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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0answers
55 views

A certain Acyclic Partition of a digraph

Has the following object been defined in the literature? What is it called? And what literature studies it? Are there other characterizations of this? What properties are known? Let $G$ be a directed ...
1
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0answers
40 views

Programmatically recognizing symmetries of a polyhedron

I asked this question on MSE a month ago, but nobody was able to answer it, so I guess the question is more difficult than I initially thought: ...
2
votes
1answer
60 views

Minimum number of transitive paths in tournament

Let $T$ be a tournament with $n$ vertices (i.e., between every pair of vertices there exists an edge in exactly one direction.) For any $k$, the vertices $A_1,A_2,...,A_k$ form a transitive path if ...
0
votes
1answer
131 views

Efficient isomorphic subgraph matching with similarity scores

I'm a computer vision PhD student, and I'm looking for an efficient approximation to the following problem, which could end up helping in image to image matching. Failing that, pointers to relevant ...
1
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0answers
68 views

Kempe chain color swaps in a partially colored map

Crossposted from math.stackexchange.com: http://math.stackexchange.com/questions/904932/kempe-chain-color-swaps-in-a-partially-colored-map Question: In this partially Tait's colored map, using ...
8
votes
1answer
335 views

What is the correct statement of this Erdös-Gallai-Tuza problem generalizing Turan's triangle theorem?

In Zsolt Tuza's Unsolved combinatorial problems I, Problem 46 is the following conjecture: Let $G$ be a graph on $n$ vertices. Let $\alpha_1$ be the maximum number of edges of $G$ such that every ...
-1
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0answers
55 views

How to find number of nonisomorphic simple graphs from a number of edges and a number of vertices? [on hold]

How many nonisomorphic simple graphs are there with __ vertices and _ edges? (Fill in the blanks with numbers, say 6 and 4) I've tried to use incident matrices to no avail.
2
votes
1answer
60 views

When does a hypergraph represent maximal independent sets?

Let $G = (V,E)$ be a simple graph. Then, we can view the set of maximal independent sets (or the set of maximal cliques) as a hypergraph $H = (V, E')$. This is quite a useful device when connecting ...
4
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1answer
209 views

How to find or constrain “particularly good” (two-sided) spectral expanders?

I'm new to graph theory, but a response to a question I asked a while ago introduced me to the concept of expander graphs. A k-regular graph (henceforth "graph") on n nodes has eigenvalues k = λ1 ≥ ...
7
votes
1answer
171 views

A conjecture about strongly regular graphs

Let $G \neq K_{v}$ be a $(v,k,\lambda,\mu)$ strongly regular graph. After perusing through Brouwer's tables of parameters I have formed the conjecture $$\lambda-\mu \leq \frac{k}{2}.$$ So far I have ...
25
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3answers
575 views

Removal of non-isomorphic edges results in the same graph

There exists a (simple unlabeled) graph on 6 nodes with a pair of non-isomorphic edges (i.e., there is no graph automorphism that sends one edge into the other) such that removal of either of them ...
1
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1answer
95 views

edge graph reconstruction conjecture : set vs multi set

Why is the edge reconstruction conjecture stated with the deck defined as the multi set of graphs formed by deleting one edge? Can someone give an example of two graphs such that the edge deleted ...
0
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2answers
110 views

Terminology for beads/necklace/bracelet problem [closed]

I'm new to mathoverflow but hopefully anyone here can point me in the right direction. The problem is as follows, imagine you have 4 beads, lets give them numbers 1,2,3,4. Now I want the unique ...
2
votes
1answer
77 views

What is the minimal girth of a cayley graph for Alt(n) in which the girth relator is not a proper power?

First, a definition: a girth relator for a Cayley graph is a word that you get by reading the edge labels along a shortest loop. The girth relator is not usually unique, though it is often unique up ...
3
votes
0answers
81 views

Full-rank factorization of the graph Laplacian

Is there a combinatorially meaningful full-rank factorization of the Laplacian matrix of a graph? The usual factorization $L=BB^{T}$, where $B$ is an oriented incidence matrix, is full-rank if and ...
10
votes
1answer
583 views

A generalization of the triangle counting problem for simple weighted graphs

One nice identity is $$tr(A^3)/6$$ which counts the number of triangles of a graph represented with adjacency matrix $A.$ It also implies that triangle counting can be performed in subcubic time. ...
3
votes
1answer
130 views

Contracting a planar graph to a (multiply-edged)-tree

Given a planar graph (no loops, no multiple edge), is it always possible to perform edge contractions* in order to obtain a graph $T$ which has no loops, but if one forgets the multiplicity of its ...
6
votes
1answer
155 views

