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, ...

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32 views

Non-negative Polynomials from Polynomial Ideal?

Related to the thread Nonnegativity conditions for a polynomial in two variables? but on more than two variables. Suppose a probability vector $p$ belongs to a compact polytope where for each entry $...
0
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1answer
60 views

Computing canonical forms from orbit partitions

Suppose we know the orbit partition of the vertices of a graph (due to the action of its automorphism group). Is it easy (as in "polynomial time") to generate a canonical form (aka "canonical labeling"...
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0answers
24 views

On balanced bipartite graphs

Of the $2^{n^2}$ balanced bipartite graphs on $2n$ vertices how many of them have $i$ perfect matchings where $i\in\Bbb N\cup\{0\}$ and $0\leq i\leq n!$ holds?
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1answer
48 views

Count Functional digraph [on hold]

Given a set of nodes, how can I count the number of different functional digraph containing a specific number of connected components? With a restriction that no node can have an edge point to itself. ...
0
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1answer
36 views

Length of longest directed circuit in random tournament

Build a random tournament $T=(V,E)$ on $V=\{1,\ldots, n\}$ in the following fashion: for $i < j\in \{1,\ldots, n\}$ let the probability be $0.5$ whether $(i,j)\in E$ or $(j,i)\in E$ (in a ...
4
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0answers
103 views

Automorphisms of an infinite graph built from a finite motif

Suppose we have a lattice $L$ in $\mathbb{R}^n$ for which we choose some fundamental domain $D\subset \mathbb{R}^n$ homeomorphic to a closed ball. Translates of $D$ by distinct elements of $L$ ...
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0answers
88 views

A color interpolation lemma

I need the following "color interpolation lemma". Actually I know a way to prove it, but I'm not very satisfied with that proof. Lemma. Let $G=(V,E)$ be a (properly) colored graph with colors $1, \...
2
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1answer
132 views

Size of automorphism group of random regular graph

If I pick a random regular graph on $n$-vertices and degree $d$ from uniform distribution what is the probability that its automorphism group is of size at least $m$? -- I want to know what is the ...
3
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0answers
64 views

Minimal algebraic degree of symmetric unit distance embedding of Heawood graph

I'm looking at embeddings of the Heawood graph in the plane as unit distance graph. Apparently the first such embedding was given by Gerbracht, 2009 and has algebraic (over the rationals) coordinates ...
0
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1answer
63 views

On some examples of critical families

I'm reading the book on Injective choice functions by Holz, Podewski and Steffens, and I find it to be at the same time well written and quite difficult. It has almost no examples - and in quite a few ...
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3answers
199 views

When can the Cayley graph of the symmetries of an object have those symmetries?

Let $P$ be an object in $\mathbb{R}^n$ with symmetry group $G$. Let $C$ be the a Cayley graph of $G$. When can $C$ be embedded in $\mathbb{R}^m$ so that the embedded graph has the same symmetry ...
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1answer
58 views

Counting and constructing some special planar graphs

We look for the property that a graph is both planar and has a trivial automorphism group. How many non-isomorphic $n$-vertex graphs have such property and is there an $O(n^\beta)$ (at least ...
4
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1answer
97 views

Connection between PageRank and Fiedler vector

This question is on graph clustering. In its simplest form, the eigenvector corresponding to the second smallest eigenvalue of the normalized Laplacian of a graph provides a relaxed solution to the ...
1
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1answer
78 views

How to compute graph ideal or cut ideal of a graph?

Graph ideals are a special case of Stanley-Reisner ideal, explained in Combinatorial Commutative Algebra book by Sturmfels, and graph ideals here. Graph ideals are generated by the minimal paths while ...
1
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0answers
59 views

Non-orientable genus of union of graphs

It is known that the orientable genus of union of two (disjoint) graphs is the sum of their genus. So, it is natural to ask What can be said about the non-orientable genus of union of two (disjoint) ...
3
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1answer
81 views

Bounds for number of edges of a graph, given girth and number of vertices

In reading a paper, I came across an affirmation "a graph of girth $g$ and $q$ vertices has at most $q^{1+(O(1)/g)}$ edges" In a previous question I asked in this site about it, I was reffered to a ...
2
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1answer
84 views

Existence of a Connectivity Polynomial for a simple graph?

