Questions tagged [topological-graph-theory]
Graph theoretical questions with a topological flavour. For example, graphs on surfaces, spatial embeddings, and geometric graphs. Use the graph-drawing tag for questions specific to graph drawing (e.g. crossing numbers).
10 questions
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Spanning trees of plane graphs containing an edge of every face
I feel sure this must be known, but can I find it??
Which connected plane graphs (graphs drawn in the plane without crossings) have a spanning tree such that at least one edge of each face is in the ...
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4
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1
answer
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genus of a finite simple undirected graph
Let $G$ be a finite simple undirected graph. Suppose there exist subgraphs $G_1,G_2,\dots,G_n$ of $G$, such that $G_i$ and $G_j$, have no common edges and have at most two common vertices, for each $...
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Why are planar graphs so exceptional?
As compared to classes of graphs embeddable in other surfaces.
Some ways in which they're exceptional:
Mac Lane's and Whitney's criteria are algebraic characterizations of planar graphs. (Well, ...
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20
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Cayley graph of $A_5$ with generators $(1,2,3,4,5),(1,4,3,2,5)$
The Cayley graph of $A_5$ with two generators of order 5 seems rather complicated. What is its graph genus (orientable or non-orientable)?
The best I could get by trial and error is an embedding ...
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13
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0
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Does there exist 2-planar graph with chromatic number 8 or 9 or 10
A 2-planar graph is a graph that can be drawn in the plane so that each edge is crossed at most twice. It is known that every 2-planar graph satisfies that $|E(G)|\le 5(|V(G)|-2)$. This implies that ...
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10
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Is there a "simplest" way to embed a graph in 3-space?
I consider embeddings of graphs into 3-space with edges embedded as arbitrary curves. In the simplest (non-trivial) case the graph $G$ is a cycle or union of cycles, in which case the embeddings can ...
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3
votes
1
answer
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Construction of planar embedding
I'm reading the following paper on universality considerations in VLSI circuits
http://www.computer.org/csdl/trans/tc/1981/02/06312176.pdf
In Theorem 2 On the second page it states there exists ...
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2
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2
answers
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Decomposing a graph into n-cycles [closed]
Suppose I have a strongly $k-regular$ graph $G$, of size $v$, where every vertex is $N>0$ $n-cycles$, for $at least$ one value of $n$ that divides $v$. Can we cut edges from $G$ in such a way ...
1
vote
1
answer
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Bounds on lengths of intervals in bounded-degree interval graphs
A graph is said to be an interval graph if its vertices can be associated with (closed) intervals on the real line $\mathbb R$ and there is an edge between two vertices if and only if the ...
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1
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1
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Bounds on lengths of boxes in bounded-degree box graphs
$\DeclareMathOperator{\box}{\operatorname{box}}$$\DeclareMathOperator{\cub}{\operatorname{cub}}$
This is a follow up and an extension of another question I asked recently.
A box graph is a graph ...
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