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, ...

learn more… | top users | synonyms

0
votes
0answers
25 views

Statistics of Length Eccess in Shortest Path Calculations

I am trying to quantify the error that arises in the following problem: let $\mathcal{T}$ be a tiling of the plane and the task is to calculate shortest paths in the network $\mathcal{N}$ of the union ...
3
votes
1answer
93 views

Spectra of undirected $d$-regular graphs

Let $G$ be a undirected $d$-regular graph, that is, a graph whose all vertices have the same degree $d$. It is known that the eigenvalues $\sigma_i$, $i=1,\cdots,n$, of the adjacency matrix are real ...
5
votes
1answer
171 views

Is there a polynomial-time algorithm to check if a signed graph contains an odd-K5 minor?

I suspect this exists, if anyone has a reference please that would be very helpful. By signed graph, I mean each edge is designated either odd or even (e.g. as in Guenin's result for weakly bipartite ...
0
votes
0answers
23 views

Grötzsch graph crossing number [duplicate]

can't find the proof that the crossing number of the Grötzsch graph is 5.
4
votes
2answers
248 views

What is the independence number of this graph which is a generalization of a Kneser graph?

Let $\mathfrak{A}=\{1,2,\dots,8\}$ and construct a graph as follows. Let the vertices of the graph be the set of all $A=(\{a,b,c\},\{d,e\})$ where $a,b,c,d,e\in \mathfrak{A}$ are distinct. Two ...
1
vote
0answers
42 views

An algorithm to compute the number of graphs of size n with a given graph as a minor

I'm looking for any results regarding computing the number of graphs of size $n$ which have a given graph $H$ as a minor. Are there any known algorithms which are more efficient than a brute force ...
4
votes
0answers
169 views

Graph spectra and topology

This is a somewhat vague question, but I'm wondering if there has been any research into connections between the spectrum of a graph and some notion of the "topology" of that graph. To give an ...
1
vote
0answers
60 views

Difference in the Four Color Theorem [closed]

How is proving that any planar graph with maximum degree of four has a four coloring, different from proving the four color theorem? If they are different, then how would one prove it?
2
votes
1answer
139 views

Counting growing tree trajectories

I am looking for help: Beginning with a single node ($\circ$), at each discrete time step I can add a node/link pair to any node currently in the tree. Nodes are unlabelled and the tree is ...
7
votes
0answers
132 views

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 ...
6
votes
1answer
150 views

Example to $2^\kappa\nrightarrow (3)^2_\kappa$, plus closed walks of odd length?

Let $\kappa$ be an infinite cardinal. Consider the following example to $2^\kappa\nrightarrow (3)^2_\kappa$. $V$ is a set of vertices, each of which is an element of $2^\kappa$. Color the edge ...
14
votes
1answer
1k views

Is deciding if one planar graph is dual to another really NP-hard (Wikipedia claim)?

Wikipedia claims (permanent link) without reference: Testing whether one planar graph is dual to another is NP-complete. Another claim with reference: For any plane graph G, the medial graph ...
1
vote
0answers
51 views

Mapping a grayscale image into a weighted undirected graph

I am looking for a method to convert an image into a network. I have found the study Z. Wu, X. Lu, Y. Deng, Image edge detection based on local dimension: A complex networks approach. Physica A ...
2
votes
0answers
97 views

Systematic treatment of folding and valued graphs

I'm going to say beforehand that this question has something of a "am I missing something?" flavor. I'm in that odd position mathematicians often find themselves, where a topic has been addressed ...
11
votes
2answers
458 views

Blinking graphs

For any simple graph $G$, assign its nodes a weight/bit of $0$ or $1$. Call this a bit assignment for $G$. Now, generate a new bit assignment as follows: Each node $x$'s bit is replaced by $1$ if the ...
6
votes
3answers
242 views

Numerical invariants for a graph or its complement that are bounded by some constant

I'm looking for numerical graph invariants that are bounded by a constant either for a graph $G$ or its complement $\bar{G}$. (The complement graph $\bar{G}$ has the same set of vertices as $G$ but ...
2
votes
1answer
106 views

Properties of bipartite graphs

For a connected bipartite graph $G$ are the two following properties equivalent: 1)Every minimal cycle in $G$ has length 4, that is every cycle of length strictly greater than 4 can be divided in ...
22
votes
10answers
1k views

What (fun) results in graph theory should undergraduates learn?

I have the task of creating a 3rd year undergraduate course in graph theory (in the UK). Essentially the students will have seen minimal discrete math/combinatorics before this course. Since graph ...
8
votes
1answer
110 views

How many uniquely colored degree two vertices in 3-coloring of subcubic graph?

