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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2
votes
1answer
148 views

Coloring vertices in a cubic lattice graph and counting edges between vertices of identical and vertices of distinct coloration

Take an $A \times B \times C$ cubic lattice graph $G$, and paint $k_1$ vertices with color $c_1$ & $k_2$ vertices with color $c_2$, where $(k_1 + k_2)$ is equal to the total vertex count. Let ...
4
votes
1answer
148 views

Which hyperbolic tilings are Cayley graphs?

I realise the question is easy but after asking to a few people (and never getting a clear answer), I thought it could be instructive to ask it here: Given a regular tiling of the hyperbolic plane is ...
3
votes
1answer
225 views

Flow of an integer

I've stumbled across this family of flow networks, and posted the sequence of maximal flows to OEIS. It doesn't appear at this time. I can't find any reference to it either. Has anyone seen it? ...
8
votes
0answers
203 views

Connections between Riemann hypothesis for curves over finite fields and Ramanujan property for graphs

this question relates to the beautiful construction of expander graphs using Cayley graphs of $PGL_2(\mathbb{F}_q)$, as exposited by Davidoff-Sarnak-Valette in their book, Elementary Number Theory, ...
13
votes
1answer
630 views

Is every graph the center of some other graph?

The center of a graph $G$ is the set of vertices that minimize the largest distance to vertices in $G$, e.g., in the graph below, that radius is $4$:           Define the ...
12
votes
3answers
276 views

How can I prove that a particular family of graphs is integral?

I'm working with an infinite family of graphs that seems to always have all integral eigenvalues, and I'd like to find some way to prove that (if it's true). Call the graphs $G_{n,k}$ and define them ...
1
vote
1answer
103 views

Non-toroidality of a simple graph

Let $G$ be a simple undirected graph and $G_1$ and $G_2$ are two subgraphs of $G$, with $E(G_1) \cap E(G_2) =\emptyset$. Which of the following conditions would imply that $G$ is not toroidal: a; ...
3
votes
1answer
109 views

The spectral radius of a modified graph

Let $H$ be a graph and let $G=H \vee K_{1}$ be obtained by creating a new vertex and joining it to every vertex in $H$. This situation has many different names: $G$ is called the cone or the ...
3
votes
1answer
180 views

Calculating a generalized inverse (Moore–Penrose pseudoinverse)

I am considering a graph with $n$ edges with the following nicely structured adjacency matrix: \begin{equation} A_n= \begin{pmatrix} 0 & 0 & 0 &\cdots & 0 & 0 & 1\\ 0 & 0 ...
3
votes
0answers
69 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 ...
4
votes
2answers
609 views

Do Random Walks on the Hexagonal Lattice have a limit?

For every positive integer $n$, consider a regular hexagon $\mathrm{H}_n$ such that the distance of each vertex from the center is $\frac{1}{\sqrt{n}}$. That in turn induces a tiling of ...
4
votes
0answers
141 views

Does the concept of connective constant make sense for any tiling of the plane?

First let me define what is the "connective constant" of a two dimensional lattice. Let $c_{n}$ denote the number of $n$ step self-avoiding walks starting from a fixed origin point in the lattice. ...
1
vote
4answers
267 views

Graphs with dangling edges

In conventional Graph Theory the role of Nodes and Edges is skewed: nodes are perfectly ok being aloof, but poor edges are always drawn between existing nodes (that is, the two maps from EDGES to ...
2
votes
2answers
126 views

Normal subgroups of automorphism group of infinite cubic tree

Let $T$ denote the unique (countably infinite) tree in which every vertex has degree $3$ and let $G=Aut(T)$ be the group of graph automorphisms of $T$. I'm interested in the properties of this ...
6
votes
1answer
158 views

What is/are the best bound/s on the sum of squares of degrees in a graph?

