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

9
votes
1answer
272 views
+100

Coherence between different ranking methods of a graph's vertices

Given a (connected) graph $G$ it is natural to want to rank its vertices, with the more "central" vertices ranked higher. Two natural ways of doing it are: By the degrees. By the entries in a ...
1
vote
0answers
32 views

Possible ways to create a graph representation from a distance matrix (through approximation)

Forgive me, Im not math professional, but a computer scientist at the beginning of my base research from my thesis, so bare with me if I miss something blatantly obvious. I have a Euclidean distance ...
5
votes
1answer
400 views

Edge Covering Shortest path

I have a graph of Points G(V,E) and I want to find the shortest path covering all the edge, I want the minimum number of edge repetitions . Which is the best way to reduce this problem to well know ...
1
vote
1answer
184 views

Categories with binary relations as objects

For the category of functions, pairs of functions making commutative diagrams are the canonical morphisms $\alpha:f\rightarrow g$. For binary relations there is an alternative, to consider the ...
1
vote
1answer
64 views

Difference between straight and piecewise linear and continuous embeddings of graphs / complexes in d-dimensional space?

Here is my main question: what is the difference between "straight" and "piecewise linear" and "continuous" embeddings of graphs/complexes in d-dimensional space? Moreover I would like to know if any ...
3
votes
0answers
46 views

Different definitions of linkless graphs

Robertson, Seymour and Thomas defined linkless embeddings of graphs as follows: An embedding of $G$ is linkless if every pair of disjoint circuits of $G$ have zero linking number (see here). However ...
4
votes
2answers
122 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 ...
2
votes
1answer
57 views

Coloring graph such that the coloring classes are not maximal independent sets

Let $G$ be a (finite or infinite) simple graph. We let $\mathrm{Ind}(G)$ be the collection of independent sets. For any cardinal $\kappa$ and coloring map $\chi: G\to \kappa$ we have ...
2
votes
0answers
68 views

second smallest eigenvalue Laplacian - submodular set function

Let $G$ be a connected unweighted undirected graph. In addition, let $\lambda_2(L(G))$ be the second smallest eigenvalue of the Laplacian matrix of graph $G$. Is $\lambda_2(L(G))$ a submodular set ...
1
vote
2answers
152 views

Cubic Cayley (undirected) graphs

The generators of a cubic Cayley graph always include at least one involution, since 3 is odd. There are then two possibilties for the other two generators: either (A) they are also (distinct) ...
0
votes
0answers
62 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
vote
0answers
52 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: ...
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
vote
0answers
83 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
340 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
votes
0answers
58 views

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

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
66 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
votes
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
177 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
votes
3answers
579 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
vote
1answer
96 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
votes
2answers
113 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 ...
4
votes
0answers
85 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
586 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
132 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
159 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
161 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
233 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
291 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
votes
0answers
569 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
votes
0answers
72 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
79 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
527 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 ...
6
votes
2answers
183 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
votes
0answers
57 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
115 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
64 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
173 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
112 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
62 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
140 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 ...