**6**

votes

**0**answers

357 views

### Graphs with graphic imbalance sequences

Let $G$ be simple undirected graph and $e=uv\in E(G)$.
The imbalance of the edge $e$ is the value $imb(e)=|d(u)-d(v)|$.
Let $M_{G}$ denotes the imbalance sequence (or more correctly, multiset of ...

**6**

votes

**0**answers

289 views

### Maximum fractional chromatic number of a 4-regular triangle-free graph (updated)

Let $G$ be a graph with maximum degree 4 and clique number 2. The fractional version of Reed's Conjecture tells us that $\chi_f(G) \leq 7/2$. But how high can $\chi_f(G)$ be?
The Chvátal graph has ...

**6**

votes

**0**answers

182 views

### When can the Cheeger constant be well-approximated by ``Hamming balls''?

Given a graph G, the Cheeger constant is defined by
$$
\DeclareMathOperator{\Vol}{Vol}
h_G := \min_{S \subseteq V, \Vol S \leq (\Vol G)/2} \frac{|\partial S|}{\Vol S}.
$$
Here, $\Vol S$ is the sum of ...

**6**

votes

**0**answers

606 views

### Is Logical Min-Cut Problem, NP-Complete?

Logical Min Cut (LMC) Problem: Suppose that G = (V, E) is an unweighted digraph, s,t are two vertices of V, and t is reachable from s. LMC Problem states that how we can make t unreachable from s by ...

**6**

votes

**0**answers

342 views

### What is this subclass of $k$-colorable graphs called?

The following property emerged naturally when I was playing with certain generalizations of Kneser graphs.
Let $k>0$ be a natural number. Consider a property $P_k$ of graphs defined as follows.
...

**6**

votes

**0**answers

1k views

### What is the best lower bound for the domination number in regular graphs of girth 5?

The following theorem is a classical result (see [Alon and Spencer, The probabilistic method, 2nd ed., Theorem 1.2.2]):
Theorem: Let $G$ be a graph on $n$ vertices with minimum degree $d$. Then ...

**6**

votes

**0**answers

652 views

### inverse eigenvalue problem on graph laplacian

I am trying to construct a graph Laplacian matrix from a set of eigenvalue. I've read several papers about inverse eigenvalue problems but to be honest I didn't understand clearly. Could somebody ...

**6**

votes

**0**answers

753 views

### Simplicial Representations of (Hyper)Graph Complexes

For graph complexes, which are families of graph [on a fixed number of vertices n] closed under the deletion of edges, there is a natural simplicial complex capturing that information. Specifically, ...

**5**

votes

**0**answers

68 views

### Diameter of the modified bubble-sort graph

The modified bubble-sort graph is the Cayley graph $Cay(S_n,S)$ of $S_n$ generated by $n$ cyclically adjacent transpositions. Thus $S = \{ (1,2),(2,3),\ldots,(n,1)\}$. I was wondering whether the ...

**5**

votes

**0**answers

96 views

### Does squaring a directed random graph more than double its out-degree?

As far as I know, it is an unsolved question
whether or not this is true:
If $G$ is a directed an oriented graph, $G^2$ always has some node whose outdegree is at least
double that of its ...

**5**

votes

**0**answers

179 views

### Unique Nash equilibrium games

Multicast network design game is a special case of a general network design game (http://www.cs.cornell.edu/home/kleinber/focs04-game.pdf) in which there is a target vertex $t$ and $n$ rational ...

**5**

votes

**0**answers

69 views

### Determinantal formulae for Tutte polynomial

Let $G$ be a connected undirected graph. Then the number $ST(G)$ of spanning trees in $G$ equals the following specific value of the Tutte polynomial of $G$: $ST(G)=T_G(1,1)$.
On the other hand, ...

**5**

votes

**0**answers

46 views

### Perfect matchings of a regular, uniform, partite hypergraph

This is in relation to the question here. What, if any, are the known conditions for the existence of a perfect matching for a $r$-regular, $r$-uniform, $r$-partite hypergraph. I specifically ...

**5**

votes

**0**answers

62 views

### Approximating a max-cut's intersection with other cuts

(This is a cross-post from the Theoretical Computer Science Stack Exchange.)
For the purposes of this question, a cut in a graph $G$ is the edge-set $\delta (S)\subseteq E(G)$ between some vertex-set ...

**5**

votes

**0**answers

103 views

### A digraph related to permutations

A finite sequence of distinct real numbers of length $n$ determines a linear order of $\{1,\ldots,n\}$, by mapping position to rank; call this the permutation of the sequence.
Consider the following ...

**5**

votes

**0**answers

67 views

### Linear intersection number and coloring (not chromatic) number

A linear hypergraph is a pair $\pi=(X, L)$ where $X\neq \emptyset$ is a set and $L\subseteq {\cal P}(X)$ has the following properties:
for $e\in L$ we have $|e|\geq 2$;
if $e_1\neq e_2 \in L$ then ...

