berge; Linial

Conjecture   The minimum $ k $-norm of a path partition on a directed graph $ D $ is no more than the maximal size of an induced $ k $-colorable subgraph.

Keywords: coloring; directed path; partition

Posted by berger
updated September 19th, 2007
2 comments
Graph Theory » Extremal G.T.

What is the smallest number of disjoint spanning trees made a graph Hamiltonian ★★

Author(s): Goldengorin

We are given a complete simple undirected weighted graph $ G_1=(V,E) $ and its first arbitrary shortest spanning tree $ T_1=(V,E_1) $. We define the next graph $ G_2=(V,E\setminus E_1) $ and find on $ G_2 $ the second arbitrary shortest spanning tree $ T_2=(V,E_2) $. We continue similarly by finding $ T_3=(V,E_3) $ on $ G_3=(V,E\setminus \cup_{i=1}^{2}E_i) $, etc. Let k be the smallest number of disjoint shortest spanning trees as defined above and let $ T^{k}=(V,\cup_{i=1}^{k}E_i) $ be the graph obtained as union of all $ k $ disjoint trees.

Question 1. What is the smallest number of disjoint spanning trees creates a graph $ T^{k} $ containing a Hamiltonian path.

Question 2. What is the smallest number of disjoint spanning trees creates a graph $ T^{k} $ containing a shortest Hamiltonian path?

Questions 3 and 4. Replace in questions 1 and 2 a shortest spanning tree by a 1-tree. What is the smallest number of disjoint 1-trees creates a Hamiltonian graph? What is the smallest number of disjoint 1-trees creates a graph containing a shortest Hamiltonian cycle?

Keywords: 1-trees; cycle; Hamitonian path; spanning trees

Posted by boris
updated September 10th, 2007
1 comment
Number Theory » Combinatorial N.T.

Davenport's constant ★★★

Author(s):

For a finite (additive) abelian group $ G $, the Davenport constant of $ G $, denoted $ s(G) $, is the smallest integer $ t $ so that every sequence of elements of $ G $ with length $ \ge t $ has a nontrivial subsequence which sums to zero.

Conjecture   $ s( {\mathbb Z}_n^d) = d(n-1) + 1 $

Keywords: Davenport constant; subsequence sum; zero sum

Posted by mdevos
updated September 8th, 2007
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Graph Theory » Coloring » Vertex coloring

Coloring and immersion ★★★

Author(s): Abu-Khzam; Langston

Conjecture   For every positive integer $ t $, every (loopless) graph $ G $ with $ \chi(G) \ge t $ immerses $ K_t $.

Keywords: coloring; complete graph; immersion

Posted by mdevos
updated September 4th, 2007
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Combinatorics

Rainbow AP(4) in an almost equinumerous coloring ★★

Author(s): Conlon

Problem   Do 4-colorings of $ \mathbb{Z}_{p} $, for $ p $ a large prime, always contain a rainbow $ AP(4) $ if each of the color classes is of size of either $ \lfloor p/4\rfloor $ or $ \lceil p/4\rceil $?

Keywords: arithmetic progression; rainbow

Posted by vjungic
updated September 1st, 2007
1 comment
Graph Theory » Basic G.T. » Matchings

The intersection of two perfect matchings ★★

Author(s): Macajova; Skoviera

Conjecture   Every bridgeless cubic graph has two perfect matchings $ M_1 $, $ M_2 $ so that $ M_1 \cap M_2 $ does not contain an odd edge-cut.

Keywords: cubic; nowhere-zero flow; perfect matching

Posted by mdevos
updated August 30th, 2007
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Number Theory » Computational N.T.

Counterexamples to the Baillie-PSW primality test ★★

Author(s):

Problem  (1)   Find a counterexample to Baillie-PSW primality test or prove that there is no one.
Problem  (2)   Find a composite $ n\equiv 3 $ or $ 7\pmod{10} $ which divides both $ 2^{n-1} - 1 $ (see Fermat pseudoprime) and the Fibonacci number $ F_{n+1} $ (see Lucas pseudoprime), or prove that there is no such $ n $.

Keywords:

Posted by maxal
updated August 7th, 2007
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Number Theory » Computational N.T.

