**3**

votes

**0**answers

109 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 ...

**3**

votes

**0**answers

73 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 ...

**3**

votes

**0**answers

47 views

### Algorithm to construct metric space endomorphism with controlled square

Given a finite metric space $(M,d)$ with parameters $K \geq 1$ and $\epsilon > 0$, I'd like to algorithmically check for the existence of a non-identity map $\phi:M \to M$ which happens to be ...

**3**

votes

**0**answers

98 views

### Node covering in a random graph

Given $N$ nodes randomly placed in a $D\times D$ area, i.e., the position of each node is randomly chosen. Assume that both $N$ and $D$ are sufficiantly large.
An agent can move in the area at ...

**3**

votes

**0**answers

66 views

### Maximal $k$-chordal subgraph

Recall that a graph is called $k$-chordal if any cycle $C$ of length $> k$ contains a chord, i.e. an edge joining to non-consecutive vertices in $C$. Let $f(n, k)$ be the minimal number of edges ...

**3**

votes

**0**answers

87 views

### Reconstructing a function from its variants that negate one argument

Call two functions $g(x_1,\ldots,x_n)$ and $h(x_1,\ldots,x_n)$ from complex numbers to complex numbers equivalent if they are the same up to the order of their arguments. Formally: there is a ...

**3**

votes

**0**answers

153 views

### Must distinct tree eigenvalues be relatively far apart?

How close to each other can two distinct eigenvalues of a tree be, as a function of the number $n$ of nodes ?
For example, the path $P_n$ exhibits a gap of order $\frac{2\pi^2}{n^2}$ asymptotically ...

**3**

votes

**0**answers

43 views

### Balancing out edge multiplicites in a graph

Let $G$ be a multigraph with maximum edge multiplicity $t$ and minimum edge multiplicity $1$ (so that there is at least one 'ordinary' edge).
Is there some simple graph $H$ such that the $t$-fold ...

**3**

votes

**0**answers

101 views

### When is an induced subgraph of a Johnson graph hamilton-connected?

The Johnson graph $J(n,k)$ has vertices which are the $k$-subsets of $\{1, 2, \dots, n\}$ where two vertices are adjacent iff their intersection has size $k-1$. A graph is hamilton-connected if every ...

**3**

votes

**0**answers

187 views

### vertex transitive and Cayley graphs

(all the graphs alluded to below are finite).
Suppose I gave you a graph, and asked you whether it was vertex-transitive. How hard is that algorithmically?
The second question is: suppose I gave you ...

**3**

votes

**0**answers

172 views

### Partitioning a cubic graph into two induced cycles of equal order

I am aware that deciding the existence of a partition of the vertices of a connected graph $G(V, E)$ into two induced cycles is $NP$-complete(Theorem 2). Induced cycle is a cycle without any chord ...

**3**

votes

**0**answers

180 views

### Hitting edges in graphs at random and let them die with honor

Let $G=(V,E)$ be a finite simple 2-connected graph. Let $B\subseteq E$ be a set of bad edges of size $k:=|B|$. For each edge $e\in E$ we toss a fair coin ($p=1/2$) and if the outcome is head, we hit ...

**3**

votes

**0**answers

78 views

### What is this expander-mixing-type graph property?

Fix $C>0$. I am interested in graphs with the following mixing property:
$$\Big|E(S,T)-\frac{1}{2}|S||T|\Big|\leq C\sqrt{|S||T|\max\{|S|,|T|\}}$$
for every disjoint $S,T\subseteq V$. Note that ...

**3**

votes

**0**answers

146 views

### Concentration inequality for function of independent Bernoulli r.v.'s (related to random graph)

Consider a random undirected graph on a set of $n$ nodes, say $\{1,2,\ldots,n\}$, such that the probability of edge between nodes $i$ and $j$ is $p_{ij}$ (we may assume $p_{ij}=o(1)$ for all $i,j$, ...

**3**

votes

**0**answers

167 views

### Bits required to encode difference between number of subgraphs with odd number of edges and number of subgraphs with even number of edges

Let $H = ( V, E )$ be a $k$-uniform connected hypergraph, with $n = |V|$ vertices and $m = |E|$ hyperedges. Let $O_w$ be the number of edge induced subgraphs of $H$ having $w$ vertices and an odd ...

**3**

votes

**0**answers

277 views

### A problem on graph theory and complex numbers!

Let ${\mathcal G} = ({\mathcal V},{\mathcal E})$ be a simple connected undirected graph with $n$ vertices. Also let $z_1, \ldots, z_n \in {\mathbb C}$ be complex numbers such that
$$
||z_1||=\ldots = ...

**3**

votes

**0**answers

155 views

### Equivalent paths in graphs

Let $G$ be a finite, planar graph. In this question I consider a path in $G$ to be a finite sequence of vertices $v_1,\dots,v_n$ of $G$, where $v_i$ is adjacent to $v_{i-1}$ for $i=2,\dots,n$. A ...

