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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5
votes
0answers
105 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 ...
3
votes
1answer
64 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 ...
0
votes
0answers
53 views

Number of graphs with M edges that does not contain K-clique [on hold]

If we consider the space of graphs $G(n,M)$ where $M$ denotes the number of edges. Is there any known way of calculating the number of graphs within this space that does not contain any k-cliques? Can ...
5
votes
1answer
74 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
176 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
40 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
1answer
74 views

Finding nodes with a particular weight in a graph

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
147 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
534 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
35 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 ...
5
votes
1answer
122 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
76 views

how to reduce 3-colorable graph to this? [on hold]

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 ...
-2
votes
0answers
42 views

when a given graph is 3-colorable? [closed]

I want to use graph 3-colorability to prove a problem is NP-complete But I'm not sure when a given graph is 3-colorable. I think if it doesn't have any node to be connected to all 3 vertices of a ...
-4
votes
0answers
62 views

About diagonal entries of the graph Laplacian [on hold]

[..in the following you can assume its a regular graph if necessary..] Is anything special known about them? Are they characterized in any other way? Is the largest diagonal entry in any power of ...
0
votes
0answers
75 views

independent subset problems [on hold]

I'm interested in the following which i suspect is probably a well studied problem. Given a set $N=\{1,2,...,n\}$ and $M=\{1,2,...,m\}$ consider a map $$f:N\rightarrow 2^{M}$$ (elements of $N$ to ...
1
vote
1answer
64 views

What is “graph-directed iterated function”?

Im translating an article about Rauzy fractal and I ran into this sentence: ...
0
votes
0answers
44 views

$\Omega(G)$ is homeomorphism to $\Omega(T)$ [closed]

Is it possible that the end set of graph $G$ is homeomorphism to the set of end for every spanning tree of $G$ where $G$ has a cycle. In other hand, for given graph $G$ and spanning tree $T$ of $G$. ...
9
votes
2answers
368 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
202 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
30 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
103 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
132 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
54 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?
0
votes
0answers
79 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 $G$. ...
2
votes
1answer
83 views

irregular pairs in half graphs - Szemeredi regularity

Szemeredi's regularity lemma is a well-known result about partitioning large graphs into pieces such that most pairs of pieces are "regular". The precise statement takes a bit of detail so I'll just ...
12
votes
0answers
163 views

Coloring the edges of a torus graph

Question:Consider the $2k \times 2k$ grid graph on a torus. Is it true that for every $2$-coloring of the edges, there is an antipodal pair of vertices connected by a path that changes colors at most ...
8
votes
1answer
173 views

Vertex-primitive graphs with two vertices having almost the same neighbourhood

Hypothesis: Let $\Gamma$ be a vertex-primitive graph with two vertices $u$ and $v$ such that $$|N(u) \cap N(v)|=|N(v)|-1$$ Question: Is it true that $\Gamma$ must either be a complete graph or have ...
0
votes
0answers
48 views

Expected number of distinct nodes visited in a directed bipartite graph [migrated]

Let $G = (V,E)$ be a directed bipartite graph with $V = \{I \cup O\}$ where $\left\vert{I}\right\vert = n$ and $\left\vert{O}\right\vert = m$. All the edges start from a vertex in $I$ and end on a ...
0
votes
1answer
30 views

Is it known whether Minimum Cost Multicut is APX-hard?

My questions is concerned with the following problem: Given an undirected graph $G = (V, E)$ and (edge costs) $c \in \mathbb{Z}^E$, $$\min \left\{ \sum_{e \in E} c_e x_e\ \middle|\ x \in \{0,1\}^E \ ...
0
votes
1answer
87 views

How to solve the opposite of max flow problem? [closed]

Each edge in graph has a minimum bound instead of a maximum capacity. The goal is to find the minimum flow such that every edge in graph has flow larger than it's lower bound. I tried to use Ford ...
7
votes
3answers
342 views

Is there a continuous analogue of Ramanujan graphs?

I think it might help to think of the following definition of a Ramanujan graph - a graph whose non-trivial eigenvalues are such that their magnitude is bounded above by the spectral radius of its ...
4
votes
0answers
106 views

minimal polynomial for a graph

I wonder if there is any result relating the degree $d$ of the minimal polynomial of a directed finite graph to any of its topological features - such as its diameter, or any other similar 'natural' ...
6
votes
2answers
224 views

Counting valid coordinates

We are given a matrix $D = (d(i,j))_{1 \leq i,j \leq n}$ such that $d(x,z) \leq d(x,y) + d(y,z)$ for each $1 \leq x,y,z \leq n$. It is also known that $d(x,y) \in \mathbb{N}$ (In this question $0 \in ...
2
votes
2answers
49 views

Deciding whether a given graph has an f-factor or not!

