Graph Theory » Topological G.T. » Genus

Consecutive non-orientable embedding obstructions ★★★

Author(s):

Conjecture   Is there a graph $ G $ that is a minor-minimal obstruction for two non-orientable surfaces?

Keywords: minor; surface

Posted by Bruce Richter
updated September 15th, 2010
2 comments
Graph Theory » Coloring » Vertex coloring

Strong colorability ★★★

Author(s): Aharoni; Alon; Haxell

Let $ r $ be a positive integer. We say that a graph $ G $ is strongly $ r $-colorable if for every partition of the vertices to sets of size at most $ r $ there is a proper $ r $-coloring of $ G $ in which the vertices in each set of the partition have distinct colors.

Conjecture   If $ \Delta $ is the maximal degree of a graph $ G $, then $ G $ is strongly $ 2 \Delta $-colorable.

Keywords: strong coloring

Posted by berger
updated November 19th, 2009
5 comments
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 » Directed Graphs

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

Posted by mdevos
updated May 31st, 2007
1 comment
Graph Theory » Coloring » Homomorphisms

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

Posted by Robert Samal
updated March 24th, 2007
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Graph Theory » Hypergraphs

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

Posted by mdevos
updated March 19th, 2007
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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
Theoretical Comp. Sci. » Complexity

Subset-sums equality (pigeonhole version) ★★★

Author(s):

Problem   Let $ a_1,a_2,\ldots,a_n $ be natural numbers with $ \sum_{i=1}^n a_i < 2^n - 1 $. It follows from the pigeon-hole principle that there exist distinct subsets $ I,J \subseteq \{1,\ldots,n\} $ with $ \sum_{i \in I} a_i = \sum_{j \in J} a_j $. Is it possible to find such a pair $ I,J $ in polynomial time?

Keywords: polynomial algorithm; search problem

Posted by mdevos
updated July 16th, 2007
10 comments
Graph Theory » Extremal G.T.

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

Posted by mdevos
updated March 18th, 2007
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