**6**

votes

**0**answers

272 views

### Have topographs been studied before?

This is my first post on MO so I hope this question is suitable. I have quite a few definitions which I will need to state before my questions at the end of this post. Please let me know if anything ...

**6**

votes

**0**answers

157 views

### Does the weak Hadwiger conjecture imply the Hadwiger conjecture?

For any cardinal $\kappa$, let $K_\kappa$ denote the complete graph on $\kappa$. We consider the following statements:
(H) If $G$ is a graph and $\chi(G) = \kappa$ then $K_\kappa$ is a minor of $G$.
...

**6**

votes

**0**answers

136 views

### What is known about the chromatic number for minimum-distance graphs in higher dimensions?

For a set of points in $\mathbb{R}^d$ with minimum distance $a$, the minimum-distance graph connect two points iff they are at distance $a$. We can also view it as the tangency graph for a set of ...

**6**

votes

**0**answers

298 views

### the length of paths in a specific graph

Let $n$ be a positive integer and $K$ be the set of all the $2$-elements subsets of $\{1,2,...,n\}$,then $|K|= \binom{n}{2}$. Define $$S=\{P\subseteq K:\bigcup_{I\in P}I=\{1,2,...,n\}\}.$$
For any ...

**6**

votes

**0**answers

146 views

### Uniformly sampling from the set of all simplicial maps

Let $K$ and $L$ be finite simplicial complexes that remain fixed throughout.
How does one efficiently sample (according to the uniform distribution) elements from the finite set of simplicial ...

**6**

votes

**0**answers

383 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

298 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

194 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

658 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

345 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

660 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

765 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

46 views

### Chromatic numbers for coloring-constrained graphs

I am interested in any and all articles about chromatic numbers applying to constrained colorings of a graph. For example, if a graph must be (properly) colored so that there is a 2-color path ...

**5**

votes

**0**answers

94 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
$$d_n=\frac{\sum_{i=1}^m \#E(H_i)}{nm}$$
...

**5**

votes

**0**answers

82 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

103 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

76 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

51 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

69 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

113 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

206 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

107 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

105 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

110 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

182 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

100 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

172 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

125 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

174 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

211 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

164 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

160 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

296 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

251 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

370 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

599 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 ...

**4**

votes

**0**answers

67 views

### A relaxation of proper coloring

I am wondering if the following relaxation of proper coloring appears somewhere. I have tried some searching and have found a few relaxations of proper coloring, but none the coincides with what I ...

**4**

votes

**0**answers

75 views

### Finding closest set of K disjoint hyperspheres to a point in $\mathbb{R}^n$ with uniform radius

I am interested in the following problem: in $\mathbb{R}^n$, we have $N$ overlapping hyperspheres all with the same radius. Given a point $p$ in $\mathbb{R}^n$, the objective is to find the $K$ non ...

**4**

votes

**0**answers

72 views

### Relationship of Weisfeiler-Lehman algorithm to weak isomorphism of coherent algebras

A coherent algebra is a matrix algebra (over $\mathbb{C}$) closed under conjugate transpose and Schur (entrywise) product, and that contains the identity matrix $I$ and all ones matrix $J$. Given ...

**4**

votes

**0**answers

55 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

89 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

111 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

41 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

98 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

88 views

### Graphs with no homomorphism and no minor relation

What is an example of two simple, undirected graphs $G,H$ such that
there are no graph homomorphisms between $G, H$, and
$H$ is not a minor of $G$, and $G$ is not a minor of $H$
?
Definition of ...

**4**

votes

**0**answers

148 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, ...