All Questions
90 questions
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Is there any known upper bound for the local crossing number of a graph drawing in the plane?
The local crossing number ${\rm LCR(G)}$ of a graph $G$ is defined as the least nonnegative integer $k$ such that the graph has a $k$-planar drawing. In other words, it is the smallest possible number ...
- 1,190
1
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0
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51
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Coarse-graining a hypergraph
$\DeclareMathOperator{\poly}{\mathrm{poly}}$I have asked this question on math.SE here, but couldn't get a satisfactory answer. I have also asked a related question on math overflow here, but haven't ...
- 277
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0
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90
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Definition of Loop in an Oriented Matroid
I had posted this on Stackexchange because I don't believe this is a particlarly difficult question, but there were no answers, so I'm posting it on here now.
I just had a quick question about the ...
- 61
4
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0
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66
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Convergence of graph geodesics to geodesics on metric spaces
Let $(X,d)$ be a compact length space metric space $\mathbb{X}_{\delta}$ be a $\delta$-packing on $X$ and, for every $k\in \mathbb{N}_+$, let $G_{k,\delta}=(\mathbb{X}_{\delta},\mathcal{E}_k,W_k)$ ...
- 364
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1
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98
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Chromatic tiling complexity and the chromatic number conjecture
Let $T$ be a finite set of tiles in $\mathbb{R}^d$. A tiling of $\mathbb{R}^d$ by $T$ is a collection of disjoint translates of tiles in $T$ whose union is $\mathbb{R}^d$. A tiling is $k$-chromatic if ...
1
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1
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115
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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 ...
- 277
1
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1
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194
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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 ...
- 277
1
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1
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141
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Covering a bounded degree graph with subgraphs of bounded sizes
Let $G$ be a connected graph on $n$ vertices with maximum degree $\Delta \ge 2$. Let $\mathcal G = \{G_1,G_2,\ldots\}$ be a collection of subgraphs of $G$ such that every edge of $G$ is contained in ...
- 277
1
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0
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42
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What lower bounds are known for pair crossing number and related questions in multigraphs?
So in terms of crossing number https://arxiv.org/pdf/1808.10480 gives a lower bound of $O(e^{2.5}/n^{1.5})$ for multigraphs with no face of length 2 with no node contained inside.
What do we know ...
- 111
1
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67
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Conjecture on the increasing efficiency of the shortest minimum-link polygonal chains covering any grids of the form $\{0,1,2\}^k$ as $k$ grows
From the well-known Nine dots problem, we know that we need a polygonal chain with at least $4$ edges to connect the $9$ points of the planar grid $G_{3,2}:=\{\{0, 1, 2\} \times \{0, 1, 2\}\} \subset \...
- 1,451
1
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1
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114
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Removing a face from 4-connected planar graph
After removing a face (vertices along with edges) of a 4-connected planar graph, is the remaining graph 4-connected? Alternatively under what conditions is this true?
- 129
2
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1
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209
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Do the dual graphs of hyperplane arrangements admit Hamiltonian paths?
Consider a simple hyperplane arrangement $H_1,\cdots,H_n$ in the Euclidean space $\mathbb{R}^d$. By "simple" we mean any $k$ hyperplanes in $\{H_1,\cdots,H_n\}$ intersect in codimension $k$. ...
- 3,187
2
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1
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106
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Exhaustive list of small graphs for which $\frac{\alpha(G)\omega(G)}{n}$ is small?
I am looking for a list of small graphs (say on less than 10 vertices) for which the parameter $p(G) = \frac{\alpha(G) \omega(G)}{n}$ is small. Here $\alpha(G)$ and $\omega(G)$ is the size of the ...
- 129
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1
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213
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Given a 3-connected graph $G$, is there an edge $e$ so that both $G-e$ and $G/e$ are still 3-connected?
Let $G$ be a 3-connected (simple) graph other than $K_4$. In Diestel's "Graph Theory" Section 3.2 we find
Lemma 3.2.2. There is an edge $e$ so that $G\mathbin{\dot-}e$ is still 3-connected (...
- 13.6k
2
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0
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87
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Computationally decomposing a complete geometric graph into forests of stars
I'm working on the following problem:
I would like to see if it possible to decompose a complete geometric graph on $8$ vertices into $5$ planar star-forests. As doing this by hand was hopeless, I ...
- 21
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76
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Is the choosability/list chromatic number of a circular arc graph equal to its chromatic number?
In 2003, Prowse and Woodall proved that for graphs $C_n^k$ which are powers of cycles,
$$\chi_\ell(C_n^k) = \chi(C_n^k).$$
They conjectured that this equality holds for the broader class of graphs ...
- 151
9
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2
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484
views
Connected geometric thickness two
A graph $G = (V,E)$ has geometric thickness two if there exists an embedding $\varphi: V \rightarrow \mathbb{R}^2$ and a decomposition $E = E_1\cup E_2$ such that $G_1 = (V,E_1)$ and $G_2 = (V,E_2)$ ...
