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Results tagged with graph-theory
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user 1532
Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.
8
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Generalizations of the four-color theorem
There are also interesting weak forms of the 4CT where the challenge is of course to give a direct proof. An immediate consequence of the 4CT is that every planar graph has an independent set of size …
- 4,713
3
votes
Generalizations of the four-color theorem
Consider a finite family of non-overlapping circles. We can ask what is the minimum number of colors needed to color the circles so that tangent circles are colored with different colors? By Koebe’s c …
- 4,713
8
votes
Connectivity of the Erdős–Rényi random graph
this is not really an answer to your question but just a related matter. As you point out the expected number of trees in a random graph is 1 already when p=c/n and is very large when p=logn/n so the …
- 4,713
18
votes
Can you determine whether a graph is the 1-skeleton of a polytope?
A few more remarks, On the bright side: To determine if a given graph is the graph of a d-polytope is decidable. Tarski's algorithm for real closed fields can be used.
In dimension 3 as Sam Nead menti …
- 35.5k
2
votes
Generalizations of the four-color theorem
Closely related conjectures are the following: An acyclic colouring of a graph is a colouring of its vertices so that the subgraph spanned on union of every two colour classes is acyclic (a forest). G …
- 24.7k
1
vote
Do degrees determine the chromatic number?
There is also a general principle one can apply: If you have any parameter $\alpha$ of graphs so that there is an efficient (polynomial-time) algorithm to compute $\alpha (G)$, then it is extremely i …
- 24.7k
26
votes
Accepted
Why is "P vs. NP" necessarily relevant?
The $P \ne NP$ problem is the best way we know to formulate the belief (which was expressed even before the problem was formally stated) that certain specific algorithmic problems (such as finding a H …
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10
votes
Can we realize a graph as the skeleton of a polytope that has the same symmetries?
There is an example of Bokowski, Ewald and Kleinschmidt of a 4-polytope with a certain symmetry of the graph that cannot be realized geometrically. The combinatorial construction is due to Kleinschmi …
- 24.7k
3
votes
Accepted
Number of homomorphisms from graph $H$ to $G$ , bounds that have to do with fractional color...
You are referring to E. Friedgut and J. Kahn, On the number of copies of one hypergraph in another, Israel Journal of Mathematics 105 (1998), 251–256.
For graphs this result is the very first paper o …
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9
votes
Accepted
Help on the following extremal problem?
If you devide your set of vertices into $k$ ($k \in \mathbb{Z}_{\geq 3}$) sets $V_1, V_2,\dots,V_k$ and take all edges from $V_i$ to $V_{i+1\ (mod\ k)}$, then you get $(n/k)^k$ holes of length $k$. Th …
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vote
What is known about the chromatic number for minimum-distance graphs in higher dimensions?
This is a very good question. Maybe I miss something but I don't know an example where the lower bound is not polynomial. There is an example of an infinite periodic configuration where the minimum de …
- 24.7k
10
votes
What are the implications of the new quasi-polynomial time solution for the Graph Isomorphis...
(a) What is the computational complexity of GI, is an example of a major question that we genuinely did not know the answer to even on a heuristic or conjectural level. Even now, whether GI is in P is …
- 24.7k
16
votes
Can one make Erdős's Ramsey lower bound explicit?
Finding explicit constructions for Ramsey graphs is a central problem in extremal combinatorics. Indeed, computational complexity gives a way to formalize this problem. Asking for a graph which can be …
- 24.7k
8
votes
Algebraic proof of 4-colour theorem?
There is an algebraic method by Alon and Tarsi which allows in certain cases to prove that certain graphs are $k$-colorable (in fact, even $k$-choosable). A famous case where this method prevails is t …
- 24.7k
4
votes
Generalizations of the four-color theorem
Let $P$ be a $d$-dimensional polytope with $n$ vertices. For every $2$-dimensional face $F$ triangulate $F$ by non crossing diagonals. So if $F$ has $k$ sides you add $(k-3)$ edges. It is known that …
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