Questions tagged [planar-graphs]

A planar graph is a graph (in the combinatorial sense) that can be embedded in a plane such that the edges only intersect at vertices. Consider tagging with [tag:combinatorics] and [tag:graph-theory].

A planar graph is a graph (in the combinatorial sense) that can be embedded in a plane such that the edges only intersect at vertices. Consider tagging with and .

A finite graph $G$ is planar if and only if no minor of $G$ is the complete graph $K_5$, or the complete bipartite graph $K_{3,3}$ (Wagner's Theorem).

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Number of simple edge-disjoint paths needed to cover a planar graph

Let $G=(V,E)$ be a graph with $|E|=m$ of a graph class $\mathcal{G}$. A path-cover $\mathcal{P}=\{P_1,\ldots,P_k\}$ is a partition of $E$ into edge-disjoint simple paths. The size of the cover is $\sigma(\mathcal{P})=k$. I am interested in upper and…
A.Schulz
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Four color theorem disproof?

My brother in law and I were discussing the four color theorem; neither of us are huge math geeks, but we both like a challenge, and tonight we were discussing the four color theorem and if there were a way to disprove it. After some time scribbling…
Doktor J
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Is this graph a planar graph or not?

I've been trying to find out if this graph is planar or not for a while and have really been coming up short when it comes to creating a planar drawing of the graph. My intuition is telling me that it's non-planar, but I cannot find any subgraph of…
John21
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Algorithms for "pleasing" drawings of planar graphs, possibly on sphere

What algorithms exist to draw planar graphs without edge crossings in a way that they are easy to interpret by humans? There are multiple algorithms that can handle any planar graph, such as Schnyder's algorithm or Chrobak-Payne. These typically…
Szabolcs
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How to prove that a simple graph having 11 or more vertices or its complement is not planar?

It is an exercise on a book again. If a simple graph $G$ has 11 or more vertices,then either G or its complement $\overline { G } $ is not planar. How to begin with this? Induction? Thanks for your help!
tamlok
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"Planar" graphs on Möbius strips

Is there an easy way to tell if a graph can be embedded on a Möbius strip (with no edges crossing)? A specific version of this: if a simple graph with an odd number of vertices has all vertices of degree 4, can it be embedded on a Möbius…
Jack Schmidt
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Checking whether a graph is planar

I have to check whether a graph is planar. The given type is $$ e ≤ 3v − 6 .$$ From Wikipedia: Note that these theorems provide necessary conditions for planarity that are not sufficient conditions, and therefore can only be used to prove a…
GorillaApe
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Can there exist an uncountable planar graph?

I'm currently revising a course on graph theory that I took earlier this year. While thinking about planar graphs, I noticed that a finite planar graph corresponds to a (finite) polygonisation of the Euclidean plane (or whichever surface you're…
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A space curve is planar if and only if its torsion is everywhere 0

Can someone please explain this proof to me. I know that a circle is planar and has nonzero constant curvature, so this must be an exception, but I am a little lost on the proof. Thanks!
Mike El Jackson
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Every planar graph has a vertex of degree at most 5.

I am trying to prove the following statement, any help!? Prove that every planar graph has a vertex of degree at most 5.
user144708
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Which graphs can be drawn using straight lines with no disjoint edges?

What is the class of graphs that can be drawn using only straight lines with no two edges disjoint? Edges are disjoint when they don't cross and they don't share a vertex. Vertices should be in general position (no three on a line). I know that for…
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Euler's formula for triangle mesh

Can anyone explain to me these two facts which I don't get from Euler's formula for triangle meshes? First, Euler's formula reads $V - E + F = 2(1-g)$ where $V$ is vertices number, $E$ edges number, $F$ faces number and $g$ genus (number of handles…
BRabbit27
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Parity on the number of rooted trees

Suppose we have a planar graph with vertices $v_0, \ldots, v_n$, where $n$ is even such that there exist a checkerboard coloring of the regions in the complement such that the black regions are $n$ (non-degerate) triangular faces having clockwise…
user895424
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Algorithm for planarity test in graphs

I am implementing a graph library and I want to include some basic graph algorithms in it. I have read about planar graphs and I decided to include in my library a function that checks if a graph is planar. I found on the web many efficient…
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Hard planar graph problem

Triangulation is called a planar graph in which every face is a triangle. $\bullet$ Prove that in every triangulation exists edge $\left\{ u,v \right\}$ such that $\deg(u)+\deg(v)\le 22$. $\bullet$ Give an example of planar graph without vertex of…
xan
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