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Results tagged with extremal-graph-theory
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user 9025
Study of graphs satisfying a property that are maximal or minimal with respect to some parameter. A classic example is Turán's Theorem, which exactly characterizes the densest graphs on $n$ vertices without a $K_t$ subgraph.
1
vote
Largest graphs of girth at least 6
It seems I have an answer to my own question, though credit belongs to my computer. Namely, $e_6(47)=118$ and the unique graph in that class is not bipartite. I wish I could offer some insight, but …
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11
votes
0
answers
309
views
How many n/2-cycles can a cubic graph have
Given a simple cubic graph with $n$ vertices (which implies that $n$ is even), what is a good upper bound on the number of cycles of length $n/2$ it can have?
A random cubic graph has $\Theta((4/3)^n …
- 37.7k
9
votes
Accepted
If many triangles share edges, then some edge is shared by many triangles
Let $N~$ be the number of figures consisting of two triangles sharing an edge, with one of the vertices not on the common edge marked.
Clearly $N=2x$. Alternatively, start with one triangle, mark a …
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22
votes
2
answers
2k
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Largest graphs of girth at least 6
Let $e_6(n)$ be the greatest number of edges in a simple graph with $n$ vertices and girth at least 6.
Let $G_6(n)$ be the set of simple graphs of order $n$ with girth at least 6 and $e_6(n)$ edges.
…
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8
votes
1
answer
728
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6-regular bipartite graphs with no 8-cycles
I'm looking for simple 6-regular bipartite graphs with no 8-cycles, as small as possible. It doesn't matter if there are 4-cycles or 6-cycles, provided there are no 8-cycles. Such graphs must exist …
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14
votes
Accepted
Spanning trees: the last darn $1/4$
Consider connected $G$ with $n$ vertices of degree $\ge 3$ and exactly one vertex $v$ of degree 1. Take an extra copy $G'$ of $G$ with $v'$ being its vertex of degree 1.
Now identify $v$ and $v'$ to …
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4
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Almost all graphs have a subgraph from a large class of graphs with constant order
We can ask not only that something appears as a subtree, but that it appears as a "limb". A limb of a tree $T$ at a vertex $v$ is a maximal subtree of $T$ that includes $v$ and a neighbouring vertex. …
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1
vote
Maximal graphs with a property that is invariant w.r.t. vertex removal
Consider a graph $G$ with $g(n)$ edges. $G$ has average degree $2g(n)/n$ so there is a vertex $v$ with degree at most
$$\delta = \left\lfloor \frac{2g(n)}{n}\right\rfloor.$$
Since by assumption $G-v$ …
- 37.7k
1
vote
3-uniform tetrahedron-free hypergraph on seven vertices
This is the "simplest" hypergraph Turán problem, where I put "simplest" in quotes because there is no such thing as a simple hypergraph Turán problem.
This paper gives a conjecture that has been prove …
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3
votes
Accepted
Vertex cover of regular graph
The complement of $S$ is an independent set, so the minimum value of $|S|$ is $n-\alpha(G)$ where $\alpha(G)$ is the size of the largest independent set. For non-complete connected regular graphs of …
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