Questions tagged [graph-theory]
Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.
1,486
questions with no upvoted or accepted answers
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Is the dominating set problem restricted to planar bipartite graphs of maximum degree 3 NP-complete?
Does anyone know about an NP-completeness result for the DOMINATING SET problem in graphs, restricted to the class of planar bipartite graphs of maximum degree 3?
I know it is NP-complete for the ...
8
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152
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Disjoint Rooted Paths with Specified Patterns
Let $S:=$ { $s_i : i \in [k]$ } and $T:=$ { $t_i : i \in [k]$ } be disjoint subsets of vertices of a graph $G$. Furthermore, let $A$ be a subset of $S_k$ (the symmetric group on $[k]$). A set of ...
8
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Simplicial Representations of (Hyper)Graph Complexes
For graph complexes, which are families of graph [on a fixed number of vertices n] closed under the deletion of edges, there is a natural simplicial complex capturing that information. Specifically, ...
7
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223
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Chip firing on hypergraphs
A (finite) hypergraph is a pair $(V, \mathcal{E})$ where $V$ is a finite set of vertices and $\mathcal{E}\subseteq\mathcal{P}(V)$ with each $E\in\mathcal{E}$ having at least two elements; a ...
7
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191
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longest path transversals
In a connected graph any two longest paths intersect at a common vertex. It is open whether any three longest paths in a connected graph intersect at a common vertex. For a connected graph $G$, let ...
7
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149
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Why is the crossing number of Tutte 12-cage 170?
From the Wikipedia entry on Tutte 12-cage , it is stated that the crossing number of Tutte 12-cage is 170, but the cited references do not seem to provide sufficient explanation for this.
Exoo, G. &...
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162
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Ring of invariants for graph automorphism
$\DeclareMathOperator\Aut{Aut}$Let $G$ be a finite simple graph with nodes numbered $1$ to $n$. Attach variables $x_1, ..., x_n$ to nodes. The graph automorphism group $\Aut G$ acts on nodes by ...
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295
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"Meritocratic" pyramid schemes
There have been a couple of times in my life when people from multi-level marketing organizations attempted to recruit me. I listened to what they had to say, and both times I did not get involved ...
7
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194
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Upper bound on the number of perfect matchings in $K_{3,3}$-free graphs
Let $G=(V,E)$ be a graph with an even number of vertices $|V|=2n$. Assume that $G$ is $K_{3,3}$-free i.e. it does not contain a graph isomorphic to biclique $K_{3,3}$. A perfect matching of $G$ is a ...
7
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380
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Has the Moore graph problem been solved?
A $d$-regular (simple finite) graph $G=(V,E)$ with diameter $k$ is a Moore graph if
$$ |V| = 1 + d \sum_{i=0}^{k-1} (d-1)^i. $$
It is known from the Hoffman-Singleton theorem and results of Damerall ...
7
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93
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What is known about chromatic polynomial of hypergraph at $-1$
Let $H$ be a hypergraph and let $P_H$ denote its chromatic polynomial. I am interested in the best results interpreting $P_H(-1)$. I am interested both in the general case (which I think is hard) as ...
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149
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Does there exist an arbitrarily large bipartite graph that is critical of dimension 4?
Here's a question in geometric graph theory that I recently kicked the tires on but was unable to resolve.
For a finite, simple graph $G$, say that $G$ is of dimension $n$, and write $\dim(G) = n$, if ...
7
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74
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Graphs all of whose cuts are positive
Let $(V, E, w)$ a weighted graph, with vertices $V$, edges $E$, and signed weight $w:E\to \mathbb R$.
I am interested to know other popular properties that are known to imply, or are equivalent to, ...
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101
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Median spaces as retracts of hypercubes
It is known (See e.g. here, Theorem 2.1) that median graphs are retracts of hypercubes.
Question: Is it also known that median metric spaces are retract of some $l¹$ product of unit intervals?
By ...
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176
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Matrix of high rank mod $2$: must it have a large non-singular minor (with disjoint rows and columns)?
Let $A$ be a $2n$-by-$2n$ matrix with entries in
$\mathbb{Z}/2\mathbb{Z}$ such that, for every $2n$-by-$2n$ diagonal
matrix $D$ with entries in $\mathbb{Z}/2\mathbb{Z}$, the matrix $A+D$
has rank $\...
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265
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Chromatic polynomial and the circle
In https://arxiv.org/pdf/1208.5781.pdf
It is proved that there is spectral sequence converging to $H^*(M^G,R)$ with the E1 page given by the graph cohomology complex $C_A(G)$ where $A:=H^*(M,R)$.
My ...
7
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84
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Universal point sets for 1-plane graphs
It is a notorious open problem to find a smallest set of $N$ points that
permit any $n$-vertex planar graph to be drawn in the plane without
crossings, using only those $N$ points as vertices, and ...
