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

0
votes
2answers
113 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
140 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
39 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
133 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
vote
1answer
48 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
169 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
389 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
47 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 ...
2
votes
0answers
53 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
197 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
309 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
43 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
93 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 $\{i,j\}$ ...
5
votes
0answers
77 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
44 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
207 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. ...
8
votes
4answers
262 views

Diameter of random segment intersection graph?

I have an even number of points $n$ randomly distributed (uniformly) in a disk. Then the points are randomly connected to form $n/2$ segments, a perfect matching. Finally, I form the intersection ...
0
votes
0answers
70 views

Gromov-Hausdorff distance measure between minimum spanning trees

I am trying to compare minimum spanning trees through time. I have two questions: 1-Is it possible to measure the similarity between two minimum spanning trees with Gromov-Hausdorff distance measure ...
2
votes
0answers
53 views

Strongly minimal covering subsets of $\text{Ind}(G)$

Let $G=(V,E)$ be any undirected, simple graph. Let $\text{Ind}(G)$ be the set of independent subsets of $V(G)$. We say that $K\subseteq \text{Ind}(G)$ is a cover (by independent subsets) if $\bigcup K ...
5
votes
0answers
67 views

Determinantal formulae for Tutte polynomial

Let $G$ be a connected undirected graph. Then the number $ST(G)$ of spanning trees in $G$ equals the following specific value of the Tutte polynomial of $G$: $ST(G)=T_G(1,1)$. On the other hand, ...
4
votes
1answer
94 views

Is it true that any $3$-uniform hypergraph that is not $k$-colorable must have $\Omega(k^3)$ edges?

What is the best lower bound in terms of $k$ on the number of edges in a $3$-uniform hypergraph that is not $k$-colorable? Thanks in advance.
0
votes
1answer
93 views

Graph automorphisms that preserve independent sets [closed]

Let $G=(V,E)$ be a graph and $\mathrm{Ind}(G)$ be the collection of its independent sets. We call a graph automorphism $f:V \to V$ of $G$ good if it is non-trivial and ...
2
votes
1answer
43 views

Graph with finite chromatic number but infinite total chromatic number

Is there a graph $G$ such that $\chi(G)$ is finite, but there is no total coloring with finitely many colours?
3
votes
1answer
144 views

Partitioning a binary tree into vertex-disjoint subtrees

Say we have a labeled, binary unrooted tree $T$, i.e. each node has either 1 or 3 neighbors. Denote by $L(T)$ the set of leaves (degree-one nodes) of $T$. For some $L \subseteq L(T)$, denote by ...
7
votes
1answer
224 views

Chromatic number of graph defined on the set of permutations

For $n\in\mathbb{N}$ let $S_n$ denote the set of permutations on the set $\{1,\ldots,n\}$. Set $$E_n = \big\{\{\pi_1, \pi_2\}: \pi_1,\pi_2\in S_n \land \exists k_1 < k_2 <\ldots <k_r\leq n: ...
3
votes
0answers
46 views

The Class of Strong Resolving Graphs of Hamiltonian Outerplanar Graphs

I'll start with a couple important definitions. I'm not sure how well-known any of them are. Firstly, if $G$ is a graph, and $u, v \in V(G)$, say that $u$ is maximally distant from $v$, denoted $u\ ...
4
votes
2answers
144 views

Can we find 3 disjoint directed Hamiltonian cycles in the cube?

Let $D$ be the digraph on $2^d$ vertices with $d2^d$ edges that we obtain by directing each edge of the $d$-dimensional hypercube in both directions. Can we partition the edges of $D$ into $d$ ...
0
votes
0answers
84 views

When does the normalized graph Laplacian have eigenvalue 1?

Let $G= (V,E)$ be a finite, undirected and unweighted graph with $V = \{v_1,\ldots, v_n\}$. Denote by $d_i$ the degree of $v_i$, i.e. the number of vertices that are adjacent to $v_i$. Let $A$ be the ...
4
votes
1answer
196 views

Does the minor graph of graphs on $\mathbb{N}$ have an uncountable independent set?

By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\}: a\neq b \in V\}$. We write $V(G) := V$. If $S, T$ are disjoint subsets of $V(G)$ we say that ...
5
votes
1answer
291 views

Uncountably many countable graphs with no homomorphism between them

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 ...
1
vote
0answers
55 views

Proper edge colorings with no bi-colored 5-paths

Consider you want to properly edge color a graph such that it has no bi-colored cycle. Denote by $\alpha'(G)$ the least number of colors required to color the edges of $G$ in such a way. It is well ...
2
votes
1answer
126 views

Is there some quantitative version of interlacing of eigenvalues of a matrix under rank-1 update?