I try to find a polynomial for an arbitrary simple graph $G$ that tells whether the graph is connected or not. A graph is st-connected if you can find a path between a vertex $s$ and a vertex $t$ -- ...
2
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0answers
57 views

Induced matchings in a bipartite graph with every matching having the same number of edges

Suppose $n,k$ are positive integers such that $k\mid n$. Consider a bipartite graph $H=(A,B,E)$ such that $|A|=|B|=n$ and the edge set $E$ consists of the union of $m(H)$ induced matchings with every ...
0
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1answer
53 views

Looking for source: Max num of edges of graph with given number of vertices and given girth

In a paper I am reading, the author states: "It is simple and well known that a graph of girth $g$ and $q$ vertices has at most $q^{1+(O(1)/g)}$ edges" He says that a proof can be found on Extremal ...
7
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1answer
148 views

Does an expander remain an expander after removing few vertices and edges?

Consider a sequence of expander graphs ($G_n$); say $G_n$ has $n$ vertices. Remove $o(n)$ vertices (and the edges emanating from these vertices) and cut $o(n)$ edges. Call $G'_n$ the largest connected ...
8
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1answer
148 views

How many edges can be added to two circles before the graph becomes Hamiltonian?

Start with two $n$-circles $(v_1\cdots v_n)$ and $(w_1\cdots w_n)$ of vertice sets $V$ and $W$, where $n\ge 5$. Add a number of vertex-disjoint edges between $V$ and $W$ (thus no chords) in a way ...
0
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1answer
119 views

Does the shortest distance between two cities of a Traveling Salesman Problem always appear in the answer? [closed]

If I had a list of 4 or more cities, then does the path between the two closest cities always appear in the final shortest route of a TSP Solution? Bill
6
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0answers
100 views

On the use of Weisfeiler-Leman refinement in Babai's GI proof

This question is for those familiar with the methods behind Babai's recent proof that graph isomorphism can be decided in quasipolynomial time. I am a newcomer to the GI problem, so I apologize if my ...
10
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1answer
287 views

Question on a paper by Benjamini/Kozma/Wormald about a “well known fact”

In "The mixing time of the giant component of a random graph" by the aforementioned authors, in the last proof on page 19 it says something along the lines of "It is well known and easy to verify ...
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0answers
78 views

Does the Ruzsa-Szemeredi Theorem also capture graphs decomposable into *nearly* induced matchings?

The well-known Ruzsa-Szemeredi Theorem states that a graph whose edges can be partitioned into $n$ induced matchings has at most $\frac{n^2}{RS(n)}$ edges, for some slow-growing function $RS(n)$. Now,...
2
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0answers
81 views

a continuous analogue of a graph theory question

I am reading a paper and it mentions a continuous analogue of a related graph theory question that people concern. The question is that suppose $E\subset Q=[0,1]^2$ has lebesgue measure $|E|>0$, is ...
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1answer
81 views

Is there currently a known way to construct an injective mapping that transforms finite graphs into discrete geometric objects? [closed]

If there is such a mapping, it seems as though it could turn the graph isomorphism problem from a purely combinatorial problem to a discrete geometric one.
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0answers
25 views

Permutation Invariant Color Class

$G$ is a $d$ regular graph, it has $n$ vertices. $S_n$ acts on $n$ vertices of graph $G$. Question: Does there exist a coloring algorithm for which color classes is invariant under all ...
4
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1answer
213 views

Number of walks on integer lattice with self-edge at zero

Consider the graph with vertices $V=\mathbb Z$ and edges $$E=\{(n,n+1):n\in\mathbb Z\}\cup\{(0,0)\},$$ that is, the usual integer lattice with a self-edge at zero. For some fixed parameters $a,b,n\in\...
0
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1answer
51 views

Bondy and Simonovits Proof for Small Graphs

In their paper, Cycles of Even Length in Graphs (http://renyi.hu/~miki/BondySimEven.pdf), Bondy and Simonovits prove that if a graph $G^n$ has $n$ vertices and at least $100kn^{1+1/k}$ edges then $G^n$...
2
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0answers
66 views

How are Polynomials of Toric ideals Studied with Exponents as ST-cuts?

Topic: Toric ideals on Expected value of Structure Functions in Random Graphs Goal: to understand the toric ideal where the exponents $h_i$ and $s_j$ are st-vertex-cuts of a digraph \begin{equation} ...
8
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3answers
578 views

Is there a 2-connected k-regular graph without Hamiltonian path?

