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
0answers
47 views

Algorithm to minimally connect line segments in Euclidean plane

Suppose you have finitely many line segments in the Euclidean plane. How do you "connect them to form one chain of line segments of minimal length?" More formally and generally, what I'm looking for ...
1
vote
0answers
31 views

finding dominating cycles in $2K_2$-free graphs

A cycle $C$ in a connected graph $G$ is called dominating if its complement $V(G)-V(C)$ is an independent set. H.J. Veldman proved in 1983 (Disc. Math. v.43, 281-96) a general result that in ...
-4
votes
0answers
57 views

Flow in graph. Proof [on hold]

Let $(S_1, \overline{S_1} ) , (S_2, \overline{S_2} )$ be minimum cuts in some network. Is it true that $(S_1 \cap S_2, \overline{S_1 \cap S_2)}$ is a minimum cut in this network?
-3
votes
0answers
33 views

Odd-cycle inequality [on hold]

Consider the stable set problem. An odd hole is a cycle with an odd number if nodes and no edges between nonadjacent nodes of the cycle. Show that if H is the node set of an odd hole, the following ...
4
votes
1answer
227 views

Endomorphisms and almost all graphs

Is it known what fraction (almost all?) of graphs have a trivial endomorphism monoid? I can't seem to find any reference to the question. Maybe it's related to the question: what fraction of graphs ...
-1
votes
0answers
49 views

Length of Paths in Graph [on hold]

There is a very large directed graph with n number of vertices, in order of millions. We are given a number p much smaller than n, and two vertices v1 and v2. What is the efficient way of finding a ...
1
vote
0answers
48 views

Is there an elliptic Harnack equality for directed graphs?

The elliptic Harnack inequality for undirected graphs was proven by Delmotte in the paper "Inegalite de Harnack elliptique sur les graphes" (French, ...
4
votes
1answer
87 views

Determining the number of hamiltonian paths of $K_n-C_n$

I would like to know information regarding the function $h(n)$ where $h(n)$ is the number of hamiltonian cycles the graph $K_n$ has after removing the edges that make up a hamiltonian cycle of $K_n$. ...
2
votes
1answer
151 views

Edge density of triangle-free graphs

Let $G$ be a finite, simple, loopless graph with $|V(G)|=n$. We define its edge density as $$ed(G) := \frac{|E(G)|}{n \choose 2}.$$ Moreover we set $$d_n := \text{max}\big\{ed(G): G \text{ is a ...
1
vote
0answers
34 views

Rate of convergence of graph-theoretic quantity to fractional graph-theoretic counterpart

Let $G^n$ denote the OR product of a graph with itself $n$ times, i.e. the graph which has an edge between distinct vertices $(v_1,v_2,\ldots,v_n)$ and $(u_1,u_2,\ldots,u_n)$ if there exists some $i$ ...
7
votes
1answer
155 views

Universal graph homomorphisms

By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\}: a\neq b \in V\}$. A graph homomorphism between graphs $G, H$ is a map $f:V(G)\to V(H)$ such ...
17
votes
1answer
269 views

Lens spaces and generalized Petersen graphs

Recently I came across this mathoverflow question, in which the number of homeomorphism classes of 3-dimensional lens spaces $L(p, q)$ is computed as a function of $p$. Using the OEIS, I found a ...
8
votes
1answer
237 views

Cauchy-Schwarz proof of Sidorenko for 3-edge path (Blakley-Roy inequality)

Is there a "Cauchy-Schwarz proof" of the following inequality? Theorem. Given $f \colon [0,1]^2 \to [0,1]$, one has $$ \int_{[0,1]^4} f(x,y)f(z,y)f(z,w) \, dxdydzdw \geq \left(\int_{[0,1]^2} f(x,y) ...
1
vote
2answers
134 views

Proving a random bipartite graph contains a perfect matching

I have the following problem consider a random bipartite with vertex classes $A$ and $B$ of size $|A|=|B|=\mathrm{log}^{2}(n)$ graph in which every possible edge is chosen independently with ...
28
votes
8answers
2k views

Generalizations of the Four-Color theorem

The four color theorem asserts that every planar graph can be properly colored by four colors. The purpose of this question is to collect generalizations and strengthenings of the four color theorem ...
4
votes
1answer
161 views

Do perfect matching(s) have signatures in the graph eigenvalues?

