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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6
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0answers
176 views

When can the Cheeger constant be well-approximated by ``Hamming balls''?

Given a graph G, the Cheeger constant is defined by $$ \DeclareMathOperator{\Vol}{Vol} h_G := \min_{S \subseteq V, \Vol S \leq (\Vol G)/2} \frac{|\partial S|}{\Vol S}. $$ Here, $\Vol S$ is the sum of ...
6
votes
0answers
595 views

Is Logical Min-Cut Problem, NP-Complete?

Logical Min Cut (LMC) Problem: Suppose that G = (V, E) is an unweighted digraph, s,t are two vertices of V, and t is reachable from s. LMC Problem states that how we can make t unreachable from s by ...
6
votes
0answers
342 views

What is this subclass of $k$-colorable graphs called?

The following property emerged naturally when I was playing with certain generalizations of Kneser graphs. Let $k>0$ be a natural number. Consider a property $P_k$ of graphs defined as follows. ...
6
votes
0answers
984 views

What is the best lower bound for the domination number in regular graphs of girth 5?

The following theorem is a classical result (see [Alon and Spencer, The probabilistic method, 2nd ed., Theorem 1.2.2]): Theorem: Let $G$ be a graph on $n$ vertices with minimum degree $d$. Then ...
6
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0answers
644 views

inverse eigenvalue problem on graph laplacian

I am trying to construct a graph Laplacian matrix from a set of eigenvalue. I've read several papers about inverse eigenvalue problems but to be honest I didn't understand clearly. Could somebody ...
6
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0answers
736 views

Simplicial Representations of (Hyper)Graph Complexes

For graph complexes, which are families of graph [on a fixed number of vertices n] closed under the deletion of edges, there is a natural simplicial complex capturing that information. Specifically, ...
5
votes
0answers
73 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 ...
5
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0answers
64 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, ...
5
votes
0answers
44 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
0answers
62 views

Approximating a max-cut's intersection with other cuts

(This is a cross-post from the Theoretical Computer Science Stack Exchange.) For the purposes of this question, a cut in a graph $G$ is the edge-set $\delta (S)\subseteq E(G)$ between some vertex-set ...
5
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0answers
101 views

A digraph related to permutations

A finite sequence of distinct real numbers of length $n$ determines a linear order of $\{1,\ldots,n\}$, by mapping position to rank; call this the permutation of the sequence. Consider the following ...
5
votes
0answers
205 views

When does a “stable” assignment of buyers into goods exist?

Consider a setting of $n$ buyers and $m$ goods. We have a value matrix $V\in[0,1]^{n\times m}$ specifying how much each buyer values each good (everything is open information here and there is no ...
5
votes
0answers
104 views

What's the variance in the Six Degrees model?

Recall the six degrees of Kevin Bacon game. You can even play the game at The Oracle of Bacon, and their search works via Breadth First Search. I interpret the punchline as saying that if I start ...
5
votes
0answers
162 views

Sets of spreads in graphs

Let $G$ be a graph. A $k$-spread is a set of cliques of order $k$ which partition the vertex set (so $k|n$, where $n$ is the number of vertices). A partial $k$-resolution of $G$ is a set of pairwise ...
5
votes
0answers
89 views

Independent domination number for grid graphs

Let $G_{n,m}$ be the $n \times m$ grid graph, i.e. $G= P_n \Box P_m$, and $T_{n,m}$ the $n\times m$ torus grid graph, i.e. $G= C_n \Box C_m$, where $P_n$ and $C_n$ indicate the path graph of length ...
5
votes
0answers
89 views

Subgraphs of planar trivalent graphs

Let's think about planar trivalent graphs. (Or you can dualize and think about triangulations if you prefer.) It's easy to come up with a list of 'planar trivalent graphs with boundary' such that at ...
5
votes
0answers
179 views

(Connected) Cayley graphs of PSL(2,q) from (2,3,n)-triples

Let $G = PSL(2,q)$. I'm interested in the Cayley graphs of $G$ generated by triples $(A,BAB^{-1},B^{-1}AB)$, where $A, B \in G$ are elements of order $2, 3$ respectively: such a triple generates all ...
5
votes
0answers
94 views

Mapping graphs to ordinals

Robertson-Seymour theorem implies that graph minor relation is a well-quasi-ordering, which means (among other things) that this relation can be extended to a well-order, and other result says that ...
5
votes
0answers
166 views

Are the roots of chromatic polynomials plus a fixed constant dense in $\mathbb{C}$?