- 479
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0
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126
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Checking existence of a non-crossing Hamiltonian path in geometric graphs
I am interested in the following computational problem. Given a geometric graph (i.e, a graph drawn in the plane so that its vertices are represented by points in general position and its edges are ...
- 1,305
2
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0
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65
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Structure Theory for Tree Decompositions
I that $G=(V,E,W)$ is a weighted graph with positive edge weights and a finite set of vertices $K$. Let $0\le k,M\le K$ be a fixed integer.
Is is known when $G$ admits the following type of ...
- 21
2
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1
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157
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Bound for a sequence of vertices in a graph
I have come across the following problem. Let $d\in\mathbb{N}$. Let $G$ be any $k$-regular connected directed graph with $n$ vertices, no parallel edges and no 2-cycles. For a vertex $v\in G$, let $...
- 167
1
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0
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54
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Chromatic number of 2-graph vs hypergraph of point-line incidences
Define the chromatic number $\chi(H)$ of a (hyper)graph $H$ as the smallest $k$ such that its vertices can be $k$-colored such that no (hyper)edge is monochromatic.
Given a finite set of points $P$ in ...
- 18.7k
0
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0
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183
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Covering discrete triangle with generalized knight jumps
Consider for $n\in\mathbb{N}, n\geq 6$ the discrete triangle $\nabla=\{(i,j)\in \{1,\ldots,n-1\}^2 \mid j\leq n-i\}$. This is basically the lower "half" of a chess board if you cut it along ...
- 91
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51
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Regular graphs of tangent spheres
Problem 1. Let several non-overlapping spheres in $\mathbb{R^3}$ are given. For which $n$ it is possible that each sphere is tangent to exactly $n$ other spheres?
Consider the smallest sphere. Since ...
- 691
3
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144
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Counting homologically non-trivial and trivial cycles in $n \times n$ square lattice torus of a given length $l \geq n$
This should be a fairly standard question but I can't really seem to find a reference.
Consider an $n \times n$ square lattice torus $\mathbb T$. Given a length $l \geq n$, what is the number of ...
- 269
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1
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312
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Pairwise intersecting circles in the plane
If I am looking at a collection $\mathcal{C}$ of circles $\{C_1,...,C_n\}$ all of which have some radii $\{r_1,...,r_n\}$ where $r_i\in\mathbb{R}^{+}$ for each $i \in[n]$. In $\mathcal{C}$, all the ...
- 135
1
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66
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Representation of $x$-non-monotone curves with one intersection each by $x$-monotone curves
Take the $y$-axis and a set of $n$ curves starting from $y$-axis, labelled as $\mathcal{C}:=\{C_1,C_2,...,C_n\}$. These curves fulfill the following conditions:
The curves all have a starting point ...
- 135
9
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3
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470
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Is it possible that every edge in a 1-planar drawing with minimum number of crossings is crossed?
A graph is 1-planar is it has drawing in the plane so that each edge is crossed at most once. Here we also assume the drawing satisfies (1) no edge is self-crossed; (2) no two adjacent edges are ...
- 557
6
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1
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142
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Embedding linklessly embeddable graphs without Borromean rings
A linklessly embeddable graph is a graph which can be embedded into $\Bbb R^3$ so that no two of its cycles are linked. For example, the Petersen graph is not such a graph.
Now, I can think of another ...
- 13.6k
0
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52
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What do optimal tours tell about finite point sets?
Let $T_{\mathrm{MIN}}$ and $T_{\mathrm{MAX}}$ denote the shortest, resp. longest Hamilton cycle through a set of $n=2k+1$ points.
Let further $S_{\mathrm{MIN}}$ and $S_{\mathrm{MAX}}$ be the "...
- 13.2k
22
votes
2
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900
views
Is every 1-million-connected graph rigid in 3D?
It is an old result that every $6$-connected graph is rigid in $\mathbb{R}^2$:
Lovász, László, and Yechiam Yemini. "On generic rigidity in the plane." SIAM Journal on Algebraic Discrete ...
- 151k
4
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1
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187
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Number of permutations with combinatorial geometric constraints
We are given a $d$-dimensional hypercube $H$, where each vertex is labeled with an integer $\ell\in\{1, 2, \ldots, 2^d\}$. Let $L$ be this labelling.
Question: How many labelling permutations $L'$ of ...
- 1,677
2
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1
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127
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The density of a tripartite 1-planar graph
1-planar graphs are those can be drawn in the plane so that there is at most one crossing per edge. We know that the maximum number of edges of an $n$-vertex 1-planar graph is at most $4n-8$, and the ...
- 1,190
16
votes
4
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1k
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Squaring a square and discrete Ricci flow
Is this a theorem?
Every $3$-connected planar graph $G$ may be represented as
a tiling of a square by squares,
one square per node of $G$, with nodes connected in $G$
corresponding to tangent squares....
- 151k
11
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5
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506
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What are efficient pooling designs for RT-PCR tests?
I realize this is long, but hopefully I think it may be worth the reading for people interested in combinatorics and it might prove important to Covid-19 testing. Slightly reduced in edit.
