Mathematics

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]. For embeddings into higher-genus spaces, use [tag:graph-embeddings].

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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Euler's formula doesn't work for null graph?

Given the null graph with no edges or vertices, we have a connected planar graph as no edges cross when this graph is drawn in the plane, and the fact that any two distinct vertices have a path between them is vacuously true. However, Euler's…
Princess Mia
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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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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 graph…
GorillaApe
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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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If a quadrilateral is partitioned into cyclic quadrilaterals, must the outer quadrilateral be cyclic?

Let (ABCD) be a (convex) quadrilateral that is partitioned into finitely many quadrilaterals. Assume: Every quadrilateral in the partition is cyclic . No vertex of any quadrilateral of the partition lies on a side of (ABCD). No vertex of any…
user1299769
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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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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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"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 strip?…
Jack Schmidt
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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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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…
Joshua Pepper
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Can you explain to me why this proof by induction is not flawed? (Domain is graph theory, but that is secondary)

Background I am following this MIT OCW course on mathematics for computer science. In one of the recitations they come to the below result: Official solution Task: A planar graph is a graph that can be drawn without any edges crossing. Also, any…
SLLegendre
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5-color coloring game.

Let there be two players, $A$ and $B$ , and a map. They now play a game such that: Player $A$ picks a region and player $B$ colors it such that the region is a different color than all adjacent regions. Player $A$ wins if at the end of the game, the…
blademan9999
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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…
Rondogiannis Aristophanes
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