Is there a graph with maximum degree three that has 3 degree two vertices that must get the same (resp. different) color in every 3-coloring of the graph? I'm interested in any similar results as ...
0
votes
0answers
37 views

How does subdividing an edge change the Tutte polynomial of graph at $x=0$?

Let $T_G(x,y)$ be the Tutte polynomial of simple graph $G$. Let $G'$ be $G$ with an edge subdivided (choose any edge). Limited experiments suggest: Conjecture 1: $T_G(0,y)=T_{G'}(0,y)$. Is ...
2
votes
1answer
62 views

Induced matching of cycle

Definition: A graph $G$ is chordal if every induced cycle in $G$ has length 3, and is co-chordal if the complement graph $G^c$ is chordal.The co-chordal cover number, denoted $cochord (G)$, is the ...
1
vote
1answer
83 views

Vectors which average to zero over any graph neighborhood

Given an undirected connected graph on $n$ nodes, let $S$ be the subspace of vectors $x \in \mathbb{R}^n$ which satisfy $$\sum_{j \in N(i)} x_j = 0,$$ for all $i=1, \ldots, n$. Here $N(i)$ is the set ...
1
vote
2answers
56 views

Maximal Minimum Weight DAGs

In the case of undirected, connected graphs the name for the maximal cycle-free subgraph of minimal weight is called Minimum Spanning Tree, and the efficient algorithms for their calculation are well ...
1
vote
0answers
59 views

A property of minimal prime ideals in rings with finite chromatic number

Let $R$ be a commutative ring with identity. There are so many ways to associate a graph to $R$. Consider this: take the elements of $R$ (All elements including zero) as vertices an two distinct ...
3
votes
1answer
57 views

Complexity of counting MAXCUT in planar graphs — seemingly contradicting claims

Confusion is likely. Appears to me two papers give contradicting claims about the complexity of counting MAXCUT in planar graphs. Exact Max 2-SAT: Easier and Faster p. 6 However, counting the ...
1
vote
0answers
48 views

Directed graph Laplacian with exactly one negative eigenvalue

Let $G$ be a digraph with adjacency matrix $A =(A_{ij})$ where $A_{ij}=1$ if and only if there is a directed edge $i \to j$ and $A_{ij}=0$ otherwise. Let $D= (D_{ij})$ be the degree matrix with ...
5
votes
1answer
87 views

Characterization of non-isomorphic graphs but isomorphic total graphs?

Given a graph $G$, the total graph of $G$, denoted $T(G)$, is the graph with vertex set $V(G) \cup E(G)$, where $a$ and $b$ are adjacent in $T(G)$ if and only if they are adjacent or incident in $G$. ...
0
votes
0answers
30 views

Linear Program for Single Source Shortest Paths Tree

This question originates in quick, however wrong, idea to calculate a shortest paths tree in the presence of negative cycles. The essential motivation was that a linear program would determine binary ...
4
votes
1answer
140 views

Probability bound for perfect matching

Let $p<1$ be a constant. Consider two sets $A,B$, each with $n$ vertices. For each pair $(a,b)\in A\times B$, the edge between $a$ and $b$ appears with probability $p$, independently of the ...
2
votes
1answer
69 views

Spectrum of Laplacian matrix of an infinite tree graph

I'm having difficulty understanding a fact stated in a research paper I'm reading. Namely, let $T$ be a tree with all nodes of degree $4$ (ie, the root has $4$ daughter nodes and all other nodes have ...
2
votes
1answer
162 views

Extracting a full rank matrix from a 0-1 matrix

If $A$ is a $n\!\times\!n$ $0$-$1$ matrix of rank $k<n$. If ever possible, what would be an efficient way of extracting a full rank $k\!\times\!k$ sub-matrix of $A$ by removing columns and rows of ...
3
votes
1answer
76 views

Generalizations of the Triangle Removal Lemma to smaller exponents

The Triangle Removal Lemma states: For all $\epsilon > 0$, there is a $\delta > 0$ such that any graph on $n$ vertices with at most $\delta n^3$ triangles may be made triangle-free by ...
1
vote
0answers
94 views

The lattice of graphs under vertex abstractions

I am curious to know if the following structure has been studied, or if anything similar is in the literature. For $n \in \mathbb{N}$, let $G = ([n],E)$ be a digraph. A partition of a subset $V$ of ...
2
votes
0answers
93 views

Detecting Negative Cycles in Undirected Graphs

I recently faced the problem of quickly detecting negative cycles in undirected, weighted graphs. Resorting to the Bellman-Ford Algorithm, as commonly suggested, turned out to be very inefficient and ...
0
votes
0answers
16 views

Does this transformation graph to multigraph keeps some (multi)graph invariants related?