7
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120
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Does the problem of recognizing 3DORG-graphs have polynomial complexity?
A 2DORG is the intersection graph of a finite family of rays directed $\to$ or $\uparrow$ in the plane. Such graphs can be recognized effectively (Felsner et al.). A 3DORG is the intersection graph of ...
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93
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Combinatorial region-halfplane incidence structures
I've seen a bunch of similar MO questions, yet hopefully this is not a complete duplicate.
Consider $n$ halfplanes in $\mathbb{R}^2$ with their borders in general position, that is, no point of $\...
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IH-moves on trivalent graphs, and a complex that might be known to low-dimensional topologists
Here is a combinatorial problem which is hard to Google but seems like it might have a solution well known to people who study finite type invariants etc.
Let $G_{g,b}$ denote the set of finite ...
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258
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Relations between Betti numbers for clique complex
Given a clique complex $K$ constructed from a discrete set of vertices (i.e. its faces are isomorphic to the set of cliques in the 1-skeleton of $K$.), it seems that the Betti numbers $\beta_k$ ...
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Why does a polynomial with discriminant $d=-163$ appear in this sequence?
Fact 1: Given the eta quotient,
$$x_d= e^{2\pi\rm{i}/48}\,\frac{\eta(\tau)}{\eta(2\tau)}$$
where $\tau=\frac{1+\sqrt{-d}}2$, then $x_d$ for $d=11,19,43,67,163$ are the roots of the simple cubics,
$$...
7
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223
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The smallest order of a 4-chromatic graph of given girth
Let $n_4(g)$ denote the smallest order of a $4$-chromatic graph with girth $g$. It is known that $n_4(4)=11$ [2] and $n_4(5)=21$ [1]. By a famous proof of Erdös, it is known that $n_4(g)$ is well-...
7
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346
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Missing count in number of perfect matchings
Let $f(G)$ give number of perfect matchings of a graph $G$.
Denote $\mathcal N_{2n}=\{0,1,2,\dots,n!-1,n!\}$.
Denote collection of all $2n$ vertex balanced bipartite graph to be $\mathcal G_{2n}$.
...
7
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227
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Has anyone seen these binary trees (Catalan-type related to the Gegenbauer polynomials and Motzkin paths)?
The OEIS entry A121448 enumerates binary trees with $n$ edges and $k$ vertices with outdegree 1.
Has anyone seen these trees?
The o.g.f. for this entry, $G(x,t)$, is essentially a discriminant ...
7
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117
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Contradicting claims about complexity of directed path graphs isomorphism
Thesis and a paper give conflicting claims about the
complexity of graph isomorphism for directed path graphs.
Since this means GI is polynomial likely I am missing something
or there is something ...
7
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186
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"Edge Density" of Infinite Planar Graphs
Given an infinite planar graph $G$, let's denote by $\{H_1,H_2,\dots,H_m\}$ all the labeled graphs on $n$ vertices that appear as subgraphs of $G$. Also let
$$d_n=\frac{\sum_{i=1}^m \#E(H_i)}{nm}$$
...
7
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196
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Determinantal formulae for Tutte polynomial
Let $G$ be a connected undirected graph. Then the number $ST(G)$ of spanning trees in $G$ equals the following specific value of the Tutte polynomial of $G$: $ST(G)=T_G(1,1)$.
On the other hand, ...
7
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358
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Have topographs been studied before?
This is my first post on MO so I hope this question is suitable. I have quite a few definitions which I will need to state before my questions at the end of this post. Please let me know if anything ...
7
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334
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Does this inequality always hold?
Denote the adjacency matrix of a given undirected graph by $g$. It is an $n$-by-$n$ symmetric Boolean matrix with elements on the diagonal to be zero ($n\geq 3$). Let $g_{12}=g_{21}=g_{13}=g_{31}=1$ ...
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740
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Does this graph property have a name?
I'm interested in a family of properties of connected simple graphs that comes up in percolation theory.
Let $G$ be a simple connected graph. Now consider the set of subgraphs of $G$ that I will call ...
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542
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Graphs with graphic imbalance sequences
Let $G$ be simple undirected graph and $e=uv\in E(G)$.
The imbalance of the edge $e$ is the value $imb(e)=|d(u)-d(v)|$.
Let $M_{G}$ denotes the imbalance sequence (or more correctly, multiset of ...
7
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179
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How quickly can we test if a graph is distance-regular?
A (simple, finite, connected) graph $G$ is distance regular if there exist integers $b_i,c_i,i=0,...,D$ such that for any two vertices $x,y$ in $G$ and distance $i=d(x,y)$, there are exactly $c_i$ ...