If the edges of a bipartite graph are such that they can be seen as a disjoint union of perfect matchings then will this somehow reflect in the eigenvalues of the Laplacian? It would be helpful to ...
0
votes
0answers
48 views

Computation on Random Bipartite graphs

I'm looking at a random bipartite graph $K_{\omega(n)}*K_{\omega(n)}$ where $\mathrm{log}(n)\leq \omega(n) \leq n^{1/2}$, in which each of the $\omega(n)^{2}$ edges is placed randomly with probability ...
3
votes
1answer
57 views

Minor ordering for finite graphs

Let $\mathcal{G}_{<\omega}$ be the set of graphs $G = (V,E)$ such that $V = \{0,\ldots,n\}$ for some $n \geq 0$ and $E \subseteq \mathcal{P}_2(V) = \{\{a,b\} : a,b \in V \textrm{ and } a\neq b\}$. ...
1
vote
0answers
57 views

Matchings in random bipartite graphs

I was wondering if anyone could point me in the direction of a text or paper which would help deal with the following problem Suppose i am given a $K_{\mathrm{log}(n)} \times K_{\mathrm{log}(n)}$ ...
4
votes
2answers
111 views

Graphs whose degree vectors coincide for all powers of their adjacency matrices

Let symmetric $A,B \in \{0, 1\}^{n \times n}$ denote the adjacency matrices of two simple graphs. Further let $\mathbf{1}$ denote the all-one-vector. Now assume that $A^k \mathbf{1} = B^k \mathbf{1}$ ...
0
votes
1answer
68 views

What is the standard name of an edge-graph

Given a graph $G=(E,V)$, I construct a graph $G'$ where the vertices of $G'$ are given by the edges of $G$ and say that two edges of $G$ are neighbors in $G'$ if they have a common vertex. Is there a ...
2
votes
2answers
138 views

“Homomorphism fingerprint” for graphs

Let $G, H$ be simple, undirected graphs without loops. We say that $G, H$ have the same homomorphism fingerprint if $|\text{Hom}(X, G)| = |\text{Hom}(X, H)|$ for all graphs $X$. (By graph ...
3
votes
0answers
96 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 ...
-1
votes
0answers
63 views

Partitioning graph in clusters

Let $H = (V, E)$ be a graph. If $v \in V$ and $r \in \mathbb{N}$, denote $S_H(v, r)$ the sphere radius $r$ with center in $v$: $S_H(v, r) = \{u \in V $: $dist_H(u, v) \leq r\}$. Algorithm for ...
4
votes
1answer
138 views

What can be said about graphs if there are homomorphisms in both directions?

Let $G,H$ be simple undirected graphs without loops such that there are graph homomorphisms $f_1:G\to H$ and $f_2: H\to G$. An easy argument shows that we have $\chi(G) = \chi(H)$, even for graphs ...
1
vote
0answers
68 views

NP hard problems on geometric graphs

I have posted this question before but i don't feel i expressed my confusion clearly enough. So i would like to try and explain again. This is a proof of the minimum vertex cover for unit disk graphs ...
2
votes
0answers
58 views

Total number of spanning trees of a set of graphs

Given an undirected graph G with $n$ nodes, we can compute its number of spanning trees in polynomial time using Kirchhoff's matrix-tree theorem. Now consider a more complicated setting, in which each ...
9
votes
4answers
252 views

Are these three different notions of a graph Laplacian?

I seem to see three different things that are being called the Laplacian of a graph, One is the matrix $L_1 = D - A$ where $D$ is a diagonal matrix consisting of degrees of all the vertices and $A$ ...
1
vote
0answers
54 views

Max flow with minimal requirements algo problem

While applying the algorithm to solve the max flow of the network with minimal requirements on edges, I have encountered a problem. The algorithm states: For graph G create an edge from target to ...
1
vote
1answer
73 views

n-cube connectivity problem

Given $n\geq 2$, let us consider the n-cube $H_n=(V,E)$, i.e. vertex set $V$ is $\{0,1\}^n$. Here, the edges of $H_n$ are directed, oriented by set inclusion, i.e., $(x,y)\in E$ iff $x\subseteq y$ and ...
6
votes
2answers
114 views

Realizing a monoid as $\mathrm{End}(G)$ for some graph $G$

This question relates to Realizing groups as automorphism groups of graphs. Given a monoid $M$, is there a graph $G$ such that the endomorphism monoid $\textrm{End}(G)$ is isomorphic to $M$?
9
votes
1answer
307 views

Could there be an exact formula for the Ramsey numbers?