The ...
- 14.4k
4
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1
answer
444
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What is the significance of ear decompositions for non-graphic matroids?
On Wikipedia there is subsection in the article on ear decompositions of graphs titled "Matroids":
Now as defined above, the circuits of a matroid can not always be listed to satisfy the ...
- 2,506
3
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0
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134
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Two questions on counterexamples to Borsuk's conjecture and ball-packings
In 1933 Karol Borsuk conjectured the following
Can every bounded subset $E$ of $\mathbb{R}^d$ be partitioned into $(d+1)$ sets, each of which has a smaller diameter than $E$?
Whilst new to this ...
- 31
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83
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Distance spectra of uniform tilings
Let a uniform tiling be defined by a vertex configuration $(n_1.n_2.\cdots.n_k)^m$, which is either spherical, Euclidean or hyperbolic. Assume that the tiling is vertex-transitive, especially that ...
- 9,720
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1
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249
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Oriented path in a graph
Let $G=(\mathcal{V}_G,\mathcal{A}_G)$ be an oriented acyclic graph. Assume that $G$ has a unique source $s\in \mathcal{V}_G$ and a unique sink $t\in \mathcal{V}_G$. Now, fix $u,v\in \mathcal{V}_G$ ...
- 353
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1
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157
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Structure of boundary labelling in Sperner‘s Lemma
Consider a triangulated polygon in the 2-dimensional plane, where each vertex is labelled green, blue, or orange. Sperner's Lemma asserts that a fully-colored triangle exists in the triangulation, if ...
- 6,917
18
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Sperner's Lemma implies Tucker's Lemma - simple combinatorial proof
Sperner’s Lemma is often called the "combinatorial analog" of Brouwer’s Fixed Point Theorem, and similarly Tucker’s Lemma is often called the combinatorial analog of Borsuk–Ulam’s Theorem.
We can ...
- 6,917
11
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1
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370
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Graph with path of length $\geq n$ along grid diagonals - a known result in graph theory?
Is the following lemma a well known result in graph theory?
I am studying a basic existence result that appears to be simple yet powerful. I have not seen it stated as an important result in graph ...
- 6,917
10
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1
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1k
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How can we find n points on a plane so that as many pairs of points as possible have the same distance?
There are $n$ points on the plane, and we need to maximize the number of pairs of points which have the same Euclidean distance.
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1
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75
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Given a vertex $u$ (of bounded degree $k$) and another vertex $v$ in a planar graph, what is the smallest number of "curves"?
Given a vertex $u$ (of bounded degree $k$) and another vertex $v$ in a planar graph $G$, what is the smallest number of "curves" in the plane drawn from $u$ to $v$ such that no $u$--$v$ path in $G$ ...
- 111
3
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2
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455
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How to generating all flats of the cycle matroid of a graph?
If $M$ is a matroid, I can use M.flats(k) in SageMath to list all the flats of rank $k$. But I hope that there is an algorithm or program to list all flats of the cycle matroid of a graph. And do not ...
- 51
3
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4
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379
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Generalization of independence complex of graphs
Let $G$ be an undirected graph with no multiple edges or loops. Recall that the independece system $\mathcal{I}(G)$ consists of all those subsets $A$ of the vertex set such that the induced subgraph $...
- 1,017
10
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1
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370
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When does a graph have a circular orientation? Or equivalently can anyone help me characterize this particular class of $3$-colorable perfect graphs?
Call an oriented digraph $D=(V,A)$ circular when for all $\small x,y,z\in V$ if $(x,y)\in A$ and $(y,z)\in A$ then $(z,x)\in A$ or equivalently if $D$ is any oriented digraph whose arc set is a ...
- 2,506
4
votes
1
answer
235
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Graphs with adjacency matrix depending on associated-vector distances
Let $G$ be a graph of order $n$ such that for each vertex $v$ there are two associated vectors, $f_v, g_v\in R^n$, where $uv\in E(G)$ if and only if $\|f_u - f_v\|^2 \ge \|g_u-g_v\|^2$.
ISGCI didn't ...
- 519
19
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2
answers
1k
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Is it possible that both a graph and its complement have small connectivity?
Let $G=(V,E)$ be a simple graph with $n$ vertices. The isoperimetric constant of $G$ is defined as
$$
i(G) := \min_{A \subset V,|A| \leq \frac n2} \frac{|\partial A|}{|A|}
$$
where $\partial A$ is ...
- 1,991
13
votes
2
answers
2k
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Counting Hamiltonian cycles in $n \times n$ square grid
I wonder if anyone has counted these curves, either exactly or asymptotically?
Let $S_n$ be an $n \times n$ subset of $\mathbb{Z}^2$ consisting of $n^2$
lattice points: a lattice square.
Define a ...
- 151k
16
votes
1
answer
546
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Chromatic numbers of infinite abelian Cayley graphs
The recent striking progress on the chromatic number of the plane by de Grey arises from the interesting fact that certain Cayley graphs have large chromatic number; namely, the graph whose vertices ...
- 19.2k