Let $G$ be a graph with degrees $d_{1},\ldots,d_{n}$. I am interested in upper bounds on $$ \sum_{i=1}^{n}{d_{i}^{2}}. $$ An example is de Caen's bound: $$ \sum_{i=1}^{n}{d_{i}^{2}} \leq ...
-2
votes
1answer
182 views

Natural constructions (not depending on parameters) [closed]

Consider graph clusterings as a prototypical example of (logical) constructions. Let a clustering of a graph $(V,E)$ be any covering of $V$, i.e. a set $C$ with $\bigcup C = V$. I am looking for a ...
7
votes
1answer
402 views

Seeming contradiction about P vs NP between graphclasses.org and at least two papers about clique in even-hole-free graphs

I believe correctness about clique in even-hole-free graphs of graphclasses.org and the paper Vertex elimination orderings for hereditary graph classes, Pierre Aboulker, Pierre Charbit, Nicolas ...
4
votes
0answers
138 views

Genus of the graph $K_{m,2,2,2}$

What is the genus of this complete $4-$partite graph, $K_{m,2,2,2}$, where $m \in \mathbb{N}$? Thanks in advance.
0
votes
1answer
117 views

Which graph topology has the greatest eigenvalue?

I am looking at comparing multiple graph topologies based on their spectra. From the set of all $N\times N$ adjacency matrices, is there any result which points to the adjacency matrix with the ...
3
votes
1answer
592 views

Big binary tree as an induced subgraph

I believe this is true: Suppose $G$ is a graph. If $G$ has a subdivision of a large binary tree, prove that $G$ has an induced subgraph which is a subdivision of a large binary tree or the line ...
10
votes
2answers
178 views

Big Mono-Chromatic Subgraphs of Vertex 2-Colourings

I'm not a graph theorist, but the following quantity came up in my work and I'm curious if it has been studied. Given a connected finite graph $\Gamma = (V,E)$ define: $$ c(\Gamma) = \min_{f : V ...
1
vote
1answer
90 views

almost equitable partitions and spectra

If a graph $G$ has an equitable partition, then its charachteristic polynomial (for the adjacency matrix) has a divisor that can be seen as the characteristic polynomial (for the adjacency matrix) of ...
1
vote
1answer
110 views

How many edges can you put in a graph such that every edge belongs to a minimal $k$-cycle?

I am trying to solve: Given $n, k$, find maximum $m$ such that there exists a graph on $n$ nodes, $m$ edges such that every edge is part of a minimal $k$-cycle. I only care about the asymptotic ...
2
votes
1answer
130 views

Non-exchangeable unimodular graph

Let $G=(V,E)$ be an countably infinite, locally finite transitive graph. Say that $G$ is exchangeable if for every two vertices $v,w \in V$ there exists a graph homomorphism that maps $v$ to $w$ and ...
3
votes
1answer
124 views

Genus of a simple graph

Let $G$ is a finite simple undirected graph. Suppose there exist subgraph $G_1,G_2,\dots,G_n$ of $G$, such that $G_i \cong K_5$ or $K_{3,3}$, $E(G_i)\cap E(G_j) = \emptyset$ and $|V(G_i)\cap V(G_j)| ...
5
votes
3answers
258 views

How hard is it to determine if a weighted graph can be isometrically embedded in R^3?

Consider a graph $G$ with nonnegative edge weights. Question: In $\mathbb{R}^3$, how hard is it to assign coordinates to vertices such that the Euclidean length of each edge is equal to its weight? ...
0
votes
0answers
52 views

A particular method of removing edges from strong di-graphs

I have been mulling over a little puzzle I gave myself involving a particular type of iterative removal of edges from a digraph and I'm stuck -- thought I'd consult experts. Start with an ...
1
vote
0answers
30 views

Harmonic Bergman spaces on graphs

Harmonic Bergman spaces on Euclidean domains are a set of harmonic functions on a domain that are from $L^{p}$ of that domain. I tried to find something on harmonic Bergman spaces on graphs because we ...
3
votes
0answers
161 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 ...
4
votes
1answer
179 views

Reference request: “unoriented composition” in generalized categories

I'm looking for a generalized notion of category (really of symmetric multicategory) which, roughly speaking, doesn't make a distinction between sources and targets. Each "morphism" in such a category ...
8
votes
1answer
318 views

A combinatorial problem concerned with logic circuits

Consider a logic circuit with two-bit gates only. The length of each gate is the number of bit lines that the gate crosses. How hard is to compute the maximum length for a given circuit? Notice that ...
4
votes
1answer
358 views

Genus of a graph

Let $G$ is a simple undirected graph. Suppose $G$ has two subgraphs $G_1$ and $G_2$, such that $E(G_1)\cap E(G_2) =\emptyset$ ($E(G_i)$, stand for the set of edges of $G_i$). Then is it true that ...
3
votes
2answers
332 views

