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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-3
votes
0answers
25 views

Calculate minimal number of nodes? [on hold]

Calculate minimal number of nodes? in a loopless simple undirected pi-partite graph. that has exacatly 144 nodes
-3
votes
0answers
27 views

Directed and undriected trees [on hold]

How many different directed trees can be obtained if we assign all possible orientation to the edges of an undirected tree having exactly 7 nodes? how many of them will be rooted(directed) trees?
0
votes
0answers
33 views

Determine number of directed trees and rooted trees obtainable [on hold]

I've been doing some exercices about graph theory and I find myself stuck on this one with no idea of to proceed. Here's the question : how many different directed trees can be obtained if we assign ...
-2
votes
0answers
45 views

Is there such a thing as cyclic Hasse diagram for posets? [on hold]

If so can you name one ? If not how to prove that there is none? Thanks !
4
votes
1answer
165 views

How to find or constrain “particularly good” (two-sided) spectral expanders?

I'm new to graph theory, but a response to a question I asked a while ago introduced me to the concept of expander graphs. A k-regular graph (henceforth "graph") on n nodes has eigenvalues k = λ1 ≥ ...
2
votes
0answers
46 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 ...
1
vote
1answer
66 views

How to calculate the maximum number of rainbows for arbitrary graphs?

This question is inspired by problem 1 of the combinatorics test of the 2012 third round iranian olympiad which is as follows: We've colored edges of $K_n$ with $n-1$ colors. We call a vertex rainbow ...
5
votes
1answer
125 views

Integral straight-line embeddings of planar graphs

Wikipedia says (in the article on Fáry's theorem), "Heiko Harborth raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers. The ...
2
votes
1answer
79 views

Distance between two networks

Suppose you have networks A and B, each with a set of nodes and edges. You want to measure how similar the networks are to each-other. None of the nodes or edges are labelled. What are the metric(s) ...
2
votes
0answers
59 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 ...
4
votes
2answers
123 views

Less general edge reconstruction problem for simple graphs

Let $G$ be a simple graph. Let $E^-(G)$ denote the set of (isomorphism classes) of subgraphs of $G$ that can be obtained by deleting a single edge of $G$. Similarly, let $E^+(G)$ be the set of ...
10
votes
1answer
532 views

A generalization of the triangle counting problem for simple weighted graphs

One nice identity is $$tr(A^3)/6$$ which counts the number of triangles of a graph represented with adjacency matrix $A.$ It also implies that triangle counting can be performed in subcubic time. ...
28
votes
1answer
3k views

An edge partitioning problem on cubic graphs

Hello everyone, I already asked this question on the TCS Stack Exchange, but it has not been resolved yet. Maybe readers of this forum will have other ideas or information, although I suspect that ...
4
votes
4answers
312 views

Determine if a graph has a large clique

This question is quite specific and practical. I hope it is still relevant for MO and will not be removed. I have a collection $\mathcal{C}$ of graphs having from 5000-6000 vertices and edge density ...
4
votes
1answer
86 views

Extremal graph theory for directed graphs

In extremal graph theory, there are results such as $$t(C_4,G)\geq t(K_2,G)^4,$$ where $G$ is an undirected graph, $C_4$ is a cycle graph on 4 nodes, $K_2$ is a complete graph of $2$ nodes, and ...
-1
votes
0answers
29 views

departure time/overlap algorithm [closed]

i'm looking for "departure time/overlap algorithm" or any other idea. Suppose you have n trains and each one has a performance profile(how much electricity they need at the current time while driving ...
2
votes
1answer
60 views

Eigenvalues of a graph and its one-edge-delation graph

Let $G$ be any graph with at least one edge and let $e$ be any edge of $G$. Let $G-e$ denote the subgraph of $G$ obtained by deletion of the edge $e$. Assume that $G$ has $n$ vertices. Suppose ...
3
votes
1answer
80 views

A problem related to routing in a graph

I have come across a new problem - I want to know whether this problem is similar to some existing problem or not. The new problem is this. There is a tourist who has a having the following ...
14
votes
1answer
1k views

What have simplicial complexes ever done for graph theory?