Duration and critical groups order in sandpile models and chip firing games

The famous chip firing game (which is closely related to sandpile models) goes like this: Place chips at the vertices of a graph. REPEATEDLY: If a vertex $v$ of degree $d_{v}$ has at least ...
8
votes
1answer
155 views

Looking for history on a theorem of clique intersections

I have a short paper I'm working on where I prove: Theorem: Every graph on (2t-1) vertices with no (t+1)-clique has a vertex that is contained in every t-clique. By "t-clique", I mean a complete ...
5
votes
2answers
227 views

Matching number and chromatic number

If $G$ is a (finite) graph, denote with $\mu(G)$ the size of any maximum matching in $G$ (this number is also called the "matching number" of $G$). For odd integers $n$ we have $n=\chi(K_n) = ...
5
votes
1answer
290 views

How does the distribution of Erdős number evolve over time ? How to build a model to fit the real data ?

Let $E(n,t)$ be the number of mathematicians with finite positive Erdős number $n$ at time $t$. As old mathematicians leave, and new mathematicians come, how does $E(n,t)$ change over time ? We can ...
0
votes
1answer
40 views

average number of cycles and closed walks length k in incomplete directed graph

I asked this question before, but formulation was poor. I've deleted previous question and reformulate it again. Let graph $G=(N,p)$ is finite simple incomplete directed graph of size $N$ (multiple ...
2
votes
1answer
146 views

Cospectrality and dimension of graphs

Firstly, I apologize if the question is long. I appreciate any helpful answers and ideas. In the following all graphs are simple and connected. Let $G$ be graph with vertex set ...
2
votes
0answers
46 views

Looking for similar centrality measurement on graph

I'm working on a graph problem somehow related to centrality measurement. Given an undirected, unweighted tree $T$ and a vertex $v$, let $D_i(v)$ be the set of vertices in $T$ that are i hops from ...
14
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0answers
567 views

A Conjecture About Directed Graphs that are the Union of Two Trees

Let D=(V,E) be a directed graph that is the union of two edge-disjoint directed spanning trees. Suppose that There no subset X of vertices so that there is precisely one directed edge from X ...
5
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0answers
70 views

Mapping graphs to ordinals

Robertson-Seymour theorem implies that graph minor relation is a well-quasi-ordering, which means (among other things) that this relation can be extended to a well-order, and other result says that ...
0
votes
1answer
78 views

Finding node-disjoint routes with mutually exclusive nodes in graphs

I have the following problem. I would like to know if it reduces to some standard problem in Graph theory. Particularly, I would like to know whether it is NP-hard, if yes, how to prove its ...
5
votes
1answer
343 views

Flow of an integer

I've stumbled across this family of flow networks, and posted the sequence of maximal flows to OEIS: A238729. I can't find any reference to it either. Has anyone seen it? Here is the description: ...
11
votes
0answers
695 views

Hamiltonian cycles and fundamental groups

I'm interested in the interplay between the Hamiltonian cycles of graphs and the compact surfaces they embed in. I was doing some reading on the Lovász conjecture for Cayley graphs, I started noticing ...
17
votes
1answer
521 views

Monomer-Dimer tatami tilings need better relationships with other math. Summary of results.

A monomer-dimer tiling of a rectangular grid with $r$ rows and $c$ columns satisfies the \emph{tatami} condition if no four tiles meet at any point. (or you can think of it as the removal of a ...
5
votes
2answers
176 views

Cubic graphs whose 2-factors all have the same cycle type

Let $G$ be a bridgeless cubic graph. I am interested in such graphs where all 2-factors are isomorphic (as graphs), i.e. have the same partition as cycle type. We'll say that this partition is ...
2
votes
0answers
47 views

When polynomial GI implies polynomial (edge) colored GI?

(edge) colored graph isomorphism is GI which preserves the colors (of edges if it is edge colored). There are several reductions using transformations/gadgets of (edge) colored GI to GI. For edge ...
2
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0answers
55 views

Properties of a smallest tournament with domination number $k$

For some tournament $T$, let $\gamma(T)$ denote the cardinality of a smallest dominating set of $T$. Denote by $f(k)$ the minimum number of vertices of a tournament $T$ having $\gamma(T) = k$. From ...
2
votes
0answers
110 views

NP-hard proof of optimization version of exact cover [closed]

Exact cover is NPC. http://en.wikipedia.org/wiki/Exact_cover#Equivalent_problems Given a collection $\mathcal{S}$ of subsets of a set $X$, an exact cover is a >>subcollection $\mathcal{S}^*$ ...
4
votes
1answer
225 views

If a graph invariant is NP-Hard, is its “deck ratio” NP-Hard as well?

