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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66 views

Is the chromatic number of the graph of the permutahedron known?

The permutahedron $\Pi_n$ is the polytope that is the convex hull of all permutations of the vector $(1,2,...,n)$. There are many results on its structure, but I couldn't find a result on the ...
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34 views

how to study the size of basins of attraction on a graph

I have a certain finite (but huge and without an apparent pattern, so that only numerical studies seem feasible) graph $G = (V,E)$, and a function $f: V \rightarrow \mathbb{R}$. On each edge $e = (u,v)...
3
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73 views

Hypergraph edge colouring

I'm interested in knowing if finding the edge-chromatic number of a $k$-uniform $k$-partite hypergraph is NP-hard for $k\geq 3$ Could anyone provide a reference for the result? By edge-chromatic ...
3
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90 views

What is the complexity of finding a third Hamilton Cycle in cubic graph?

According to Smith Theorem: if a cubic graph has a hamilton circuit then it must have a second one. SMITH : Given a Hamilton circuit in a 3-regular graph, find a second Hamilton circuit. It is known ...
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49 views

Number of Hamiltonian paths in the Hoffman-Singleton graph/Moore graphs?

Does anyone happen to know how many Hamiltonian paths are in the Hoffman-Singleton graph, or have any idea on how I could count such paths?
3
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99 views

Systematic treatment of folding and valued graphs

I'm going to say beforehand that this question has something of a "am I missing something?" flavor. I'm in that odd position mathematicians often find themselves, where a topic has been addressed ...
3
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0answers
76 views

Graph adjacency grouping with geometric criteria

I start with a list of adjacent tetrahedra, where there are tight seals to one another along faces for two tetrahedra that are adjacent. The vertices belonging to these faces for both tetrahedra are ...
3
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80 views

Is there a Havel-Hakimi for geometric graphs?

Suppose that we are given $n$ points in the plane, with a degree prescribed for each, and the question is whether we can place a geometric graph on them. Is there an efficient algorithm for this? ...
3
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130 views

Finiteness for 2-dimensional contractible complexes

While thinking about graph-complex and related operadic stuff, I found a quite interesting (at least for me) question. However, I'm a novice in the algebraic topology, so I'm unable to resolve it by ...
3
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107 views

Construction of algebraic curves using line bundles on graphs

In this paper http://arxiv.org/abs/0707.1309 Matthew Baker and Serguei Norine, construct a analogue of the Riemman Roch formula for Lineal Systems defined on graphs. In the paper http://arxiv.org/abs/...
3
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127 views

In a random graph which one is more probable, $k$-clique or $k$-core?

Recall that the $k$-core of a graph $G$ is the unique maximal subgraph of $G$ with minimum degree at least $k$. In an Erdos-Renyi random graph, where the edge selection is independent with ...
3
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107 views

Group Travel Salesman Problem

For the following Group Traveling Salesman problem, I'd like to know if there exists some poly-time approximation algorithm with constant approximation factor. Group TSP is defined as follows: Take a ...
3
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139 views

Edit distance vs. canonical adjacency matrix distance

Let $G$ and $G'$ be two simple random graphs on the same set of nodes. Let $d_{edit}$ be the edit distance between $G$ and $G'$. Let $\mathbf{A}$ and $\mathbf{A'}$ be the adjacency matrices of the ...
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104 views

Synonyms for “labeling” of a graph

In Preprint 1 we write numerical labels 0 or 1 at each vertex of a Dynkin diagram $D$. We call it a labeling of the graph (Dynkin diagram) $D$. In Preprint 2 we consider an extended (affine) Dynkin ...
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104 views

The degree/diameter problem for even girth graphs starting with upper bound

I posted this on stackexchange but due to a lack of response there I am posting here. Let $G$ be a graph with girth $g$, minimal degree $\delta$, maximal degree $\Delta$, and diameter $D$. Define $$...
3
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99 views

Does this notion of “$\mathcal{F}$-digraph” appear in the literature?

By a digraph, I mean a quiver with no multiple edges. So in particular: Loops are okay. An infinite set of vertexes is okay. Furthermore, I will tend to identify each digraph with its underlying ...
3
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63 views

Small degree vertices in an epsilon-tough graph

We say that a graph is t-tough if by deleting a set if vertices $S$, the resulting graph will have at most $|S|/t$ connected components. We say that a graph is minimally t-tough if the deletion of an ...
3
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72 views

The Class of Strong Resolving Graphs of Hamiltonian Outerplanar Graphs

I'll start with a couple important definitions. I'm not sure how well-known any of them are. Firstly, if $G$ is a graph, and $u, v \in V(G)$, say that $u$ is maximally distant from $v$, denoted $u\ ...
3
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153 views

Polynomial-time algorithm solving approximately a generalization of the travelling salesman problem

Take a graph $G$ and a number of sets of nodes of $G$. The problem is to find the shortest path passing through at least one node in each node set. If each node set consists of only one node, the ...
3
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67 views

Is the size of maximum matching in vertex transitive 3-uniform hyper-graph on $n$ vertices always $\Omega(n)$?

What is the best known lower bound on the size of the maximum matching in a vertex transitive $3$-uniform hyper-graph?
3
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183 views

Characterizing graphs with $k$ edge-disjoint minimum diameter spanning trees

Henneberg [1] and Laman [2] characterized graphs which have, after adding any edge, 2 edge-disjoint spanning trees. This was generalized to $k$ edge-disjoint spanning trees by Frank and Szegõ [3]. ...
3
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170 views

What mathematical models can analyze and optimize systems based on gossip?

I look for a mathematical model that can accommodate, analyze and suggest optimizations for a system that can be humanly described as people gossiping about stuff. System description: We have a ...
3
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260 views

The Bilu-Linial conjecture and Ramanujan graphs

The Bilu-Linial conjecture claims that every $d-$regular graph has a $2-$lift such that for the signing matrix has its eigenvalues between $[-2\sqrt{d-1},2\sqrt{d-1}]$ (the ``signing matrix" is the ...
3
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98 views

What is the significance of the median eigenvalue?

When I look at the spectral density plots of my (usual) laplacian graphs, they spike at the median eigenvalue. But what significance for the graph/matrix (which originates from a network) does the ...
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79 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 ...
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52 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 $K$-...
3
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0answers
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
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70 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 s....
3
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88 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
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0answers
159 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 ...
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44 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
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121 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 ...
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195 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
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206 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
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187 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
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0answers
79 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
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152 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 $i,j$, i....
3
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0answers
189 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
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305 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
0answers
86 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
0answers
157 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
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0answers
243 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 ...
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67 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
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309 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
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0answers
81 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 (\...
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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
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0answers
449 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
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0answers
162 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 ...
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132 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
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122 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 (R_1,R_2,...,...