(I am asking in a somewhat tongue-in-cheek fashion, of course, but nevertheless...) Are there examples of results in "classical" [*] graph theory that have been achieved by using simplicial ...
3
votes
1answer
581 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 ...
4
votes
2answers
113 views

Is There a Graph that is Ramsey for $P_{2n}$ but is $C_{2n+1}-$free

Write $F\to G$ to mean that for every two coloring of the edges of $F$, there exists a monochromatic copy of $G$. Nesetril and Rodl proved that for a graph $G$, there exists a graph $F\to G$ with ...
0
votes
1answer
101 views

Spectral Graph Theory

Let G be an undirected graph, then Laplacian Matrix(L(G)) = Degree Matrix (D(G)) - Adjacency Matrix (A(G)). What is the relationship between laplacian and adjacency spectrum of undirected graphs?
0
votes
1answer
62 views

Laplacian matrix of a graph with negative weights

I am trying to calculate the Laplacian and Adjacency matrix of a graphs for positive and negative weights. If a graph be simple with only non-negative weight it is easier. But in my graph I have some ...
7
votes
4answers
287 views

Extremal examples for a folklore lemma on subgraphs of large minimum degree

It's a well known fact that a graph $G$ of average degree $d$ has a subgraph $G'$ of minimum degree at least $d/2$ and that the constant $1/2$ cannot be improved. The proof I know, which proceeds by ...
3
votes
0answers
118 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 ...
13
votes
3answers
475 views

Complexity of equitable partitions

We are talking about undirected simple graphs and partitions of their vertex sets into disjoint non-empty cells. Such a partition is equitable if for any two vertices $v,w$ in the same cell, and any ...
1
vote
1answer
128 views

Degree Sequence Problem on $k$-Partite Graphs

The general Degree Sequence Problem asks for a simple undirected graph (that is a graph without self-loops and with no more than one edge between any pair of nodes) for which it holds that the degrees ...
4
votes
1answer
114 views

Graph presentation of Lexicographic shifts

Consider a finite alphabet $\{0,1, \ldots, n-1\}$. Let $\Sigma_n = \mathop{\prod}\limits_{j=1}^{\infty}\{0, \ldots n-1\}$ be the set of infinite one sided sequences and $\prec$ the lexicographic ...
4
votes
1answer
168 views

Do graphs with large number of paths contain large chain minor?

Definition: A "$k$-chain" is a multi-graph obtained from a path of length $k$ by duplicating every edge. Note that the number of paths between two endpoints of a $k$-chain is $2^k.$ Question: Let ...
1
vote
0answers
29 views

Euclidean embedding of a graph based on 1-ring neighborhood distances only

Consider a graph $(V,E)$, $\vert V \vert = n$ and weights $\{l_{ij}\}$, where $l_{ij}>0$ iff there is an edge connecting vertices $v_i$ and $v_j$. Distances beyond the 1-ring neighborhood are not ...
14
votes
3answers
800 views

Subposets of small Dushnik-Miller dimension

The Dushnik–Miller dimension of a partial order $(P,{\leq})$ is the smallest possible size $d$ for a family ${\leq_1},\ldots,{\leq_d}$ of total orderings of $P$ whose intersection is ${\leq}$, ...
0
votes
0answers
23 views

“Mutant knots” generalizable to “mutant tangled graphs”?

Just in case: Take a link L (drawn into the plane with over- and undercrossings), draw a closed loop C on it which cuts L in four points, rotate the inside of C around 180° (align the cut points on ...
9
votes
3answers
698 views

Could a perfect squared square be split into two perfect squared squares?