**3**

votes

**0**answers

85 views

### Usefulness of polynomial ideals for graphs

Given a graph $G=(V,E)$ on $n$ vertices, one can associate to it the polynomial $f \in k[x_1,\dots,x_n]$ given by $f = \prod_{\{i,j\}\in E}(x_i - x_j)$. Then one can map various graph properties into ...

**3**

votes

**0**answers

150 views

### Maximize weighted vertex sequence subject to neighbour inclusion

Let $G=(V,E)$ be a simple undirected graph on $n$ nodes, with node weights $W = [w_1,w_2,\dots,w_n] \in \mathbb{R}^n$. Define the weight for a sequence of nodes $v_1,v_2,\dots,v_k$ by the average ...

**3**

votes

**0**answers

231 views

### Beating Kadane's Algorithm

I am seeking some reference on already existing work for the following problem.
Given an $n$-dimensional square matrix $A=DP$ where $D$ is a diagonal and $P$ is a permutation matrix (think of Gaussian ...

**3**

votes

**0**answers

64 views

### Is the $d$-dimensional Arrangement of Trees still $NP$-hard?

The $d$-dimensional Arrangement Problem for general graphs is known to be $NP$-hard since the special case $d=1$ (OLA) already is (Garey et al, [1976]). For Trees however, the one dimensional case can ...

**3**

votes

**0**answers

275 views

### How many trees can be constructed from k vertices using an LCA operator?

Consider the class of rooted trees and suppose I have at my disposal a lowest common ancestor (LCA) operator given by
$$
\textrm{lca}(u,v) = \text{the lowest common ancestor of $u$ and $v$}
$$
for all ...

**3**

votes

**0**answers

71 views

### Is there precedent in the literature for a variant of a random geometric graph where vertices are the centroids of discs placed by random sequential adsorption (RSA)?

Imagine I form a random graph by simulating random sequential adsorption (RSA) of discs (each with the same radius $r$) on $[0,1]^2$ until I cover the plane at a density $\leq U$, where $U \leq ...

**3**

votes

**0**answers

166 views

### Is there a flow criterion for a graph being planar?

Given the set od propagators (say momenta flows) on a Feynman diagram (flow network), I would like to decide whether this diagram is planar or not.
I know that non-planar diagrams manifest different ...

**3**

votes

**0**answers

417 views

### Minimum weight bipartite graph clique covering

I was wondering if anyone here could give me any pointers as to how to solve the following problem.
Let $B=(L,R,E)$ be a bipartite graph, and $\forall u\in L\cup R$, let $c_u$ be a cost associated to ...

**3**

votes

**0**answers

161 views

### Generalised de Bruijn Graph

I have encountered sets of the following type, consisting of words over a finite aphabet $A$.
If $S$ is such a set, then
$S$ is finite,
No word in $S$ is part of another element of $S$, and
every ...

**3**

votes

**0**answers

130 views

### The mean number of vertices in small connected components of random geometric graphs

I place $N$ points on a circular plane of radius $R$, and draw edges to connect points that are less than or equal to some distance $D$ to form a set of graphs or cliques $G_i$. As a function of $N$, ...

**3**

votes

**0**answers

120 views

### On the faces of the multicommodity flow polytope

Consider the multicommodity flow problem on an undirected graph with k source-destination pairs and specified capacity constraints on the edges. The set of concurrently achievable flows ...

**3**

votes

**0**answers

187 views

### Bounds on number of small minimal cut-sets in graph?

David Karger developed an algorithm for estimating graph reliability; a key lemma in this algorithm is that if a graph has minimal cut-set size $c$, then the number of cut-sets of size
$\alpha c$ is ...

**3**

votes

**0**answers

192 views

### Minimum sum of distances of G for spanning tree of G

Let $G=(V,E)$ be a undirected simple graph, and $\Omega(G)$ denote the set of spanning trees of G. Does the next funcion
$$l(G) = \min_{T \in \Omega(G)}\quad \sum_{uv \in E} \mbox{dist}_T(u,v)$$
have ...

**3**

votes

**0**answers

120 views

### Polynomials in graphs

I have part of a physical simulation which I've realised can be modelled using a directed graph where each node is a polynomial. I then calculate this graph by functional composition and summing to ...

**3**

votes

**0**answers

246 views

### Minimal-edge graph with diameter 2 and bounded max degree

Consider all connected simple graphs with diameter $d = 2$ and maximal vertex degree $\Delta$. In my particular practical case $\Delta = 4$, but general problem is much more interesting — probably ...

**3**

votes

**0**answers

240 views

### Domination number of hamming graphs

The problem of finding the domination number of Hamming graph $H(3, 2n)$ ($n$ is an integer) was given as a homework for my discrete math class. I didn't manage to solve the question. But later the ...

**3**

votes

**0**answers

304 views

### Computing Quantum Dimensions

Hi,
in "Jaeger’s Higman-Sims state model and the B2 spider" by Greg Kuperberg
(arxiv:math9601221v1, 1996) there are some quantum dimensions listed in the
"Discussion" part. Evidently quantum groups ...

