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, ...

learn more… | top users | synonyms

1
vote
1answer
255 views

Expression for summation involving factorial

It is known that $ \sum_{k = 0}^{n} {n \choose k} = 2^n$ and $ \sum_{k = 0}^{n} {n \choose k} (!k)= n!$. But is it known what $ \sum_{k = 0}^{n } {n \choose k}(k!)$ is equal to?
4
votes
0answers
72 views

Existence a class of graphs with special property

In the following, suppose all graphs are simple and finite. For a given graph $G$, we denote its complement by $\overline{G}$. Let $*$ be a binary operation among graphs, such as Cartesian product, ...
0
votes
0answers
140 views

Graph Coloring: Two adjacent vertices share same color

Consider, subgraphs $G_1, G_2,...... G_x$ of graph $G$. Each subgraph has $k$ vertices. Now, Fix subgraph $G_1$ and consider another subgraph $G_k$ where $1 <k \le x$. The edge set ...
3
votes
1answer
59 views

Is transitive reduction for a direct acyclic graph really unique? [closed]

According to Wikipedia, "If a given graph is a finite directed acyclic graph, its transitive reduction is unique" Here is what I think might be a counter-example: Imagine a diamond-shaped DAG where ...
2
votes
0answers
79 views

Isomorphism with fixed number of Permutations [closed]

Suppose, I have a fixed number of permutations for each sub-graph to determine isomorphism of whole graph. Is it possible to determine efficiently ? For example, $G, H$ are isomorphic graphs. For ...
-1
votes
1answer
58 views

A plane graph problem [closed]

Let G be a planar graph, with edges colored red and blue. Show that there is a vertex v such that going round the vertex in a clockwise direction we encountered no more than two change of colors. Has ...
0
votes
0answers
76 views

Generating set of Graph-Automorphism from Direct Product

Notation: $H$ is the adjacency matrix of graph $\mathcal{H}$ . $$H = \begin{bmatrix} H_{(3)} & R_{(3, 2)} & R_{(3,1)} \\ R_{(3,2)} & H_{(2)} & R_{(2,1)} \\ R_{(3,1)} & R_{(2,1)}...
0
votes
0answers
119 views

Closed form solution of a complex recurrence relation

I am looking for a closed form expression for $ST(n, k)$ defined as $$ ST(n, k) = \sum_{s = 0}^{n - k} {{n - k} \choose s} QT( k + s, k + s - 2, k), $$ where $QT( n, m, k)$ is defined by the ...
3
votes
0answers
67 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 ...
7
votes
2answers
207 views

Graphs with prescribed numbers of k-cliques

Let $(a_1,a_2,\dots, a_n)$ be a sequence of non-negative integers. Q. When does there exists a simple graph $G$ such that its number of $k$-cliques is $a_k$ (that is $G$ has $a_1$ vertices, $a_2$ ...
3
votes
1answer
77 views

Triange-free graph and its complement has Lovász number > 3

I found an example by the method in the paper Explicit Ramsey graphs and orthonormal labelings by Noga Alon 1994. The graph is around $10^6$ vertices, anyone knows smaller graph which is Triangle-free ...
1
vote
3answers
130 views

Hadwiger number and minimal degree

Suppose $G$ is a finite simple graph and $\eta(G)$ is the maximal $n\in\mathbb{N}$ such that $K_n$ is a minor of $G$. If $\delta(G)$ is the minimal degree, do we have $\delta(G)\leq\eta(G)$?
1
vote
0answers
24 views

History of the Vertex Disjoint Cycle Cover with Minimal Edgeweight Sum

Questions: who first posed the problem of determining a collection of (directed) cycles, whose edgeweight sum is minimal and, for which each vertex belongs to exactly one of the cycles? who came up ...
-1
votes
1answer
54 views

Hadwiger number of total graph

Let $G=(V,E)$ be a finite, simple, undirected graph, and let $T(G)$ be its total graph. The Hadwiger number $\eta(G)$ is the maximum $n\in\mathbb{N}$ such that $K_n$ is a minor of $G$. Is there an ...
1
vote
1answer
59 views

Asymptotic formula for average geodesic length on graph?