**5**

votes

**0**answers

35 views

### Isomorphic Hadwiger graphs of connected infinite graphs

Let $G$ be a graph, then we define its Hadwiger graph $\textrm{Hadw}(G)$ in the following way:
$V(\textrm{Hadw}(G)) = \{S\subseteq (V(G): S\neq \emptyset\textrm{ and } S \textrm{ is connected}\}$;
...

**5**

votes

**0**answers

205 views

### When does a “stable” assignment of buyers into goods exist?

Consider a setting of $n$ buyers and $m$ goods.
We have a value matrix $V\in[0,1]^{n\times m}$ specifying how much each buyer values each good (everything is open information here and there is no ...

**5**

votes

**0**answers

105 views

### What's the variance in the Six Degrees model?

Recall the six degrees of Kevin Bacon game. You can even play the game at The Oracle of Bacon, and their search works via Breadth First Search.
I interpret the punchline as saying that if I start ...

**5**

votes

**0**answers

165 views

### Sets of spreads in graphs

Let $G$ be a graph. A $k$-spread is a set of cliques of order $k$ which partition the vertex set (so $k|n$, where $n$ is the number of vertices).
A partial $k$-resolution of $G$ is a set of pairwise ...

**5**

votes

**0**answers

115 views

### Cores of infinite graphs

Let $\kappa$ be a cardinal and let $\textrm{Grph}(\kappa)$ be the set of graphs $G = (V,E)$ such that $V \subseteq \kappa$ and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\} \subseteq V: a\neq b\}$.
We ...

**5**

votes

**0**answers

97 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 ...

**5**

votes

**0**answers

96 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 ...

**5**

votes

**0**answers

179 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 ...

**5**

votes

**0**answers

96 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 ...

**5**

votes

**0**answers

171 views

### Are the roots of chromatic polynomials plus a fixed constant dense in $\mathbb{C}$?

Alan Sokal proved that chromatic roots are dense in the whole complex plane. I.e., if $P(G;z)$ denotes the chromatic polynomial of a finite simple graph $G$ evaluated at $z \in \mathbb{C}$, then ...

**5**

votes

**0**answers

90 views

### Implementations of Tutte polynomial [reference request, of a kind]

This question is not a 100% fit for MO, but it is a serious question that can be viewed as a sort of reference request, and I think fits here more than elsewhere.
I have been asked to write a chapter ...

**5**

votes

**0**answers

123 views

### Complexity of finding three perfect matchings with no edge in common in a bridgeless cubic graph

According to a conjecture:
Conjecture (Fan & Raspaud, 1994) Every bridgeless cubic graph contains three perfect matchings with no edge in common.
Equivalent statement here
Main question:
...

**5**

votes

**0**answers

170 views

### A linear optimization problem on a graph

Let $G=(V,E)$ be a finite graph and let $f$ be any positive function defined on the vertices. Put weights on the vertices $v_{i}$, way $w_{i}$ so that $\sum_{i=1}^{n}w_{i}\leq 1$. Assume that every ...

**5**

votes

**0**answers

205 views

### Graphs with many positive eigenvalues of their distance matrix

Let $G$ be a simple connected graph $D(G)$ its distance matrix and $n_{+}(G), n_{-}(G)$ the number of positive and negative eigenvalues of $D(G)$ respectively.
We call a graph $G$ optimistic if ...

**5**

votes

**0**answers

161 views

### Diameter of subset sum graph

We have a finite set $X$, a weight function $w: X\rightarrow \mathbb{Z}^+$, and constants $k\leq c\in\mathbb{N}$.
Let the weight $w(S)$ of a set $S\subseteq X$ be the sum of the weights of its ...

**5**

votes

**0**answers

99 views

### making a graph well-covered without changing its Shannon capacity

This strongly relates to an earlier question of mine.
Let $G$ be a graph, $\alpha(G)$ its independence number and $\Theta(G)$ its Shannon capacity.
Question: can one 'add new vertices' to $G$ such ...

**5**

votes

**0**answers

150 views

### How does the number of self-avoiding paths between two points scale, in a square/cubic lattice?

Consider two different infinite graphs, whose vertices are drawn from $\mathbb Z^2$ or $\mathbb Z^3$. Let $P_d : \mathbb Z^d \times \mathbb N \to \mathbb N$ for $d \in \{2,3\}$ be the function such ...

**5**

votes

**0**answers

290 views

### Sequence of graphs with small $\lambda_1$ (the smallest nonzero eigenvalue of a regular finite graph)

The combinatorial laplacian on a finite graph $G$ can be defined as $ \Delta: \mathbb{C}^G \to \mathbb{C}^G$ sending the function $f:G \to \mathbb{C}$ to $(\Delta f)(v) = \sum_{v' \sim v} \big( ...

