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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0answers
104 views

Efficiently counting all paths of length n in a graph with vertex visitation contraints [closed]

I have a graph G with two classes of vertices. The first class represents no resource limitation entities and can be visited an unlimited number of times in any path traversal. The second class of ...
0
votes
2answers
134 views

“Nice” and “nasty” partitions in graphs

Let $G=(V,E)$ be a simple, undirected graph, that is $V$ is a set and $E \subseteq [V]^2 = \{\{v,w\}: v,w \in V \land v\neq w\}$. For $v\in V$ and $S\subseteq V$ we set $$N(v,S) = \{w\in S: \{v,w\} ...
1
vote
0answers
33 views

One sided Satisfactory Partition problem

The Satisfaction Partition problem is to decide if a graph has a vertex partition (U,V) into non empty parts where each vertex has as many neighbours in its part as in the other part. This problem is ...
1
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0answers
367 views

Possibility of Disconnected Subgraphs of a $k$ Connected $r$ regular Graph under a given condition

Context: Given a adjacency matrix A of a $r$-regular graph $G$ (not complete graph $K_{r+1}$) . $G$ is $k$ connected. The matrix A can be divided into 4 sub-matrices based on adjacency of vertex $x ...
1
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0answers
35 views

Weak law for component count of Erdos-Renyi random graphs

Penrose and Yukich derive a weak law for functionals of binomial point processes, which implies a law of large numbers for the component count of random geometric graphs. Do similar results exist for ...
8
votes
2answers
232 views

Dividing the edges and diagonals of a polygon among disjoint sub-polygons

Let $P$ be a convex $n$-gon ($n$ is odd and $n \geq 5$). Determine the smallest $m$ such that all edges and diagonals of $P$ can be covered by the edges of $m$ convex sub-polygons of $P$ which ...
1
vote
0answers
98 views

From Planar Graphs To Tangent Circles

I have a conjecture: "For each planar graph with vertices $V_1, V_2,\ldots, V_n$ there exist disjoint circles $w_1,w_2,\ldots,w_n$ in the plane, such that for every $i,j$, $w_i$ is tangent to $w_j$ ...
12
votes
3answers
1k views

A question on representation of graphs

Take a complete graph $K_n$. You want to assign a vectors from $\Bbb F_2^d$ to every edge such that sum of vectors in every simple cycle does not sum to $0$ vector. The question is what is minimum $d$ ...
5
votes
1answer
153 views

Modification of matching

Suppose i have an $n \times n$ random bipartite graph and suppose that i repeat the following process $n$ times. At the start (stage 1) each edge is selected independently with probability $p(n)$, and ...
2
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1answer
80 views

Images of interval edge coloring

I found out the definition of interval edge colorings, concept put by Kamalian in various papers but could not find a graph depicting an example. Where can I find pictures of explicit examples of ...
8
votes
1answer
213 views

Smallest strongly regular graph whose automorphism group is not vertex transitive?

I'm looking for a small strongly regular graph whose automorphism group is not vertex-transitive. This answer to a different question shows that the Chang graphs on 28 vertices are such graphs. Is ...
1
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0answers
135 views

Cayley graphs with special subgraphs and some related problems

I asked some questions about finite Cayley graphs with special type of subgraphs which has been answered by Dear Prof. Godsil. It can be seen in the MO page with address: Cayley graphs and its ...
1
vote
0answers
58 views

Interpreting (Fiedler) spectral bisectioning

I would appreciate help on how to interpret the results of spectral bisectioning of a graph. Given a $G=(V,E)$ with size $N$ represented by $Q$ its Laplacian matrix where the eigenvalues are ordered ...
-2
votes
1answer
107 views

Two graph structures on $\text{Hom}(G,H)$

By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq [V]^2 := \{\{a,b\}: a\neq b \in V\}$. A graph homomorphism between graphs $G, H$ is a map $f:V(G)\to V(H)$ such that $\{v, ...
5
votes
1answer
89 views

Resource Constrained Routing with Refueling

What are good algorithms (resp. models) for calculating optimal or near optimal routes while taking into account fuel consumption, options for refueling and, limited tank capacity? Especially modeling ...
12
votes
1answer
156 views

Reconstruction Conjecture: are almost all digraphs reconstructible?