If $G$ is a finite, connected simple graph then is there an expression for the average geodesic length? That is suppose I know two nodes $n_1$ and $n_2$, the number of edges in my graph and at those ...
-1
votes
1answer
95 views

Total chromatic number and total clique number

Let $G=(V,E)$ be a finite, simple, unconnected graph. We define the total graph $T(G)$ of $G$ as follows: $V(T(G)) = (V\times\{0\}) \cup (E\times\{1\})$, $E(T(G)) = E_v \cup E_e \cup E_{v+e}$, where ...
0
votes
0answers
42 views

Generating tournaments inductively

This is a somewhat vague question, but I'm interested in ways to create a strong tournament from one or more smaller tournaments. Obviously, the disjoint union of two tournaments is a new tournament, ...
3
votes
1answer
106 views

Zero-one law in binomial random graph model $G(n,p)$

Consider the binomial random graph model $G(n,p)$ with $0<p<1$. We say that $G(n,p)$ satisfies the Zero-One law if for every first order property $Q$ one has $\lim\limits_{n \rightarrow \infty} ...
0
votes
0answers
47 views

Reconstructing a graph from set of sequences of edges

I have the following problem to solve: Given a set of sequences of edges of an undirected, planar, connected graph, find a "reasonable" reconstruction of the graph. There is an unknown number of ...
0
votes
0answers
24 views

is any closed form relation that can state the error probability of code versus its variable and check node degree distributions?

In Low Density parity check code design, when bit (or frame) error probability of code is the objective of the design, we need a closed form relation between error probably (or even an approximate or ...
0
votes
0answers
64 views

Binary operations on graphs

Are there "binary operations" on graphs like in (https://en.wikipedia.org/wiki/Graph_product), which make the set of all graphs ("under consideration") a (abelian) group or a (commutative) ring or a ...
1
vote
3answers
214 views

Existence of special graph

Let $G$ be a $n$-vertices graph and $\lambda_1$ is the largest eigenvalue of this graph. If $\lambda_1$ is an integer value, we can easily find the $\lambda_1$- regular graph with $n$ vertices. Now, ...
2
votes
1answer
34 views

Edge-disjoint paths avoiding some subgraphs

Let $G$ be a directed graph on $n$ vertices. Let $H_1$, ..., $H_k$ be marked subgraphs of $G$. (Specifically, each $H_i$ consists of a subset of the vertices of $G$ and a subset of the edges of the ...
2
votes
0answers
63 views

Number of $(2n-1)$-edge-colorings of the complete graph $K_{2n}$

I just started reading about graph theory and have a question (which might be trivial). How many $(2n-1)$ edge colorings of $K_{2n}$ are there? A vaguer question: can I write $K_{4n}= K_4 + K_4 +.......
1
vote
0answers
69 views

Missing count in number of perfect matchings

Let $f(G)$ give the number of perfect matchings of a graph $G$. Consider set $\mathcal N_{2n}=\{0,1,2,\dots,n!-1,n!\}$. Consider collection of all $2n$ vertex balanced bipartite graph to be $\...
4
votes
1answer
77 views

Degree of neighbors in a simple graph (friendship paradox variant)

Context: this question is a translation of a common informal phrasing of the friendship paradox ("Most people have fewer friends than most of their friends"). Note that the question is similar to, but ...
0
votes
1answer
75 views

Choosing directed subgraph in a triangulation

Consider triangulation $T.$ Is it always possible to choose such a subgraph $H$ of $T$ that has a common edge with every face of $T$ and can be directed in such way that indegrees of all vertices of ...
3
votes
1answer
66 views

Maximal acyclic subgraph

It is well known that the problem of finding a maximal acyclic subgraph of a digraph is NP-complete. Is this the case also when the digraph is symmetric ,i.e. if $(a,b)$ is a link, then $(b,a)$ is ...
2
votes
0answers
50 views

Universal path function for all small trees

Let $f$ be a function $f: [k]^2 \rightarrow [k]$ (Where $[k]$ is the set $ \{0,1,\dots,k-1\}$). A function $f$ is called $n$-universal path function if for every tree $T$ with $n$ vertices there ...
0
votes
3answers
104 views

How to do a clockwise ordering of a planar graph in order to define its faces?

I am currently making an algorithm for planar graphs that I need to triangulate so they become maximally planar (that is triangulated and planar) given only the lists of neighbors for each node : no ...
3
votes
2answers
145 views

Characterizing graphs whose Incidence Matrix has the all ones vector in its row span

Suppose we have a simple connected graph $G=(V,E)$. Then let $A$ be its $|E|\times |V|$ incidence matrix. Here I am considering the unoriented incidence matrix. I want to known when the row span of $A$...
3
votes
0answers
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)...
1
vote
0answers
59 views

bounded degree graph colouring.

