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Results tagged with graph-theory
Search options questions only
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user 4558
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.
11
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Algorithm for embedding a graph with metric constraints
Suppose I have a graph $G$ with vertex set $V$, edge set $E \subseteq {V \choose 2}$, a poistive integer $d$, and a weight function $w:E \to \mathbb{R}^{+}$. Is there a nice algorithmic way to decide …
- 13.2k
18
votes
2
answers
1k
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In an Erdős–Rényi random graph, what is the threshold for the property "every edge is contai...
Let $G(n,p)$ denote the Erdős–Rényi random graph, where $n$ is the number of nodes and $p$ is the probability for each edge. I'm interested in precisely what range of $p$ the random graph has at least …
CommunityBot
- 1
10
votes
2
answers
441
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Higher-dimensional Fáry's theorem?
Fáry's theorem says that every finite simple planar graph admits a planar embedding with straight line edges.
For which $(k,d)$ is it true that every finite $k$-dimensional simplicial complex embeddab …
- 1,446
20
votes
2
answers
881
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Is there an analogue of the Erdős–Gallai theorem for simplicial complexes?
The Erdős–Gallai theorem gives a necessary and sufficient condition for a finite sequence of natural numbers to be the degree sequence of a simple graph.
In particular $d_1 \ge d_2 \ge \dots \ge d_n$ …
- 1,446
57
votes
4
answers
15k
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Connectivity of the Erdős–Rényi random graph
It is well-known that if $\omega=\omega(n)$ is any function such that $\omega \to \infty$ as $n \to \infty$, and if $p \ge (\log{n}+\omega) / n$ then the Erdős–Rényi random graph $G(n,p)$ is asymptoti …
- 4,713
27
votes
1
answer
3k
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Monochromatic triangles in every two-coloring of the plane?
An old problem (possibly due to Erdős and Graham?): given a triangle $T$ and a two-coloring of the plane, does there necessary exist a monochromatic congruent copy of $T$? Here "monochromatic" means …
- 1,318
14
votes
5
answers
663
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Do there exist sparse graphs with large crossing number?
Does there exist a sequence of graphs $\{ G_n \}$ such that
$G_n$ has $n$ vertices,
the number of edges of $G_n$ is $O(n)$, and
the crossing number of $G_n$ is $\Omega(n)$?
In particular, do rando …
- 6,101
4
votes
0
answers
263
views
What are the best bounds to date on the maximum girth of a cubic graph?
The girth of a graph is the length of its smallest cycle. Studying the maximum possible girth for a $k$-regular graph on $n$ vertices is a very well-studied problem.
In the 1988 paper "Ramanujan grap …
- 7,857
10
votes
1
answer
455
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For what range of edge probability does the following property hold for random graphs?
Let $G(n,p)$ denote the Erdős–Rényi model of random graph. For a given function $p = p(n)$ we say that $G \in G(n,p)$ asymptotically almost surely has property $\mathcal{P}$ if
$$\mbox{Pr}[G \mbox{ …
- 6,711
6
votes
2
answers
724
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Has the following kind of (minimum degree $d$) random graph been studied?
The following random construction is simple enough that I am guessing it must have been studied. Fix $d \ge 3$, and let $n > d$. For each of the $n$ vertices, pick exactly $d$ other vertices to conne …
- 85.6k
21
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11
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What are some good examples of non-monotone graph properties?
It seems that many, if not almost all, of the properties studied in graph theory are monotone. (Property means it is invariant under permutation of vertices, and monotone means that the property is e …
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