The Reconstruction Conjecture for simple graphs remains unresolved. Most attempts I've seen at resolving the conjecture aim at proving it to be true (or partially true). I don't believe there is a ...
4
votes
0answers
48 views

Existence of certain graph gadget related to coloring odd hole free graph

Wondering about the existence of graph gadget related to coloring (or 3-coloring) odd hole free graphs. Let $G$ be simple $k$-chromatic connected graph with two vertices $u,v$. Is it possible $G$ to ...
1
vote
1answer
163 views

“Canonical” graph structure on $\text{Hom}(G, H)$

By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq [V]^2 := \{\{a,b\}: a\neq b \in V\}$. A graph homomorphism between graphs $G, H$ is a map $f:V(G)\to V(H)$ such that $\{v, ...
1
vote
0answers
146 views

Complexity of reordering a matrix which consists independent sub matrices

Introduction: Given a matrix A of a $k$ regular graph G. The matrix A can be divided into 4 sub matrices based on adjacency of vertex $x \in G$. $A_x$ is the symmetric matrix of the graph $(G-x)$, ...
2
votes
0answers
45 views

Maximum cardinality general factor of a graph

Given a graph $G=(V,E)$ and a set of integers $B(v)$ associated to each vertex, a general factor of $G$ is a set of edges $F\subseteq E$ such that the degree of each vertex $v\in V$ in the graph $(V, ...
0
votes
0answers
42 views

Enumeration of simple graphs with given degree distribution/sequence [duplicate]

Is there any exact formula for asymptotic/exact enumeration of simple graphs with given degree sequence? I just found some results about it, but the formula is hold on for some conditions, for example ...
1
vote
0answers
109 views

What are constructions for induced $C_5$-free graphs?

During a recent workshop, the question came up whether there are some constructions for graphs that are induced $C_5$-free, but they contain "everything else," so we don't want to forbid $C_5$'s, ...
1
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0answers
33 views

Small degree vertices in an epsilon-tough graph

We say that a graph is t-tough if by deleting a set if vertices $S$, the resulting graph will have at most $|S|/t$ connected components. We say that a graph is minimally t-tough if the deletion of an ...
4
votes
0answers
88 views

Prove or disprove a claim about covering a polytope by convex polytopes in a certain way

Here is the claim: Given a polytope $K$ in a unit ball in $\mathbb{R}^d$, there exists a universal constant $C(d)>0$ depending only on $d$ and a countable collection of convex polytopes ...
3
votes
1answer
86 views

$P_3$-factors for 3-regular, 3-connected cubic graphs

Suppose that $G=(V,E)$ is a simple graph. We know if $G$ is 3-regular, 3-connected and $|V|=4k$ for some $k\in \mathbb{N}$, then $G$ has a $P_4$-factor. Question. Let $G=(V,E)$ be 3-regular, ...
0
votes
0answers
17 views

Complexity of edge coloring graphs of sufficiently large maximum degree

I am interested in the complexity of edge coloring graphs with $\Delta(G) > |V(G)|/3$. This is closely related to the Overfull conjecture (OC). Conjecture/Question: If a simple graph G with n ...
-1
votes
1answer
58 views

How to generate computational data in graph theory?

For a given number of nodes how many non-isomorphic graphs are available? Might be this is an open problem. For less number of vertices some computational statistics available. I want to get all ...
5
votes
2answers
258 views

Four Dimensional Rook Domination

Let $\gamma(G)$ denote the domination number of a graph, and $G\,\square\,H$ denote the cartesian product of two graphs. Then $K_8\,\square\, K_8$ is the rook graph, whose vertices are the squares of ...
4
votes
0answers
98 views

Application of finding shortest paths on Cayley graphs

For a fixed integer number $m$, Consider Cayley graph defined by all m-cycles in Symmetric group $Sym(n)$. I know that for $m=2$, there are some applications of finding shortest paths (or distance ...
2
votes
0answers
35 views

Is this infinite family of non-trivial snarks resulting from the first Celmins-Swart?

Non-trivial snark is cubic graph with chromatic index $4$, girth at least $5$ and doesn't to contain three edges whose deletion results in a disconnected graph, each of whose components is nontrivial. ...
9
votes
1answer
277 views

Travelling Salesman Problem: Can the nearest neighbor algorithm be $n$ times longer than the optimal solution?

This is inspired by a recent question. Given a positive integer $n\in\mathbb{N}$, is there a setting of finitely many points and a designated "starting point" $s$ in $\mathbb{R}^2$ such that the ...
6
votes
1answer
189 views

Length of nearest neighbor path in travel salesman problem

Given $n$ nodes uniformly distributed in $[0,1]^2$, consider the nearest neighbor algorithm to solve traveling salesman problem, i.e., each time I select the nearest neighbor not visited so far as the ...
0
votes
0answers
73 views

What is the diameter of the Cayley graph of $S_n$ w.r.t the generator $\{(12),(23), . . . ,(n − 1 n) \}$? [duplicate]

Given a symmetric group $S_n $ and the generator set : $S = \{(12),(23), . . . ,(n − 1 n) \}$ is there any closed form expression for the diameter of the Cayley graph generated by this set of ...
0
votes
2answers
114 views

Generate all non-isomorphic partitions $\pi = \{ \{1, …, n-1\}, \{n\} \}$ for all graphs of order $n$