A structure of the group of automorphisms of an infinite binary tree

My friend asked me to ask his question here. Where he can find (a paper or a book) containing a complete description (with the proof) of a structure of the group of automorphisms of an infinite binary ...
3
votes
2answers
195 views

Making a graph claw-free by adding as few edges as possible

Independent set is polynomial in claw-free graphs, so I am wondering if this can approximate independent set. By adding enough edges to $G$ and gets claw-free $G'$. IS in $G'$ is IS in $G$, so this ...
0
votes
1answer
102 views

Reverse optimization of a minimum cost flow network

Given an undirected graph $(V,E)$, with $W$ as the weight of each edge, and a convex cost function $F(X)$, such as $|X|^k$ ($k>1$). The cost to send $x$ unit of flow through edge $e_i$ is defined ...
2
votes
0answers
82 views

Is there an universal (dis)similarity measure between two structures?

I'm always wondering is there an universal (dis)similarity measure between two structures (let's say between two undirected simple graphs)? I mean, not "the measure with universal parameter that we ...
4
votes
2answers
215 views

Cubic graphs decompositions

There are many interesting computational problems related to connected cubic graph decomposition. For instance, decomposition of cubic graph into a perfect matching and a connected 2-factor ...
0
votes
0answers
153 views

What is the number of connected subgraphs with $n$ vertices of a labelled connected simple graph with $n$ vertices?

Suppose $G$ is a connected simple labeled graph. Let $n$, $e$, and $k$ be its number of vertices, edges, and the upper bound of the degree of a vertex, respectively. How many connected sub-graphs ...
2
votes
0answers
105 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 ...
6
votes
1answer
128 views

Hamiltonicity criteria for sparse graphs

Given a sparse graph, how can one go about proving that it is Hamiltonian? (Assuming it actually is, of course). There are three main classes of criteria for Hamiltonicity that I am aware of: ...
1
vote
0answers
50 views

Perfect Matchings in Biclique Decompositions of Multigraphs

Suppose you have the $K_{2n}$ covered by a multigraph consisting of $2n-1$ bicliques, each consisting of a partition of the vertex set into two sets of equal size. Here is a picture of $K_{6}$ with 5 ...
3
votes
1answer
56 views

Are all (non-constant) symmetric submodular functions non-monotone?

I am trying to show (if possible) that symmetric submodular functions are non-monotone (excluding constant sub-modular functions). Recall that a submodular function $f : 2^{\Omega} \rightarrow R$ is ...
4
votes
2answers
144 views

class 1 vs class 2 in regular graphs

Vizing's theorem states that a graph can be edge-colored in either $\Delta$ or $\Delta+1$ colors, where $\Delta$ is the maximum degree of the graph. A graph with edge chromatic number equal to ...
3
votes
0answers
159 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 ...
2
votes
1answer
195 views

Upper bound on the number of vertex transitive graphs

Is there a known upper bound on the number of vertex transitive graphs on $n$ vertices?
6
votes
0answers
190 views

Permutation Group Question

A question about permutation groups: I wonder if someone who is expert in permutation group theory could answer the following question. Let $x \in S_n$ (the symmetric group) be an involution which ...
1
vote
0answers
87 views

On 'Very Movable' Geometric Configurations (Configurations with a large degree of freedom)

Let $C$ be an $(n_r, b_k)$ combinatorial configuration that admits a geometric realization in the plane. I'm interested in the maximum number of points/lines $M$ of $C$ we can place in general ...
4
votes
2answers
139 views

Details of generation programs supplied with nauty

The program nauty comes with gtools which contains, among others, several generation programs like geng, genbg, ... I was ...
2
votes
3answers
205 views

Regular graphs whose neighbourhoods induce matchings

Studying some problem I've arrived to the following notion. Let a $2r$-regular graph $G$ be called neighbour-matching if $N(v) = rK_2.$ In other words, the neighbourhood of any vertex induces a ...
1
vote
4answers
229 views

Counting simple 4-cycles in an undirected graph [closed]

I'm looking for an algorithm which just counts the number of simple and distinct 4-cycles in an undirected graph labelled with integer keys. I don't need it to be optimal because I only have to use it ...