A sextic counterexample to Euler's sum of powers conjecture ★★

Author(s): Euler

Problem   Find six positive integers $ x_1, x_2, \dots, x_6 $ such that $$x_1^6 + x_2^6 + x_3^6 + x_4^6 + x_5^6 = x_6^6$$ or prove that such integers do not exist.

Keywords:

Posted by maxal
updated August 5th, 2007
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Graph Theory » Basic G.T. » Cycles

Geodesic cycles and Tutte's Theorem ★★

Author(s): Georgakopoulos; Sprüssel

Problem   If $ G $ is a $ 3 $-connected finite graph, is there an assignment of lengths $ \ell: E(G) \to \mathb R^+ $ to the edges of $ G $, such that every $ \ell $-geodesic cycle is peripheral?

Keywords: cycle space; geodesic cycles; peripheral cycles

Posted by Agelos
updated August 4th, 2007
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Graph Theory » Coloring » Vertex coloring

Oriented chromatic number of planar graphs ★★

Author(s):

An oriented colouring of an oriented graph is assignment $ c $ of colours to the vertices such that no two arcs receive ordered pairs of colours $ (c_1,c_2) $ and $ (c_2,c_1) $. It is equivalent to a homomorphism of the digraph onto some tournament of order $ k $.

Problem   What is the maximal possible oriented chromatic number of an oriented planar graph?

Keywords: oriented coloring; oriented graph; planar graph

Posted by Robert Samal
updated August 4th, 2007
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Number Theory » Combinatorial N.T.

Covering systems with big moduli ★★

Author(s): Erdos; Selfridge

Problem   Does for every integer $ N $ exist a covering system with all moduli distinct and at least equal to~$ N $?

Keywords: covering system

Posted by Robert Samal
updated August 4th, 2007
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Number Theory » Combinatorial N.T.

Odd incongruent covering systems ★★★

Author(s): Erdos; Selfridge

Conjecture   There is no covering system whose moduli are odd, distinct, and greater than 1.

Keywords: covering system

Posted by Robert Samal
updated August 4th, 2007
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Graph Theory » Infinite Graphs

Hamiltonian cycles in powers of infinite graphs ★★

Author(s): Georgakopoulos

Conjecture  
    \item If $ G $ is a countable connected graph then its third power is hamiltonian. \item If $ G $ is a 2-connected countable graph then its square is hamiltonian.

Keywords: hamiltonian; infinite graph

Posted by Robert Samal
updated July 24th, 2007
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Graph Theory » Infinite Graphs

Hamiltonian cycles in line graphs of infinite graphs ★★

Author(s): Georgakopoulos

Conjecture  
    \item If $ G $ is a 4-edge-connected locally finite graph, then its line graph is hamiltonian. \item If the line graph $ L(G) $ of a locally finite graph $ G $ is 4-connected, then $ L(G) $ is hamiltonian.

Keywords: hamiltonian; infinite graph; line graphs

Posted by Robert Samal
updated July 24th, 2007
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Graph Theory » Basic G.T. » Cycles

Hamiltonian cycles in line graphs ★★★

Author(s): Thomassen

Conjecture   Every 4-connected line graph is hamiltonian.

Keywords: hamiltonian; line graphs

Posted by Robert Samal
updated July 24th, 2007
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Graph Theory » Infinite Graphs

Infinite uniquely hamiltonian graphs ★★

Author(s): Mohar

Problem   Are there any uniquely hamiltonian locally finite 1-ended graphs which are regular of degree $ r > 2 $?

Keywords: hamiltonian; infinite graph; uniquely hamiltonian

Posted by Robert Samal
updated July 24th, 2007
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Theoretical Comp. Sci. » Complexity

Linear-size circuits for stable $0,1 < 2$ sorting? ★★

Author(s): Regan

Problem   Can $ O(n) $-size circuits compute the function $ f $ on $ \{0,1,2\}^* $ defined inductively by $ f(\lambda) = \lambda $, $ f(0x) = 0f(x) $, $ f(1x) = 1f(x) $, and $ f(2x) = f(x)2 $?

Keywords: Circuits; sorting

Posted by KWRegan
updated July 19th, 2007
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