Alan Sokal proved that chromatic roots are dense in the whole complex plane. I.e., if $P(G;z)$ denotes the chromatic polynomial of a finite simple graph $G$ evaluated at $z \in \mathbb{C}$, then ...
5
votes
0answers
90 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 ...
5
votes
0answers
120 views

Complexity of finding three perfect matchings with no edge in common in a bridgeless cubic graph

According to a conjecture: Conjecture (Fan & Raspaud, 1994) Every bridgeless cubic graph contains three perfect matchings with no edge in common. Equivalent statement here Main question: ...
5
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0answers
169 views

A linear optimization problem on a graph

Let $G=(V,E)$ be a finite graph and let $f$ be any positive function defined on the vertices. Put weights on the vertices $v_{i}$, way $w_{i}$ so that $\sum_{i=1}^{n}w_{i}\leq 1$. Assume that every ...
5
votes
0answers
188 views

Graphs with many positive eigenvalues of their distance matrix

Let $G$ be a simple connected graph $D(G)$ its distance matrix and $n_{+}(G), n_{-}(G)$ the number of positive and negative eigenvalues of $D(G)$ respectively. We call a graph $G$ optimistic if ...
5
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0answers
159 views

Diameter of subset sum graph

We have a finite set $X$, a weight function $w: X\rightarrow \mathbb{Z}^+$, and constants $k\leq c\in\mathbb{N}$. Let the weight $w(S)$ of a set $S\subseteq X$ be the sum of the weights of its ...
5
votes
0answers
98 views

making a graph well-covered without changing its Shannon capacity

This strongly relates to an earlier question of mine. Let $G$ be a graph, $\alpha(G)$ its independence number and $\Theta(G)$ its Shannon capacity. Question: can one 'add new vertices' to $G$ such ...
5
votes
0answers
145 views

How does the number of self-avoiding paths between two points scale, in a square/cubic lattice?

Consider two different infinite graphs, whose vertices are drawn from $\mathbb Z^2$ or $\mathbb Z^3$. Let $P_d : \mathbb Z^d \times \mathbb N \to \mathbb N$ for $d \in \{2,3\}$ be the function such ...
5
votes
0answers
288 views

Sequence of graphs with small $\lambda_1$ (the smallest nonzero eigenvalue of a regular finite graph)

The combinatorial laplacian on a finite graph $G$ can be defined as $ \Delta: \mathbb{C}^G \to \mathbb{C}^G$ sending the function $f:G \to \mathbb{C}$ to $(\Delta f)(v) = \sum_{v' \sim v} \big( ...
5
votes
0answers
238 views

Any approximation algorithms for self-avoiding walks?

I've a graph whose edges are weighted by probabilities, perhaps all equal. I would like to compute the overall probability of traveling between vertices x and y in the graph after I delete each edge ...
5
votes
0answers
362 views

A binary bipartite graph - reference request

Consider the bipartite graph whose partite sets are two disjoint copies of $\{0,1\}^n$, with an edge joining $u$ and $v$ if and only if there is no position in which both $u$ and $v$ have $1$; that ...
5
votes
0answers
399 views

“sum over labelings” representations of graph polynomials

It seems that there's a general way to go from "recursive" definition of a graph polynomials to "subset expansion" formulas. Furthermore, polynomials with subset expansion formulas often have a ...
5
votes
0answers
592 views

Counting graphs whose vertex induced subgraphs are members of a fixed set

Let $C$ be the class of graphs of girth $g$ or more. $C$ can alternatively be characterized as: $G \in C$ iff each of $G$'s vertex induced subgraphs on less than $g$ nodes is a forest. We can ...
5
votes
0answers
439 views

Natural models of graphs?