7
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472
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Maximum fractional chromatic number of a 4-regular triangle-free graph (updated)
Let $G$ be a graph with maximum degree 4 and clique number 2. The fractional version of Reed's Conjecture tells us that $\chi_f(G) \leq 7/2$. But how high can $\chi_f(G)$ be?
The Chvátal graph has ...
7
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340
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How does the number of self-avoiding paths between two points scale, in a square/cubic lattice?
Consider two different infinite graphs, whose vertices are drawn from $\mathbb Z^2$ or $\mathbb Z^3$. Let $P_d : \mathbb Z^d \times \mathbb N \to \mathbb N$ for $d \in \{2,3\}$ be the function such ...
7
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302
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When can the Cheeger constant be well-approximated by ``Hamming balls''?
Given a graph G, the Cheeger constant is defined by
$$
\DeclareMathOperator{\Vol}{Vol}
h_G := \min_{S \subseteq V, \Vol S \leq (\Vol G)/2} \frac{|\partial S|}{\Vol S}.
$$
Here, $\Vol S$ is the sum of ...
7
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279
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Is there an analog of Khovanov homology for edge deletion-contraction-extraction?
Motivated by Khovanov's categorification of the Jones polynomial, several authors have worked on the categorification of graph invariants. For the chromatic polynomial some references are:
"A ...
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385
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Sequence of graphs with small $\lambda_1$ (the smallest nonzero eigenvalue of a regular finite graph)
The combinatorial laplacian on a finite graph $G$ can be defined as $ \Delta: \mathbb{C}^G \to \mathbb{C}^G$ sending the function $f:G \to \mathbb{C}$ to $(\Delta f)(v) = \sum_{v' \sim v} \big( f(v)-f(...
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396
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What is this subclass of $k$-colorable graphs called?
The following property emerged naturally when I was playing with certain generalizations of Kneser graphs.
Let $k>0$ be a natural number. Consider a property $P_k$ of graphs defined as follows.
...
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The uniform odd and even subgraph of $\mathbb{Z}^2$
Given a (first finite and later infinite) graph $G =(V,E)$ the uniform even graph is the uniform probability measure on the set of spanning even subgraphs. That is subgraphs (V, E') with $E' \subset E$...
6
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Do vertex-maximal paths in 4-connected graphs intersect?
Call a path in a (possibly infinite) graph vmax (for vertex-maximal) if there is no path that covers a containmentwise larger subset of vertices.
For example, in any spider graph the union of any two ...
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224
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Reference to a definition of a graph homology
Let $G$ be a graph, and define $C_k$ to be the free abelian group on the homomorphisms from graphs $H$ such that $K_k$ is a minor of $H$ without needing to do any vertex deletions, only edge ...
6
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132
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Graph-theoretic quasi-crystals?
I have recently been interested in the following purely graph-theoretic notion that weakens the assumption of transitivity in a similar way to how quasi-crystals have "(possibly) aperiodic long-...
6
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408
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Example of graph with strange property
I've also posted this problem in Math Stack Exchange (here).
Note: Whenever I mention a coloring of a graph I'm referring to a proper coloring over its vertices using the least amount of colors.
...
6
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366
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Circle numbers on edges of a graph
Let $k$ vertices in a graph be given. Some pairs of vertices are connected by an edge, each edge is labeled either $\{1,2\}$, $\{1,3\}$, or $\{2,3\}$. We can circle some of the numbers on the edges. ...
6
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242
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Catalan numbers from matchings?
There are several examples of interpreting the Catalan numbers as non-nesting or non-crossing matchings of some graph.
My question is:
Is there a family of graphs $G_1,G_2,\dotsc$ with the number of ...
6
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62
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Vertex cover in bipartite graphs with bounds on cost and size
Suppose we have a bipartite graph $G$ with non-negative integer vertex costs. We would like to find a vertex cover of cost at most $C$ and size (number of vertices) at most $S$, where $C$ and $S$ are ...
6
votes
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134
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Hamilton cycles in random graphs with just enough connectivity
What is the asymptotic probability that $G$ has a Hamilton cycle if $G$ is a random $n$ vertex $\frac{4}{3}n$ edge graph, with minimum degree 2 and without degree 2 vertices at distance 1 or 2 to each ...
6
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0
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213
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Lindström-Gessel-Viennot from properties of the $Alt^k$ functor?
Let $A$ be the directed adjacency matrix of an acyclic directed graph, with variables as its nonzero entries (one for each edge). The $(a,b)$ entry of the matrix $(I-A)^{-1}$ is the sum over all paths ...
6
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183
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Maximum number of Hamilton paths in a tournament on $n$ vertices
Recall that a tournament is a directed graph $T$ such that for every pair of distinct vertices $\{v,w\}$, exactly one of the ordered pairs $(v,w)$, $(w,v)$ is an arc of $T$.
A tournament is strongly ...