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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1answer
93 views

Finding a sufficiently large complete bipartite subgraph using matrix counting

I'm trying to reconstruct the proof using matrix counting that there exists two subsets $A,B$ of $\{1,\cdots,N\}$ with $\#A=\#B$ such that for any $a\in A$ and $b\in B$, $a+b$ is prime, and $\#A=\#B$ ...
3
votes
0answers
86 views

Counting Problems where Labeled is Known but Unlabeled is Not

Cayley's formula states that the number of labeled trees on $n$ vertices is $n^{n-2}$. There are many nice proofs of this compact formula. To contrast, counting unlabeled trees is considerably ...
1
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3answers
100 views

Is there a formula for the number of labeled forests with $k$ components on $n$ vertices?

Cayley's formula states that the number of labeled trees on $n$ vertices is $n^{n-2}$. My question is: Is there a generalization of this formula for forests? Let $f_{n,k}$ denote the number of ...
0
votes
2answers
50 views

Coloring maximal independent sets with 1 color

Let $G$ be a graph and $M\subseteq V(G)$ be a maximal independent set. Is there a coloring $c:V(G)\to\chi(G)$ such that $c$ is constant on $M$? (The answer is positive for graphs with infinite ...
1
vote
1answer
40 views

Difference between modularity and clustering in graphs

The definition of modularity given on wikipedia is as follows. http://en.wikipedia.org/wiki/Modularity_(networks) Modularity is one measure of the structure of ...
2
votes
1answer
132 views

Directed Minimal Cuts in a DAG

I'm looking for information about minimal directed cuts (dicuts) in (connected) DAGs (directed acyclic graphs). A dicut in a directed graph, is a cut $(P_1,P_2)$ in which all edges in $E(P_1,P_2)$ ...
1
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0answers
111 views

lower bound for difference between max cut and min cut

Let $G=(V,E)$ be a graph on $n$ vertices with edge weights $w=(w_e)_{e\in E}$. Let $M^+$ and $M^-$ be the maximum and the minimum cut values, i.e. ...
9
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5answers
304 views

Do there exist sparse graphs with large crossing number?

Does there exist a sequence of graphs $\{ G_n \}$ such that $G_n$ has $n$ vertices, the number of edges of $G_n$ is $O(n)$, and the crossing number of $G_n$ is $\Omega(n)$? In particular, do ...
5
votes
0answers
71 views

Independent domination number for grid graphs

Let $G_{n,m}$ be the $n \times m$ grid graph, i.e. $G= P_n \Box P_m$, and $T_{n,m}$ the $n\times m$ torus grid graph, i.e. $G= C_n \Box C_m$, where $P_n$ and $C_n$ indicate the path graph of length ...
4
votes
1answer
139 views

Expectation of ratio of functions of i.i.d. Bernoullis: a concentration question

Consider the following $n \times n$ symmetric matrix of i.i.d. Bernoulli random variables, $X_{ij}$. For $i=1,...,n$ and $i<j\le n$. Let $X_{ij} \sim \text{Bernoulli}(p)$ when $i \ne j$ ($p$ ...
1
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2answers
80 views

Regular graphs with strongly regular edge colorings

Given a positive integer $c>1$, for what parameters $(v,k,\lambda,\mu)$ does there exist a $c k$ regular graph on $v$ vertices that can be given an edge coloring with $c$ colors, such that the ...
1
vote
1answer
88 views

Can we count isospectral graphs?

On n-vertices, how many isospectral graphs exist? [..I saw this previous "historic" discussion between two of the stalwarts in this field - Operation on Isospectral graphs ] Given a graph are ...
6
votes
2answers
117 views

Number of edges in linklessly embeddable graphs

Consider graphs over $n$ nodes. What is the maximum number of edges of a linklessly embeddable graph? A more general question is the following. Given $\mu$ what is the maximum number of edges of ...
3
votes
0answers
122 views

What do we know about isospectral Cayley graphs?

Let $\Gamma_1=\Gamma_1(G_1,S_1)$ (respectively $\Gamma_2=\Gamma_2(G_2,S_2)$) be a Cayley graph for the group $G_1$ and the finite generating set $S_1$ (respectively $G_2$ and the finite generating set ...
0
votes
0answers
51 views

Induced subgraphs on a Laminar family of vertices with constant diameter

$X$ is a family of subsets of $V$. $X$ is called a Laminar family on $V$ if for all $A,B\in X$, either $A\cap B=\emptyset$, $A\subset B$ or $B\subset A$. Let $X$ be a family of subsets of $V$. A ...
0
votes
1answer
70 views

Paper on unit disk graphs

I was wondering if anybody knows of a 'link' to the paper by Marathe 1995 et al on analysis of the greedy algorithm for finding a Max independent set in Unit Disk Graphs?
2
votes
0answers
31 views

“drift” of a random graph $G(n,p)$ with $p=\alpha\ln{n}/n$

Suppose $G\sim G(n,p)$ with $p=\alpha\ln{n}/n$ for some large constant $\alpha$. I wish to show a certain "drift" property of $G$, which can roughly be phrased as follows: if $u$ is "not that far" ...
1
vote
1answer
76 views

Coloring algorithm maximising color difference between neighbors

Consider a graph and a set of ordered colors ${\cal C} = \{1,2,\cdots,C\}$. I want to color each node $i$ with a color $c_i\in{\cal C}$ so as to maximize the minimum color difference between two ...
2
votes
4answers
337 views