Let $R(k)$ denote the diagonal Ramsey number, i.e. the minimal $n$ such that every red-blue colouring of the edges of $K_n$ produces at least one monochromatic $K_k$. Is it possible that there ...
4
votes
1answer
85 views

Strongly asymmetric graphs

Asymmetric graphs are graphs that have a trivial automorphism group $\textrm{Aut}(G)$, i.e. the only graph isomorphism from $G$ to itself is the identity. Let's call a graph $G$ strongly asymmetric ...
5
votes
1answer
83 views

Isomorphic Hadwiger graphs

Let $G$ be a graph, then we define its Hadwiger graph $\textrm{Hadw}(G)$ in the following way: $V(\textrm{Hadw}(G)) = \{S\subseteq (V(G): S\neq \emptyset\textrm{ and } S \textrm{ is connected}\}$; ...
2
votes
1answer
188 views

Counting Ribbon graphs

Let $G$ be a ribbon graph (sometimes called fat graph) with $v$ vertices and $e$ edges. Furthermore each vertex is of degree $d$. Q) What is the number of $G$ with the above properties? I mean does ...
2
votes
0answers
46 views

relationship of max-sat and min-cut in theory and practice

I have been using MAX-SAT solver to obtain the exact ground state of ising spin glass model: For 1D periodic model, for systems with 50 binary variables and interaction range of 15th nearest ...
0
votes
0answers
79 views

Finding nodes with a particular weight in a graph [migrated]

Say that an edge $e$ is incident to a node $v$ if one of its two extremes is $v$. Then we can also say that $v$ is hit by $e$. We might define the notion of "weight of a node $v$" as the sum of all ...
3
votes
1answer
160 views

Determinant of the oriented adjacency matrix of a tree

Let $(V,E)$ be a finite oriented directed graph, with vertices and edges ordered, and $M$ the $|V|\times |E|$ matrix with entries $$ m_{ve} = \begin{cases} 1 &\text{if $e$ points at $v$}\\ -1 ...
2
votes
3answers
564 views

Koebe–Andreev–Thurston theorem - where can I find a proof?

Koebe–Andreev–Thurston theorem (known also as the circle packing theorem) says that any planar graph can be realized by a set of (interior-) disjoint disks corresponding to vertices, such that two ...
1
vote
0answers
38 views

Solutions for n such that the path number to the n-th node is also n in a complete, rooted, ordered k-ary tree

The path to find the $n$-th node within a complete, rooted, ordered $k$-ary tree can be represented by a number to the base $(k+1)$ that contains no zero digits. See A245905 in OEIS as an example for ...
7
votes
1answer
210 views

Number of median graphs?

What is the number of $n$-vertex median graphs? These graphs generalize hypercubes and trees, and have many applications. It seems unlikely that a closed form expression is known, so I would also be ...
-2
votes
1answer
86 views

how to reduce 3-colorable graph to this? [closed]

suppose we have a finite set X and a set S of subsets of X and we want to determine is there a subset S' of S such that all members of X belong to exactly one set in S' I think the best problem to ...
0
votes
0answers
11 views

matching Robinson-Foulds distance and way to compute RF dist in Phylip

In Comparison of Phylogenetic Trees, Robinson D.F. and Foulds L.R., didn't show how to compute the RF distance between trees, counting the different partition generated by the removing of an internal ...
1
vote
1answer
71 views

What is “graph-directed iterated function”?

Im translating an article about Rauzy fractal and I ran into this sentence: ...
9
votes
2answers
405 views

Can a parent and child node have the same type in a well-founded digraph tree?

$\newcommand\toward{\rightharpoonup}$It would help me to understand something in a current research project if someone could provide an example of directed graph $\langle G,\toward\rangle$ with the ...
7
votes
4answers
218 views

4-regular graph with every edge lying in a unique 4-cycle

What are all 4-regular graphs such that every edge in the graph lies in a unique-4 cycle? Among all such graphs, if we impose a further restriction that any two 4-cycles in the graph have at most one ...
0
votes
1answer
38 views

Petersen 2-factor decomposition theorem for directed graphs

Petersen proved that every 2k-regular graph can be decomposed into k disjoint 2-factors. I would like to know that is it true that if G is a directed regular graph (d_out(v)=d_in(v)=k), then can G be ...
1
vote
1answer
122 views

NP hard problems on UD graphs

I'm reading up on NP hard problems in Unit Disk graphs. I'd like to point out i'm fairly new to this NP hard stuff so i'm trying to get around how to prove something is NP hard. ...
2
votes
1answer
135 views

Graph automorphism that swaps two pairs of nodes

Suppose we have two automorphisms on a graph $G$ such that each one swaps a separate pairs of vertices. Is it possible to construct (or prove the existence of) a third automorphism that swaps both ...
0
votes
1answer
55 views

Intersection graphs

Does anybody know of a paper which proves that finding the maximum independent set in geometric intersection graphs is NP hard? Even general intersection graphs?