This is a geometric puzzle though it might conceivably also define a special class of Pythagorean triples. A perfect squared square PSS is a square (as a plane figure) partitioned into smaller ...
5
votes
1answer
215 views

Bipartite Graphs arising from two k-partitions of a given Graph

Let $G$ be an $n$-chromatic connected graph. Let $(V_1, V_2, \cdots, V_n)$ and $(U_1, U_2, \cdots, U_n)$ be two partitions of $V(G)$ corresponding to proper n-colorings of $G$. Consider the bipartite ...
5
votes
0answers
72 views

Implementations of Tutte polynomial [reference request, of a kind]

This question is not a 100% fit for MO, but it is a serious question that can be viewed as a sort of reference request, and I think fits here more than elsewhere. I have been asked to write a chapter ...
2
votes
1answer
224 views

Is there a graph with 99 vertices in which every edge belong to a unique triangle and every nonedge to a unique quadrilateral?

99-Graph:Is there a graph with 99 vertices in which every edge(i.e. pair of joined vertices) belong to a unique triangle and every nonedge(pair of unjoined vertices) to a unique quadrilateral?
2
votes
1answer
109 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 ...
0
votes
1answer
82 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
1answer
45 views

Graph construction to double coloring & Hadwiger number

For any graph $G$ let $\eta(G)$ be the Hadwiger number of $G$. Is there for every graph $G$ a graph $2G$ such that -- $\chi(2G) = 2\chi(G)$, and -- $\eta(2G) = 2\eta(G)$? For each one of the above ...
3
votes
0answers
63 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 ...
0
votes
0answers
133 views

Induced graphs of cayley graph

I have a Cayley graph $Cay(G,S)$, its group presentation $G=< S | R >$ and it is a metric graph by assigning a length equal to 1 to each edge. I also have an induced subgraph of that Cayley ...
0
votes
1answer
65 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; ...
0
votes
1answer
54 views

Additivity of genus of a graph [duplicate]

Let $G$ be a finite simple undirected graph. Suppose there exist subgraph $G_1,G_2,\dots,G_n$ of $G$, such that $E(G_i)\cap E(G_j) = \emptyset$ and $|V(G_i)\cap V(G_j)| \leq 2$, for $i\neq j$. Then, ...
1
vote
1answer
116 views

Upper-bound for maximal-cliques on perfect graphs

It has been proved by Moon and Moser in 1965 that any finite simple graph has at most $3^{|V|/3}$ maximal cliques. Still, some hereditary classes of graphs have very few maximal cliques in comparison ...
3
votes
2answers
121 views

Definition of the Moebius Ladder Graph

I found two different definitions of the Moebius Ladder Graph, whose essential difference is, whether the smallest one shall be $K_4$ or $K_{3,3}$. according to Wikipedia ...
1
vote
0answers
57 views

Recovering Spherical Harmonics from Discrete Samples

Consider a collection of $N$ points on the 2-sphere chosen uniformly at random. Let's say that there's an edge between two such vertices if their geodesic distance is less than $r_N$. The resulting ...
2
votes
2answers
114 views

Which graphs generate a matroidal independence complex?

The independence complex $I(G)$ of a graph $G=(V,E)$ has as point set the vertex set $V$ and as simplices the independent sets of $G$. Now, if $G$ is a well-covered graph (where all maximal ...
1
vote
1answer
65 views

Is undirected short-simple-path-through-3-vertices decidable in polynomial time?

Consider the following language: $L=\{\langle G=(V,E),s,v,t,l\rangle\;|\;s,v,t\in V, l\in \mathbb{N} \wedge $ There exists a simple path from $s$ to $t$, going through $v$ of length $\leq l\}$. ($G$ ...
5
votes
1answer
825 views

Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants

Zeta functions abound in mathematics. Audrey Terras describes in Zeta Functions and Chaos three zeta functions--the zeta fct. of a projective non-singular algebraic variety; the Artin-Mazur zeta ...
0
votes
0answers
40 views

Preferential Attachment and salton similarity in directed networks

Preferential Attachment similarity between two nodes in an undirected graph is the degree of the first node multiplied by the degree of the second node. But what about directed graphs? Which degree ...