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Graph Theory » Directed Graphs

Arc-disjoint out-branching and in-branching ★★

Author(s): Thomassen

Conjecture   There exists an integer $ k $ such that every $ k $-arc-strong digraph $ D $ with specified vertices $ u $ and $ v $ contains an out-branching rooted at $ u $ and an in-branching rooted at $ v $ which are arc-disjoint.

Keywords:

Posted by fhavet
updated March 2nd, 2013
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Graph Theory » Coloring » Edge coloring

Strong edge colouring conjecture ★★

Author(s): Erdos; Nesetril

A strong edge-colouring of a graph $ G $ is a edge-colouring in which every colour class is an induced matching; that is, any two vertices belonging to distinct edges with the same colour are not adjacent. The strong chromatic index $ s\chi'(G) $ is the minimum number of colours in a strong edge-colouring of $ G $.

Conjecture   $$s\chi'(G) \leq \frac{5\Delta^2}{4}, \text{if $\Delta$ is even,}$$ $$s\chi'(G) \leq \frac{5\Delta^2-2\Delta +1}{4},&\text{if $\Delta$ is odd.}$$

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Posted by fhavet
updated March 1st, 2013
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Graph Theory » Directed Graphs

Long directed cycles in diregular digraphs ★★★

Author(s): Jackson

Conjecture   Every strong oriented graph in which each vertex has indegree and outdegree at least $ d $ contains a directed cycle of length at least $ 2d+1 $.

Keywords:

Posted by fhavet
updated March 1st, 2013
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Graph Theory » Directed Graphs

Splitting a digraph with minimum outdegree constraints ★★★

Author(s): Alon

Problem   Is there a minimum integer $ f(d) $ such that the vertices of any digraph with minimum outdegree $ d $ can be partitioned into two classes so that the minimum outdegree of the subgraph induced by each class is at least $ d $?

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Posted by fhavet
updated March 1st, 2013
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Graph Theory

Stable set meeting all longest directed paths. ★★

Author(s): Laborde; Payan; Xuong N.H.

Conjecture   Every digraph has a stable set meeting all longest directed paths

Keywords:

Posted by fhavet
updated March 1st, 2013
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Graph Theory » Directed Graphs

Ádám's Conjecture ★★★

Author(s): Ádám

Conjecture   Every digraph with at least one directed cycle has an arc whose reversal reduces the number of directed cycles.

Keywords:

Posted by fhavet
updated March 1st, 2013
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Graph Theory » Directed Graphs

Caccetta-Häggkvist Conjecture ★★★★

Author(s): Caccetta; Häggkvist

Conjecture   Every simple digraph of order $ n $ with minimum outdegree at least $ r $ has a cycle with length at most $ \lceil n/r\rceil $

Keywords:

Posted by fhavet
updated February 28th, 2013
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Graph Theory » Directed Graphs

Directed path of length twice the minimum outdegree ★★★

Author(s): Thomassé

Conjecture   Every oriented graph with minimum outdegree $ k $ contains a directed path of length $ 2k $.

Keywords:

Posted by fhavet
updated February 28th, 2013
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Graph Theory » Directed Graphs

Antidirected trees in digraphs ★★

Author(s): Addario-Berry; Havet; Linhares Sales; Reed; Thomassé

An antidirected tree is an orientation of a tree in which every vertex has either indegree 0 or outdergree 0.

Conjecture   Let $ D $ be a digraph. If $ |A(D)| > (k-2) |V(D)| $, then $ D $ contains every antidirected tree of order $ k $.

Keywords:

Posted by fhavet
updated February 26th, 2013
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Graph Theory » Directed Graphs » Tournaments

Decomposing an even tournament in directed paths. ★★★

Author(s): Alspach; Mason; Pullman

Conjecture   Every tournament $ D $ on an even number of vertices can be decomposed into $ \sum_{v\in V}\max\{0,d^+(v)-d^-(v)\} $ directed paths.

Keywords:

Posted by fhavet
updated February 26th, 2013
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Graph Theory » Directed Graphs

Oriented trees in n-chromatic digraphs ★★★

Author(s): Burr

Conjecture   Every digraph with chromatic number at least $ 2k-2 $ contains every oriented tree of order $ k $ as a subdigraph.