I was wondering if anyone could provide references on the following: Is determining the chromatic number of a bounded degree graph APX-complete? 2.I've seen the result that states it is NP-hard ...
3
votes
0answers
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 ...
4
votes
1answer
83 views

Choice number of embedded graphs

For given $g$, consider the family of graphs which may be embedded to the compact orientable surface of genus $g$. For this family, consider maximal clique $\alpha(g)$, maximal chromatic number $\chi(...
-3
votes
1answer
69 views

Connected homogeneous graphs [closed]

Let's call a simple, undirected graph $G=(V,E)$ homogeneous if for every $v,w\in V$ there is a graph isomorphism $\varphi:G\to G$ such that $\varphi(v)=w$. It is clear that every finite homogeneous ...
3
votes
1answer
179 views

A spectral graph theory problem

Let $S$ be a zero-free subset of the group ${\bf Z}_2^n$ and $\Gamma={\rm Cay}({\bf Z}_2^n,S)$ be a bipartite Cayley graph. For some choices of $S$, the graph $\Gamma$ has $4$ distinct eigenvalues, ...
0
votes
1answer
64 views

Reference for Turan Density

I am working a 3-graph problem. I convert it to calculate Turan density, that is $lim_{n\to \infty}\frac{ex_3(n,F)}{\binom{n}{3}}$, where F is a3-graph. I'd like to know are there some methods and ...
4
votes
1answer
102 views

Behaviour of eigenspaces of adjacency matrices after a single change to the graph

Say I know the eigenvalues and eigenvectors of an adjacency matrix of an unweighted graph. Can I say anything about the eigenvalues and eigenvectors of an adjacency matrix of a graph with one extra ...
-3
votes
1answer
53 views

Hamiltonian path in countable connected graph such that $\text{deg}(v)=\omega$ for all $v$ [closed]

Is there a countable connected graph $G=(\omega, E)$ such that $\text{deg}(v)=\omega$ for all $v\in\omega$, but there is no Hamiltonian path in $G$?
19
votes
3answers
587 views

Which paths in a graph are orthogonal to all cycles?

Start with some standard stuff. Suppose we have a directed graph $\Gamma$. I'll write $e : v \to w \,$ when $e$ is an edge going from the vertex $v$ to the vertex $w$. We get a vector space of 0-...
3
votes
1answer
113 views

What is the densest bipartite graph with unique Hamiltonian cycle?

In a prior post regarding perfect matching, it was stated that the densest graph with a unique perfect matching cannot have more than $n^2$ edges, if graph has $2n$ vertices. Analogously, what is the ...
1
vote
1answer
135 views

Number of eigenvalues of a Cayley graph

Let $G=Z_2^n$ and $S\subset G$. Is there any relation for number of distinct eigenvalues of $\Gamma=Cayley(G,S)$ graph depending on $n$ and $|S|$, or at least diameter of $\Gamma$? If you have any ...
3
votes
1answer
165 views

Alternative parallel paths

There are $n$ non-intersecting strings (with ends $x_1,\dots, x_n$ and $y_1,\dots, y_n$). An additional string intersects the first $n$ strings somehow. All the intersections are simple (vertices of ...
2
votes
1answer
111 views

Size of connected components of a graph via its spectrum

I know that we can determine the number of connected components of a graph from the eigenvalues of its Laplacian matrix. My question is: Is there a way to understand the size of each connected ...
7
votes
1answer
287 views

Choosing two-colorable subgraph in a triangulation

Consider a planar graph $G$ which is a triangulation. Is it possible to find a two-colorable subgraph $H$ of $G$ which has a common edge with every face of $G$? It is known that it is not always ...
0
votes
2answers
96 views

Shortest path in a weighted graph with coloured edges

I have a weighted undirected graph with $N$ veritces and $M$ edges with 10 different colours in the whole graph. Each time I pass edges of different colour I have to pay additional fee equal to $K$. ...
6
votes
1answer
105 views

Which graphs embedded in surfaces have symmetries acting transitively on vertex-edge flags?

A vertex-edge flag in a graph is a vertex together with an edge incident to that vertex. Given a graph $\Gamma$ embedded in a compact oriented surface $S$, when does the group of homeomorphisms of $S$...
3
votes
1answer
191 views

Self-containing trees

Suppose that $r^2-r-1=0$ and that $T$ is the tree with root $1$ such that the children of each node $x$ are $rx$ and $x+1$. Remove all duplicates as they occur, and let $T(r)$ denote the remaining ...
2
votes
0answers
85 views

Maximum number of $4$-cycles

Suppose we have a balanced bipartite planar maximum degree $k$ graph. How many such graphs on $2n$ vertices have at most $f(n)$ maximum number of $4$ cycles for a given function $f:\Bbb R^+\...