Given a graph $G$ with $n$ vertices and a function $f$ from $\{1,2,...,n\}$ to non-negative integers, Does there exist an efficient (for example polynomial time) algorithm, that decides whether $G$ ...
1
vote
1answer
136 views

Probability of each edge in K-clique [closed]

For $c \in R$ and $k \in N$, $k \geq 3$ let $p_{k,c} := n^{\frac{−2}{k+1}}log^c(n)$. I would like to prove that exists $c\in R$ such that every edge in the random graph $G(n,p_{k,c})$ lies in a copy ...
0
votes
1answer
69 views

Using upper bound information in graph search

I am using A* (A-Star) to search a graph. A* algorithm takes advantage of the information $h(x)$, which is a lower bound of the distance between a vertex $x$ and the destination vertex. In other ...
2
votes
2answers
77 views

About the roots of the matching polynomial

Can someone kindly give me an expository reference on matching polynomial and its roots? (there is a proof that they are always real?) I saw these two related discussions, Roots of matching ...
2
votes
0answers
60 views

Are there two-sided $\varepsilon$-expanders with independent sets of size $(1-\varepsilon)n$?

Terry Tao's notes on expander graphs has the following exercise: Exercise 13 Let $G$ be a $k$-regular graph on $n$ vertices that is a two-sided $\epsilon$-expander for some $n > k \geq 1$ and ...
1
vote
1answer
204 views

Probability of connected graph on torus

Let $G = (V, E)$ be a graph on n vertices constructed in the following way: Each vertex $v \in V$ is positioned uniform randomly in $[0, 1] × [0, 1]$. Connect two vertices $u, v \in V$ if $d(v,u) ≤ ...
2
votes
0answers
51 views

Fixing (non)-independency of a the subfamilies of finitely many events.

I'm would be interesting in any construction of a probability space with n events (n is given), where for every subset of these events, it is also given whether or not, the family is mutually ...
1
vote
0answers
48 views

clustering in a graph on boolean functions

Fix some n and k. Consider the following directed graph: vertices are all functions $2^n\rightarrow 2^n$ and a vertex f has an edge to a vertex g for every h such that $f=h\circ g$ and h depends only ...
2
votes
2answers
203 views

Find all faces in a graph from list of edges

I have the information from a undirected graph stored in a 2D array. The array stores all of the edges between nodes, e.g. graph[3] might be equal to [1,8,30] and represents the fact that node 3 ...
3
votes
2answers
256 views

Graph game minimum vertex degree

Consider the following graph game, given a graph $G=(V,E)$ on $n$ vertices with minimum degree $ >> log(n)$. Players are BR and MA (BR moves first): BR claims an unclaimed edge from $E$, adds ...
2
votes
1answer
148 views

Solving assignment problem using Hungarian method vs min cost max flow problem

The traditional solution for the assignment problem is the Hungarian method - it's complexity is O(V^4) or O(V^3) if using Edmonds method. However, it can also be reduced to a min cost max flow ...
5
votes
3answers
103 views

Minimize distance between centroids of subsets of points

In a n-dimensional space, I want to divide a set of m points into v (non-empty) subsets. I want to minimize the sum of the pairwise Euclidean distances between the centroids of the resulting subsets. ...
3
votes
1answer
138 views

Polygamous stable marriage/ assignment problem

I'm not sure under which 'algorithm' it falls under, but here is the problem: I need to match each person to 5 people from the opposite gender (each guy gets 5 girls, each girl gets 5 guys). Not all ...
-1
votes
1answer
43 views

How to construct a semi-positive definite matrix in this form: (L=D-A')

As known, the graph Laplacian $L = D - A$ is semi-positive definite. What if there is a matrix $A'$ where $$ A'_{ij} = \begin{cases} A_{ij}, \quad if A_{ij} >0 \\ -\varepsilon, \quad if A_{ij} = ...
3
votes
1answer
97 views

Existence of subgraphs when given its degree sequence

For a given simple graph $G$ with $n$ vertices $v_1,v_2,\dots v_n$, the corresponding degree sequence is $d_1,d_2,\cdots,d_n$. My qusetion is: How to determine whether there exist subgraphs in $G$ ...
4
votes
0answers
175 views

Navigation in a graph

The problem Let $G=(V,E)$ be a graph. $k = O\left(\log(|V|)\right)$ distinct vertices are picked randomly from $V$. We call the set of chosen $k$ vertices $T$. Assumptions about the graph: You may ...