Keywords:

Posted by fhavet
updated February 25th, 2013
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Theoretical Comp. Sci. » Complexity

Discrete Logarithm Problem ★★★

Author(s):

If $ p $ is prime and $ g,h \in {\mathbb Z}_p^* $, we write $ \log_g(h) = n $ if $ n \in {\mathbb Z} $ satisfies $ g^n = h $. The problem of finding such an integer $ n $ for a given $ g,h \in {\mathbb Z}^*_p $ (with $ g \neq 1 $) is the Discrete Log Problem.

Conjecture   There does not exist a polynomial time algorithm to solve the Discrete Log Problem.

Keywords: discrete log; NP

Posted by cplxphil
updated February 25th, 2013
3 comments
Graph Theory » Coloring » Labeling

Good Edge Labelings ★★

Author(s): Araújo; Cohen; Giroire; Havet

Question   What is the maximum edge density of a graph which has a good edge labeling?

We say that a graph is good-edge-labeling critical, if it has no good edge labeling, but every proper subgraph has a good edge labeling.

Conjecture   For every $ c<4 $, there is only a finite number of good-edge-labeling critical graphs with average degree less than $ c $.

Keywords: good edge labeling, edge labeling

Posted by DOT
updated February 16th, 2013
1 comment
Number Theory

Special Primes ★

Author(s): George BALAN

Conjecture   Let $ p $ be a prime natural number. Find all primes $ q\equiv1\left(\mathrm{mod}\: p\right) $, such that $ 2^{\frac{\left(q-1\right)}{p}}\equiv1\left(\mathrm{mod}\: q\right) $.

Keywords:

Posted by maththebalans
updated February 7th, 2013
2 comments
Graph Theory » Coloring

Three-chromatic (0,2)-graphs ★★

Author(s): Payan

Question   Are there any (0,2)-graphs with chromatic number exactly three?

Keywords:

Posted by Gordon Royle
updated February 6th, 2013
1 comment
Graph Theory » Coloring » Vertex coloring

Choice Number of k-Chromatic Graphs of Bounded Order ★★

Author(s): Noel

Conjecture   If $ G $ is a $ k $-chromatic graph on at most $ mk $ vertices, then $ \text{ch}(G)\leq \text{ch}(K_{m*k}) $.

Keywords: choosability; complete multipartite graph; list coloring

Posted by Jon Noel
updated February 2nd, 2013
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Number Theory » Analytic N.T.

The Riemann Hypothesis ★★★★

Author(s): Riemann

The zeroes of the Riemann zeta function that are inside the Critical Strip (i.e. the vertical strip of the complex plane where the real part of the complex variable is in ]0;1[), are actually located on the Critical line ( the vertical line of the complex plane with real part equal to 1/2)

Keywords: Millenium Problems; zeta

Posted by eric
updated January 27th, 2013
4 comments
Number Theory » Analytic N.T.

Euler-Mascheroni constant ★★★

Author(s):

Question   Is Euler-Mascheroni constant an transcendental number?

Keywords: constant; Euler; irrational; Mascheroni; rational; transcendental

Posted by Juggernaut
updated January 21st, 2013
1 comment
Graph Theory » Basic G.T.

Graham's conjecture on tree reconstruction ★★

Author(s): Graham

Problem   for every graph $ G $, we let $ L(G) $ denote the line graph of $ G $. Given that $ G $ is a tree, can we determine it from the integer sequence $ |V(G)|, |V(L(G))|, |V(L(L(G)))|, \ldots $?

Keywords: reconstruction; tree

Posted by mdevos
updated January 16th, 2013
5 comments
Logic » Finite Model Theory

Vertex Cover Integrality Gap ★★

Author(s): Atserias

Conjecture   For every $ \varepsilon > 0 $ there is $ \delta > 0 $ such that, for every large $ n $, there are $ n $-vertex graphs $ G $ and $ H $ such that $ G \equiv_{\delta n}^{\mathrm{C}} H $ and $ \mathrm{vc}(G) \ge (2 - \varepsilon) \cdot \mathrm{vc}(H) $.

Keywords: counting quantifiers; FMT12-LesHouches

Posted by dberwanger
updated October 2nd, 2012
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