Given a real symmetric matrix $A$ and a vector $v$ of the same dimension we know that the eigenvalues of $A + vv^T$ are left interlaced by the eigenvalues of $A$. But do we have any quantitative ...
6
votes
1answer
112 views

Infinite graphs with “similar” Hom-sets

Let $G,H$ be infinite simple undirected graphs with the property that for any graph $X$ we have $|\text{Hom}(X,G)| = |\text{Hom}(X,H)|$. Does this imply that $G$ is isomorphic to a subgraph of $H$, ...
1
vote
0answers
35 views

Find paths in a graph that any 2 vertices can be reached through N of them

Given a undirected weighted graph. I would like to find a finite set of paths (consecutive vertices and edges) each shorter than L any two vertices can be reached through at most N(in my case N=4) ...
2
votes
0answers
118 views

Polynomial-time algorithm solving approximately a generalization of the travelling salesman problem

Take a graph $G$ and a number of sets of nodes of $G$. The problem is to find the shortest path passing through at least one node in each node set. If each node set consists of only one node, the ...
7
votes
1answer
116 views

Does the Tutte polynomial of iterated cone graphs detect isomorphism?

Let $T_G(x,y)$ denote the Tutte polynomial of a graph. Of course we may have $T_G(x,y) = T_H(x,y)$ for $G$ and $H$ non-isomorphic graphs. Now let $c(G)$ denote the cone graph of $G$, i.e., the graph ...
1
vote
1answer
60 views

Graph classes which are not perfect but the stability number = clique cover numer?

I have a result for graphs whose stability number=clique cover number, which naturally includes the perfect graphs, but I'm curious about if there are other known and well-definable graph classes ...
7
votes
2answers
144 views

Isomorphism problem on the class of induced subgraphs of a hypercube

A problem that I am currently studying translates to the problem of deciding whether two induced subgraphs of the hypercube $Q_k$ are isomorphic. Now it feels to me that this class of graphs is "too ...
0
votes
0answers
21 views

Cardinality of maximum independent set for a given degree distribution

Consider undirected graph $G(V,E)$. Assume that $f_n(k)$ be the probability mass function of degree of a vertex in $G$. Further, assume that $f_n(k)$ is an strictly decreasing function of $k$ with ...
-2
votes
1answer
75 views

How does deletion-contraction affect chromatic number? Can it increase chromatic number? [closed]

Question: In graph theory, contracting an edge or deleting an edge are basic operations in many topics such as graph minors or Wagner's theorem on planar graphs. And I'm interested in how these ...
2
votes
0answers
27 views

Minimality of maximal expansions of a hypergraph cover

This is a follow-up question to Maximal expansions of strongly minimal covers of hypergraphs. Let $H = (V,E)$ be a hypergraph, that is $V$ is a set and $E \subseteq {\cal P}(V)$. We assume $\bigcup E ...
2
votes
1answer
26 views

Maximal expansions of strongly minimal covers of hypergraphs

Let $H = (V,E)$ be a hypergraph, that is $V$ is a set and $E \subseteq {\cal P}(V)$. We assume $\bigcup E = V$. Moreover we assume that every $e\in E$ is contained in some maximal member $e'\in E$ ...
7
votes
3answers
271 views

Borel coloring of a graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$

The following question was asked in a comment by Joel David Hamkins in Graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$. Let $V$ be the set of all functions $f:\mathbb{N}\to\mathbb{N}$. ...
2
votes
1answer
253 views

Graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$

Let $V$ be the set of all functions $f:\mathbb{N}\to\mathbb{N}$. Let two distinct functions $f,g:\mathbb{N}\to\mathbb{N}$ form an edge if and only if they differ in exactly one input $n\in\mathbb{N}$. ...
11
votes
2answers
317 views

Graph $G$ with $\omega(G) = 2$ but $\chi(G) \geq \aleph_0$

Given an infinite cardinal $\kappa$, is there a graph $G$ that has no clique consisting of more than 2 points, but $\chi(G) = \kappa$?
5
votes
0answers
46 views

Perfect matchings of a regular, uniform, partite hypergraph

This is in relation to the question here. What, if any, are the known conditions for the existence of a perfect matching for a $r$-regular, $r$-uniform, $r$-partite hypergraph. I specifically ...
5
votes
1answer
132 views

Regular epimorphisms in the category of simple undirected graphs

Let $\textbf{Grph}$ be the category whose objects are graphs $G = (V,E)$ such that $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\} \subseteq V: a\neq b\}$. We sometimes write $E(G)$ for ...
7
votes
1answer
300 views

Under $\neg CH$, have countable unions of rationally independent numbers inner measure zero?

In their 1943 paper On non-denumerable graphs, Erdos and Kakutani suggest as likely the following proposition. (EK*) Suppose CH fails and $\lbrace M_n : n \in \omega \rbrace$ is a countable family of ...
23
votes
1answer
1k views

Algebraic proof of Five-Color Theorem using chromatic polynomials by Birkhoff and Lewis in 1946

I'm guessing everyone is familiar with Four Color Theorem which was proved by Appel and Haken using computers. A weaker version of this theorem is Five Color Theorem which states that a planar graph ...