Let $G$ be any connected, undirected, and unweighted graph of order $n$. Let $\pi = \{ \{ 1, ..., n-1 \}, \{ n \} \}$ be partitioning of $G$ such that always $n-1$ vertices are in the first cluster ...
5
votes
1answer
143 views

partition of a convex set into squares

Let $P$ denote the perimeter function. It's not hard to prove that for any rectangle $R$ in $\mathbb{R}^2$, $R$ can be partitioned into a countable collection of squares $\{Q_k\}_{k=1}^{\infty}$ such ...
4
votes
0answers
40 views

Digraph weak connectivity in $O(V)$ space and $O(V+E)$ time

A digraph is called weakly connected if its underlying undirected graph is connected. You are given a digraph $G$ with $V$ vertices and $E$ edges as a read-only data structure consisting of lists of ...
1
vote
1answer
141 views

How many triangles can a connected graph with $n$ vertices and $m$ edges have?

I am very interested in the maximum number of triangles could a connected graph with $n$ vertices and $m$ edges have. For example, if $m\leq n−1$, this number is $0$, if $m=n$, this number is $1$, if ...
-1
votes
1answer
50 views

Sizes of maximum matchings in a finite, simple, undirected graph

Let $G=(V,E)$ be a finite, simple, undirected graph. We say that a matching $M\subseteq E$ is a maximum matching if for all $e\in (E\setminus M)$ the set $M\cup\{e\}$ is not a matching any more. ...
5
votes
0answers
182 views

Unique Nash equilibrium games

Multicast network design game is a special case of a general network design game (http://www.cs.cornell.edu/home/kleinber/focs04-game.pdf) in which there is a target vertex $t$ and $n$ rational ...
6
votes
2answers
397 views

Embedding of planar graphs

I've recently come across the following lemma. Lemma (Valiant): A planar graph $G$ with maximum degree $4$ can be embedded in the plane using $O(|V|)$ area in such a way that its vertices are at ...
0
votes
1answer
52 views

Extracting path information for a directed acyclic graph

For a research problem I am tackling, I have a directed acyclic graph $G(V,E)$. With every node in $V$, I have a variable $y$ associated. Now, given two nodes $i$ and $j$, I would like to have the sum ...
0
votes
0answers
55 views

Linear intersection number of a product of graphs

A linear hypergraph is a pair $\pi=(X, L)$ where $X\neq \emptyset$ is a set and $L\subseteq {\cal P}(X)$ has the following properties: for $e\in L$ we have $|e|\geq 2$; if $e_1\neq e_2 \in L$ then ...
8
votes
1answer
203 views

Cheeger Numbers for 3-regular Graphs

A student wanted a challenging Graph Theory programming project and I had him try to determine the maximum value of the Cheeger number (isoperimetric number) among all 3-regular graphs of order $n$, ...
3
votes
2answers
313 views

Geometry, Number Theory and Graph Theory of n-gon, permutation and graph labeling?

Given $n$ and $t$ lengths $ l_i, 1\leq l_1<l_2<\cdots<l_t\leq n-1$, of directed diagonals within an $n$-gon such that $l_1+\cdots+l_t\neq 0 \pmod n)$. Does it exist a directed path within ...
1
vote
0answers
48 views

Is there a polynomial time algorithm for Poly-trees (Oriented trees) isomorphism?

In terms of graph isomorphism complexity classes Trees have a polynomial time algorithm and Directed Acyclic Graphs (DAG's) do not (so far). What about Poly-trees (oriented trees)? These are DAG's ...
4
votes
0answers
94 views

Counting vertex-permutations of a finite tree which rip all edges

Given a finite tree $T$ with $n$ vertices labelled $1,\dots,n$, we say that a permutation $\sigma$ of $1,\dots,n$ rips all edges if $\{\sigma(i),\sigma(j)\}$ is never an edge for every edge $\{i,j\}$ ...
4
votes
0answers
80 views

Graphs with no homomorphism and no minor relation

What is an example of two simple, undirected graphs $G,H$ such that there are no graph homomorphisms between $G, H$, and $H$ is not a minor of $G$, and $G$ is not a minor of $H$ ? Definition of ...
0
votes
0answers
45 views

Expected length of minimum spanning trees

For a simple, finite, connected and complete graph $K_n = (V(K_n), E(K_n))$ with vertex set $V(K_n)$ and edge set $E(K_n)$, we assign a non-negative independent and identical distributed random weight ...
0
votes
0answers
66 views

Probabilities in a directed graph

Given a directed graph of "n" vertices, having on average "m" out-edges each, what is the probability that an arbitrarily chosen vertex will belong to a unique circuit? Also, how does that ...
4
votes
3answers
222 views

How networks with high largest eigenvalues are more robust?

In the literature, it is sometimes indicated that network with high value of largest eigenvalue (either adjacency matrix or its Laplacian counterpart) are more robust against link/node removals. ...