**5**

votes

**0**answers

240 views

### Any approximation algorithms for self-avoiding walks?

I've a graph whose edges are weighted by probabilities, perhaps all equal. I would like to compute the overall probability of traveling between vertices x and y in the graph after I delete each edge ...

**5**

votes

**0**answers

363 views

### A binary bipartite graph - reference request

Consider the bipartite graph whose partite sets are two disjoint copies of $\{0,1\}^n$, with an edge joining $u$ and $v$ if and only if there is no position in which both $u$ and $v$ have $1$; that ...

**5**

votes

**0**answers

399 views

### “sum over labelings” representations of graph polynomials

It seems that there's a general way to go from "recursive" definition of a graph polynomials to "subset expansion" formulas.
Furthermore, polynomials with subset expansion formulas often have a ...

**5**

votes

**0**answers

592 views

### Counting graphs whose vertex induced subgraphs are members of a fixed set

Let $C$ be the class of graphs of girth $g$ or more. $C$ can alternatively be characterized as: $G \in C$ iff each of $G$'s vertex induced subgraphs on less than $g$ nodes is a forest.
We can ...

**5**

votes

**0**answers

444 views

### Natural models of graphs?

Motivation
I want to capture the notion of natural models of finite graphs: How can natural predicates and natural relations on a given natural base class $D$ be defined? If this succeeds the ...

**4**

votes

**0**answers

30 views

### What are some useful invariants for distinguishing between random graph models?

Quite a few probabilistic algorithms for generating random graphs exist in the literature, such as:
The Erdos-Renyi model
The Stochastic Block model
The Watts-Strogatz model
The Barabasi-Alber model
...

**4**

votes

**0**answers

67 views

### “Edge Density” of Infinite Planar Graphs

Given an infinite planar graph $G$, let's denote by $\{H_1,H_2,\dots,H_m\}$ all the labeled graphs on $n$ vertices that appear as subgraphs of $G$. Also let
$$d_n=\frac{\sum_{i=1}^m \#E(H_i)}{nm}$$
...

**4**

votes

**0**answers

48 views

### Existence of certain graph gadget related to coloring odd hole free graph

Wondering about the existence of graph gadget related to coloring
(or 3-coloring) odd hole free graphs.
Let $G$ be simple $k$-chromatic connected graph with two
vertices $u,v$.
Is it possible $G$ to ...

**4**

votes

**0**answers

88 views

### Prove or disprove a claim about covering a polytope by convex polytopes in a certain way

Here is the claim:
Given a polytope $K$ in a unit ball in $\mathbb{R}^d$, there exists a universal constant $C(d)>0$ depending only on $d$ and a countable collection of convex polytopes ...

**4**

votes

**0**answers

96 views

### Application of finding shortest paths on Cayley graphs

For a fixed integer number $m$, Consider Cayley graph defined by all m-cycles in Symmetric group $Sym(n)$.
I know that for $m=2$,
there are some applications of finding shortest paths (or distance ...

**4**

votes

**0**answers

40 views

### Digraph weak connectivity in $O(V)$ space and $O(V+E)$ time

A digraph is called weakly connected if its underlying undirected graph is connected.
You are given a digraph $G$ with $V$ vertices and $E$ edges as a read-only data structure consisting of lists of ...

**4**

votes

**0**answers

94 views

### Counting vertex-permutations of a finite tree which rip all edges

Given a finite tree $T$ with $n$ vertices labelled $1,\dots,n$, we say that a permutation $\sigma$ of $1,\dots,n$ rips all edges if $\{\sigma(i),\sigma(j)\}$
is never an edge for every edge $\{i,j\}$ ...

**4**

votes

**0**answers

134 views

### Connection between connectivity and cohesion of a graph

Tutte [1] proved that, for every $3$-connected graph $G$ and vertices $u$ and $v$, there exists a nonseparating $uv$-path.
A graph $G$ is $t$-cohesive if $G$ is connected, has at least two vertices, ...

**4**

votes

**0**answers

143 views

### What is known about the complexity of this covering problem?

Let $G=(V,E)$ be a graph. A vertex set $X\subseteq V$ is called critical if $X\neq\emptyset$ and no vertex in $V\setminus X$ is adjacent to exactly one vertex in $X$. The problem is to find a vertex ...

**4**

votes

**0**answers

126 views

### Hypercube edge-coloring problem

Question: Is there a pairing (a fixed point free involution) of the vertices of the $n$-dimensional cube graph, and a $2$-coloring of its edges such that the number of color changes needed to get from ...

**4**

votes

**0**answers

127 views

### minimal polynomial for a graph

I wonder if there is any result relating the degree $d$ of the minimal polynomial of a directed finite graph to any of its topological features - such as its diameter, or any other similar 'natural' ...