This question is inspired by the Graph Reconstruction Conjecture. Suppose that $\psi$ is some graph invariant and that it is NP-Hard. There is a plethora of examples, of course. Now define ...
5
votes
1answer
297 views

Bipartite Graphs arising from two k-partitions of a given Graph

Let $G$ be an $n$-chromatic connected graph. Let $(V_1, V_2, \cdots, V_n)$ and $(U_1, U_2, \cdots, U_n)$ be two partitions of $V(G)$ corresponding to proper n-colorings of $G$. Consider the bipartite ...
0
votes
1answer
61 views

N random walkers that hit node v in a graph

Consider a finite, undirected graph G, with uniform edge weights. Assume that there are n number of random walkers that will start at different nodes (lets say n=3, hence the random walkers will start ...
2
votes
1answer
172 views

Coloring vertices in a cubic lattice graph and counting edges between vertices of identical and vertices of distinct coloration

Take an $A \times B \times C$ cubic lattice graph $G$, and paint $k_1$ vertices with color $c_1$ & $k_2$ vertices with color $c_2$, where $(k_1 + k_2)$ is equal to the total vertex count. Let ...
0
votes
1answer
111 views

Reverse optimization of a minimum cost flow network

Given an undirected graph $(V,E)$, with $W$ as the weight of each edge, and a convex cost function $F(X)$, such as $|X|^k$ ($k>1$). The cost to send $x$ unit of flow through edge $e_i$ is defined ...
1
vote
1answer
61 views

Two definitions of genus for circle graphs

In the (very nice) article of Goldstein and Turner untitled Applications of Topological Graph Theory to Group Theory, the following definitions can be found: Definitions: A circle graph is a pair ...
6
votes
1answer
138 views

Smallest Connected Graph for Given Degree Sequence

For a given integer sequence $(d_1, d_2,...,d_n)$, a natural question is if such a sequence is graphical, i.e. is a degree sequence of some graph. According to Erdős–Gallai theorem, A sequence of ...
6
votes
1answer
54 views

Is a connected graph uniquely determined by its weighted 2-step graph?

This is an extension of a previous question: Is a graph uniquely determined by its weighted 2-step graph?. In that question I asked about arbitrary graphs; in this question I restrict to connected ...
7
votes
2answers
247 views

What is the number of noncrossing acyclic digraphs?

A noncrossing graph on $n$ vertices is a graph drawn on $n$ points numbered from $1$ to $n$ in counter-clockwise order on a circle such that the edges lie entirely within the circle and do not cross ...
3
votes
1answer
209 views

How hard is a variant of graph automorphism problem?

I'm interested in a variant of graph automorphism problem (which is prime candidate for $NP$-Intermediate problem). Restricted GA Input: Given an undirected graph $G(E, V)$, and $\epsilon |V|/2$ ...
0
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1answer
41 views

Resource on characterizations or properties of traceable graphs

I am looking for some resources that provide information on traceable graphs(paths containing a hamiltonian path). I have found a lot of information on hamiltonian graphs, but none on traceable ...
2
votes
0answers
158 views

Computing the chromatic polynomial of graph modulo $x-3$

The chromatic polynomial of graph $P(G,x)$ is univariate polynomial which counts the number of colorings of $G$ with $x$ colors for natural $x$. Graph is not $k$ colorable iff $P(G,k)=0$. The ...
0
votes
0answers
69 views

asymptotic notation with graph colouring

This is my first ever post so I hope this is an appropriate question. Basically I am looking at the paper here: http://homepages.math.uic.edu/~mubayi/papers/biclique.pdf Namely theorem 5. Now, feel ...
30
votes
5answers
2k views

Does every bipartite graph with 512 edges have an induced subgraph with 256 edges?

Suppose we have a (simple) bipartite graph with $2^k$ edges. Is it true that there is a subset of the vertices such that their induced subgraph has exactly $2^{k-1}$ edges? I know that the answer is ...
11
votes
0answers
370 views

Is this graph polynomial known? Can it be efficiently computed?

I am a physicist, so apologies in advance for any confusing notation or terminology; I'll happily clarify. To provide a minimal amount of context, the following graph polynomial came up in my research ...
6
votes
0answers
66 views

Set system with prescribed intersection sizes

Questions: What is the asymptotic maximal size of a $4$-uniform (every set has 4 elements) set system $\mathcal{A}$ of subsets of $[n]$ such that, no two sets have size of their intersection $2$? In ...