**3**

votes

**0**answers

207 views

### For Ising models on finite graphs, is the gradient of Z (w/r/t coupling and field) easier to compute than Z?

Suppose we have a graph $G$ with $n$ vertices, edgeset $E$, $\mathcal{X}=\{1,-1\}^n$. The partition function of the spin-1/2 Ising model on $G$ is
$$Z(J,h)=\sum_{x\in \mathcal{X}} \exp\left(J ...

**3**

votes

**0**answers

129 views

### Dimension of convex arrangements for hypergraphs

Suppose you have a hypergraph H on n vertices. Let d be the smallest integer such that we can find an arrangement A of convex subsets in Rd so that H represent the intersections of sets in A.
Has ...

**3**

votes

**0**answers

412 views

### Graph recognition software

ISGCI lists a lot of graph classes, many of which are recognizable in polynomial time. Is anyone here aware of actual implementations of these algorithms?

**3**

votes

**0**answers

408 views

### cell complexes and higher graph theory

Suppose that, on an intuitive basis, one defines a "2-graph" $(V,E,F,\partial)$ as a collection of vertices, oriented edges and oriented faces, all of which should be considered as abstract objects ...

**2**

votes

**0**answers

51 views

### Planar triangulations for which all distinct 4-colorings consist of exactly 6 Kempe chains

Are there any internally 6-connected planar triangulations other than the icosahedron all of whose distinct 4-colorings consist of exactly 6 Kempe chains, one for each of the 6 color-pairs?
Addendum: ...

**2**

votes

**0**answers

133 views

+50

### A path optimisation problem

Consider a graph of $n$ nodes randomly located in $[0,1]^2$. Each node moves following a path randomly chosen from the set of all possible paths. Regard nodes as attackers. A policeman seeks an ...

**2**

votes

**0**answers

51 views

### The numbers of edge colorings and partitions into vertex covers

Let $G = \langle X \cup Y, E\rangle$ be a $d$-regular bipartite graph such that $|U|=|V|=n$, and assume that $d$ divides $n$. Let $F(G)$ denote the number of proper edge colorings of $G$ using $d$ ...

**2**

votes

**0**answers

275 views

### Maps between graphs defined through laplacian operations

Edit: The views/answers ratio on this question tells me that it was too long. As such, I've stripped out examples and now ask the question in brief. For examples, please ask in the comments or look at ...

**2**

votes

**0**answers

60 views

### Non-existence of commutative rings with many nilpotent elements with some restrictions where matrix powers are efficient

At the moment can't find better reference than "Cycle Enumeration using Nilpotent Adjacency Matrices with Algorithm Runtime Comparisons"
though certainly there are others.
Consider the following ...

**2**

votes

**0**answers

97 views

### Obtaining a quasi-isometry of the 'boundary'

It is well-known that a quasi-isometry induces a homeomorphism on the space of ends of say a locally finite graph for simplicity. Clearly the converse is not true. In other words the concept of ends ...

**2**

votes

**0**answers

63 views

### Even cycle constrained edge coloring

Is minimum colors needed to assign colors to edges of complete graph $K_n$ so that every $2t$ simple cycle where $t\in\Big\{1,\dots,2\Big\lfloor\frac{n}2\Big\rfloor\Big\}$ contains atleast $t+1$ ...

**2**

votes

**0**answers

102 views

### From Planar Graphs To Tangent Circles

I have a conjecture:
"For each planar graph with vertices $V_1, V_2,\ldots, V_n$ there exist disjoint circles $w_1,w_2,\ldots,w_n$ in the plane, such that for every $i,j$, $w_i$ is tangent to $w_j$ ...

**2**

votes

**0**answers

46 views

### Maximum cardinality general factor of a graph

Given a graph $G=(V,E)$ and a set of integers $B(v)$ associated to each vertex, a general factor of $G$ is a set of edges $F\subseteq E$ such that the degree of each vertex $v\in V$ in the graph $(V, ...

**2**

votes

**0**answers

38 views

### Is this infinite family of non-trivial snarks resulting from the first Celmins-Swart?

Non-trivial snark is cubic graph with chromatic index $4$, girth
at least $5$ and doesn't to contain three edges whose deletion results in a disconnected graph, each of whose components is nontrivial.
...

**2**

votes

**0**answers

54 views

### Decomposing a weakly chordal graph into disjoint union of co-chordal graphs

A graph G is said to be co-chordal if it is $\bar C_n$-free for any $n \ge 4$. It is weakly chordal if it is $C_n$ and $\bar C_n$ free for all $n\ge 5$. Assume that the induced matching number of $G$ ...

**2**

votes

**0**answers

63 views

### Edge-disjoint path-systems in infinite digraphs

Let $ D=(V,A) $ be a directed graph without backward-infinite paths and let $ \{ s_i \}_{i<\lambda},U \subset V $ where $
\lambda $ is some cardinal. Assume that for all $ u\in U $ there is a ...