Algorithm to find the “optimal” path in a given graph

Assume that $G=(V,E)$ is an undirected connected graph and that $H: V \to \mathbb R$ is a function that assign at each vertex $v \in V$ its height $H(v)$. Think of the pair $(G,H)$ as an energy ...
2
votes
1answer
79 views

Existence of neighborhood inclusion for 4-chordal graphs

Let $N(v)$ be the (open) neighbourhood set of a vertex $v$, and let $N[v]$ be the closed neighbourhood set of $v$. A graph $G$ is called 4-chordal if $G$ has no induced cycle with five or more ...
2
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3answers
513 views

a colouring / matching problem

While trying to find a bijection which preserves various combinatorial statistics, I was led to the problem below. Very much to my surprise, a closely related question, Coloring summands of given ...
0
votes
1answer
176 views

Coin graph is 4-colorable

How can we prove that a coin graph is 4-colorable???Also, can we find any example of an non-3-colorable coin graph.
3
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1answer
106 views

Independent Sets in random geometric graphs

I was wondering if a lot is known about independent sets in Random Geometric graphs? Most google searches don't bring up much. In addition I'm interested in any algorithms used for finding independent ...
1
vote
2answers
137 views

Non-DS circulant graphs

Let $p$ be a prime number greater than or equal to 11. Are there any cospectral non-isomorphic graphs with circulant graphs on $p$ vertices. Which circulant graphs over prime number of vertices ...
2
votes
0answers
48 views

Weighted graph similarity

I have the following problem. Consider an undirected biconnected graph with $n$ vertices and $m$ edges ($n \leq m \leq n(n-1)/2$). The $m$ edges of the graph are then "populated" by integer weights ...
2
votes
1answer
195 views

On the maximum number of $t$-subset of $\{1,\ldots, n\}$ having pairwise singleton or empty intersections

Let $n$, $t \in \mathbb N$ two natural numbers such that $0<t<n$, and let $A$ be a set of $n$ elements. We call a quasi-partition or q-p of $A$ a subset $W \subset \mathcal P(A)$ such that we ...
4
votes
0answers
74 views

Subgraphs of planar trivalent graphs

Let's think about planar trivalent graphs. (Or you can dualize and think about triangulations if you prefer.) It's easy to come up with a list of 'planar trivalent graphs with boundary' such that at ...
1
vote
2answers
115 views

Minimum number of unlabeled planar graphs

Does anybody know if there is any research on a lower bound on the number of (non-isomorphic) unlabeled planar graphs with maximum node degree $d$? Alternatively, a lower bound on the number of all ...
-4
votes
1answer
66 views

Survey on Hypergraph Theory [closed]

I want to learn about hypergraphs.Can anyone help me with some links, articles and surveys. Thanks
3
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0answers
78 views

More 3-connected cubic graphs with all 2-factors of same cycle type?

The setup is as in this question: Let $G$ be a 3-connected cubic graph. If all 2-factors of $G$ are isomorphic (as graphs), i.e. all have the same partition $\pi$ as cycle type, we'll say that $G$ is ...
2
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0answers
66 views

Problems Solvable/Decidable by Counting Shortest Paths in Graphs

This questions is based on a dispute, whether it would be possible to calculate 'nice' routes in Manhattan, if the road network is assumed to be a rectangular grid and, that 'nice' means that there is ...
5
votes
1answer
305 views

Are there number-theoretic graphs that are far from being isomorphic

I say that two graphs $G_1=(V_1,E_1)$, $G_2 = (V_2,E_2)$ with the same number of vertices, edges, are $\epsilon$-far from being isomorphic, if for any bijection between $V_1$ and $V_2$, the fraction ...
0
votes
1answer
90 views

Comparing ideals in posets

Consider a partially ordered set $P$, and two upper sets $U_1$, $U_2$ in this poset. What are some natural ways to measure how equal these two upper sets are? This question arise naturally in the ...
5
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0answers
160 views

(Connected) Cayley graphs of PSL(2,q) from (2,3,n)-triples

Let $G = PSL(2,q)$. I'm interested in the Cayley graphs of $G$ generated by triples $(A,BAB^{-1},B^{-1}AB)$, where $A, B \in G$ are elements of order $2, 3$ respectively: such a triple generates all ...
1
vote
0answers
76 views

Disks in Flat Embeddings of Graphs in $\mathbb{R}^3$

Robertson, Seymour and Thomas proved that any linkless graph $G$ has a flat embedding in $\mathbb{R}^3$ (see for example A survey of linkless embeddings). An embedding of $G$ is flat if for any cycle ...
2
votes
1answer
68 views

Hamiltonicity of random graphs with high girth

We say that $G\sim G_{n,f}$ (for $f=f(n)$) if $G$ is chosen uniformly at random from all graphs on $n$ vertices with girth $g(G)\ge f(n)$. Is there any threshold function $F(n)$ such that when $f\ll ...
1
vote
0answers
75 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 ...
3
votes
0answers
64 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 ...
1
vote
1answer
100 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 ...
2
votes
2answers
365 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 ...
3
votes
1answer
79 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
85 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
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2answers
172 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) ...
4
votes
2answers
160 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 ...
1
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0answers
122 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 ...
0
votes
0answers
67 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 ...
2
votes
1answer
77 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 ...
7
votes
1answer
206 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 ...
0
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2answers
157 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
94 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 ...