In this paper (Construction 2.6 p860) the authors have built examples of connected $k$-regular graph without Hamiltonian path, but with a cut-vertex (i.e. it is not $2$-connected). Question: Is ...
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2answers
124 views

Relaxed path decomposition of a graph

Definition Given a directed connected graph $G$ without multiple edges or self loops. We call a final path of $G$ a path ending with a vertex with no successor (the path can not be extended anymore) ...
1
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1answer
378 views

Doing graph theory after a thesis in pure mathematics [closed]

I've just went through the 1st year of my PhD in France, it is related to Floer Homology. I didn't know what it was really about at that time, I chosed this subject because I thought it would combine ...
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0answers
25 views

Example/ Explanation of Weisfeiler-Lehman method

I am trying to read An Optimal Lower Bound on the Number of Variables for Graph Identification. On page 3 (4th paragraph), it is written- It might color vertices and edges implicitly by using $k$-...
4
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2answers
287 views

Create a graph with a specified number of spanning trees

I read that one of the current challenging problems in mathematics is constructing a minimal graph with a specified number of spanning trees (say, $k$). However, is there a quick way to create some ...
1
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0answers
51 views

Computing subgraph orbits

I have group $G$ acting on a 4-regular 120 node graph $\Gamma$. I want to compute equivalence classes of connected subgraphs of $\Gamma$, where by equivalent I mean in the same orbit. More ...
10
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0answers
116 views

Asymptotics of subgraph densities in graphons

In Pittel (1989)'s solution to a problem of Knuth (1976) on the expected number of stable matchings between $n$ men and $n$ women under uniform random preferences, it was shown that, as $n \to \infty$,...
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0answers
47 views

Intersection Of Valentine Convex Sets

A set X is said to be m-convex , m integer >=2, if for each set of m points at least one of the associated line segments lies in X. A 3-convex set is sometimes also known as Valentine convex after ...
3
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1answer
249 views

Comparison nauty vs. bliss of canonical form of bipartite graphs

I need to compute canonical forms of many (~10^6-10^8) vertex-facets incidence graphs of polytope. Two rather big examples I want to consider are the 600-cell with 120 vertices and 600 facets (...
1
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1answer
67 views

best known bounds for spectral radius [closed]

There are many bounds for the spectral radius of graphs in terms of no. of vertices, maximum degree, chromatic number etc. I wish to know till date what are the best lower and upper bound for the ...
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0answers
55 views

Recognizing cubic graphs decomposable into 2-factor with given cycle type

Petersen's theorem states that every cubic, bridgeless graph contains a perfect matching. It implies that the edge set $E$ can be partitioned into a perfect matching and a 2-factor. Determining the ...
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0answers
35 views

max-flow at max-cost

I have a flow network with gains. In practical terms, a gain is the opposite of a cost. So, I interested in finding the maximal gain of a network flow, what could be interpreted as finding a maximum ...
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votes
1answer
88 views

Expression for a complex summation involving factorial [closed]

It is known that $\sum_{k = 0}^{n } {n \choose k}(k!) = \lfloor e \cdot n! \rfloor $ But is it known what $\sum_{p = 0}^{n} \sum_{q = 0}^{n - p} {n \choose p}{{n - p} \choose q} p! \cdot q! \cdot (n-p-...
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0answers
37 views

How to find faces of graph? [duplicate]

I have à planair graph and I want to find an algorithm that will find all of the faces of the graph. Thanks you in advance for your answers.
0
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1answer
39 views

Augmention property of matroid along perfect matching

Let M be a matroid of rank k, B a base, X a set of rank rank(X) < k, and P a perfect matching of the complete bipartite graph (X, B). Is it true that there exists an edge (x, b) of P augmenting X (...
3
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1answer
105 views

Base decomposition of matroids

I want to find a generalization of the idea that, in a graphic matroid, every base can be decomposed on the stars (edges adjacent to a vertex). For example one could say that a matroid $M$ of rank $k$...
8
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1answer
276 views

How many chromatic polynomials of planar maps are there?

Let P(n) be the set of polynomials that can occur as the chromatic polynomial of a planar map with n countries. What is known or conjectured about the growth of |P(n)|? PS: Thanks Gerry and Noam, ...
0
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0answers
55 views

Does anyone have a reference for a proof of expansion for this construction?

http://people.seas.harvard.edu/~salil/pseudorandomness/expanders.pdf "Construction 4.26: p-cycles with inverse chords.... The proof of expansion relies on the “Selberg 3/16 Theorem” from number ...
1
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1answer
255 views

Expression for summation involving factorial

It is known that $ \sum_{k = 0}^{n} {n \choose k} = 2^n$ and $ \sum_{k = 0}^{n} {n \choose k} (!k)= n!$. But is it known what $ \sum_{k = 0}^{n } {n \choose k}(k!)$ is equal to?