Consider the following transformation graph $G$ to multigraph $G'$. $V(G')=V(G)$. For the edges of $G'$ add a clique of $V(G')$. For each edge $e \in E(G)$ add parallel edge $e'$. So $G'$ is clique ...
3
votes
0answers
73 views

Graph adjacency grouping with geometric criteria

I start with a list of adjacent tetrahedra, where there are tight seals to one another along faces for two tetrahedra that are adjacent. The vertices belonging to these faces for both tetrahedra are ...
-3
votes
2answers
143 views

Does every 3-regular bridgeless graph have a perfect matching? [closed]

Let $G$ be a simple $3$-regular (every vertex has degree $3$) $2$-edge connected graph. Does $G$ contain a perfect matching?
0
votes
0answers
15 views

Complexity of computing the multivariate Tutte polynomial of clique where each edge have distinct label

The multivariate Tutte polynomial $Z_G(q,v)$ is generalization of the Tutte polynomial and each edge is labelled by variable $v_e$. $Z_G(q,v)$ is linear in $v_i$. Let $G$ be a clique where each edge ...
1
vote
0answers
20 views

Complexity of computing the Tutte polynomial of multigraph when the Tutte polynomial of the underlying simple graph is known

Let $G$ be multigraph with $l$ loops and $m$ multiple edges and $G'$ be the underlying simple graph (loops and multiple edges removed). Assume the Tutte polynomial of $G'$ is given. Q1 What is ...
0
votes
0answers
30 views

Non-adjacent Pair of Edges with Minimal Weight Sum

Given an weighted, undirected Graph $G(V,E)$ without loops or parallel edges, what is the complexity of determining a pair of non-adjacent edges, whose sum of weights is w.l.o.g. minimal? ...
2
votes
1answer
115 views

Name for the set of vertices with the same neighborhood as another vertex

Suppose $\Gamma$ is a simple graph and $N_{\Gamma}(g)=\{x\in V(\Gamma)|x\sim g\}$ is the neighborhood of $g\in V(\Gamma)$. Then consider $$\mathbb{S}=\{y\in V(\Gamma)|N_{\Gamma}(y)=N_{\Gamma}(g)\}.$$ ...
1
vote
0answers
26 views

Some confusion regarding the definition of NPO reduction

I've seen the following definition in a paper on approximation preserving reductions. Definition:Let $\pi_{1}$ and $\pi_{2}$ be two NPO maximization problems. Then we say that $\pi_{1} \leq_{R} ...
2
votes
1answer
86 views

Orthogonal embeddings and edge lengths

I'm interested in orthogonal embeddings of graphs into the 2-dimensional, i.e where vertices are placed at integer co-ordinates and edges are routed along the grid lines and are not allowed to ...
0
votes
1answer
57 views

Reference Request: Graph Edge Density

I was curious if there was a reference which answers the question, What is the maximum number of edges in a graph $G$ with $n$ vertices which does not contain a $5$-cycle? $k$-cycle? The analogous ...
2
votes
0answers
110 views

Formulating shortest path as submodular minimization

I'm curious about the general question of whether any combinatorial optimization problem with polynomial time solution can necessarily be reformulated as minimizing a submodular function. The answer ...
2
votes
1answer
64 views

Minimum Edge Density given a particular condition

Consider a graph with $n$ vertices such that if one takes any 4 vertices there are at most 4 edges among these 4 vertices (Notice that there are 6 "possible" edges among these 4 vertices). What is the ...
2
votes
1answer
187 views

Variants of Szemeredi's regularity lemma

I've noticed that the name 'Szemeredi's regularity lemma' is used for several closely related yet different statements about graphs. Specifically, I'm interested in the distinction between two of ...
2
votes
1answer
175 views

Existence of Spanning Tree implies Well Ordering Principle

Every connected graph has a spanning tree. Every non-empty set can be well ordered. Basically I am trying to show that statement 1 implies statement 2. What I tried is as following: Let $X \ne ...
4
votes
1answer
120 views

Edge Reconstruction Conjecture

I have seen this question asked at least once before, but not with any real answers. I was reading about the various reconstruction conjectures and equivalents, and I saw that the reconstruction ...
3
votes
0answers
79 views

Is there a Havel-Hakimi for geometric graphs?

Suppose that we are given $n$ points in the plane, with a degree prescribed for each, and the question is whether we can place a geometric graph on them. Is there an efficient algorithm for this? ...