Motivation I want to capture the notion of natural models of finite graphs: How can natural predicates and natural relations on a given natural base class $D$ be defined? If this succeeds the ...
4
votes
0answers
30 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 ...
4
votes
0answers
113 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 ...
4
votes
0answers
92 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
127 views

Connection between connectivity and cohesion of a graph

Tutte [1] proved that, for every $3$-connected graph $G$ and vertices $u$ and $v$, there exists a nonseparating $uv$-path. A graph $G$ is $t$-cohesive if $G$ is connected, has at least two vertices, ...
4
votes
0answers
108 views

What is known about the complexity of this covering problem?

Let $G=(V,E)$ be a graph. A vertex set $X\subseteq V$ is called critical if $X\neq\emptyset$ and no vertex in $V\setminus X$ is adjacent to exactly one vertex in $X$. The problem is to find a vertex ...
4
votes
0answers
119 views

Hypercube edge-coloring problem

Question: Is there a pairing (a fixed point free involution) of the vertices of the $n$-dimensional cube graph, and a $2$-coloring of its edges such that the number of color changes needed to get from ...
4
votes
0answers
124 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' ...
4
votes
0answers
194 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 ...
4
votes
0answers
88 views

Actions of amenable groups on graphs with uncountably many ends

Let $G$ be a finitely generated amenable group acting transitively on an amenable Schreier graph $S$. Is it possible for $S$ to have uncountably many ends? An amenable graph with uncountably many ends ...
4
votes
0answers
132 views

What do we know about isospectral Cayley graphs?

Let $\Gamma_1=\Gamma_1(G_1,S_1)$ (respectively $\Gamma_2=\Gamma_2(G_2,S_2)$) be a Cayley graph for the group $G_1$ and the finite generating set $S_1$ (respectively $G_2$ and the finite generating set ...
4
votes
0answers
45 views

What is the probability of interpolating the Tutte polynomial of a planar graph from the values at the two hyperbolas?

The Tutte polynomial is a bivariate polynomial with positive integer coefficient which is a graph invariant and can be defined recursively. Evaluating it is $\#P$-complete even when restricted to ...
4
votes
0answers
155 views

Does the concept of connective constant make sense for any tiling of the plane?

First let me define what is the "connective constant" of a two dimensional lattice. Let $c_{n}$ denote the number of $n$ step self-avoiding walks starting from a fixed origin point in the lattice. ...
4
votes
0answers
144 views

Genus of the graph $K_{m,2,2,2}$

What is the genus of this complete $4-$partite graph, $K_{m,2,2,2}$, where $m \in \mathbb{N}$? Thanks in advance.
4
votes
0answers
259 views

Unique Domino Tiling

Question: how does one enumerate all star-convex $2n$-vertex sublattices of the plane that have the unique domino-tiling property? Definitions: A subset S of the xy-plane is star-convex if there is ...
4
votes
0answers
517 views

Does this graph property have a name?

I'm interested in a family of properties of connected simple graphs that comes up in percolation theory. Let $G$ be a simple connected graph. Now consider the set of subgraphs of $G$ that I will call ...
4
votes
0answers
108 views

What are the best bounds to date on the maximum girth of a cubic graph?

The girth of a graph is the length of its smallest cycle. Studying the maximum possible girth for a $k$-regular graph on $n$ vertices is a very well-studied problem. In the 1988 paper "Ramanujan ...
4
votes
0answers
477 views

Questions about dessin d'enfants, trees and their Shabat polynomials

This will be a series of questions, a few of which have been troubling me for quite a while now. Before I jump right in, let me first introduce a few notions which I will assume. (Note: All of these ...
4
votes
0answers
169 views

Rough structure of the double coset space/Graph bijections up to automorphisms

I am dealing with bijective maps $\pi:\Gamma_1\to \Gamma_2$ between two graphs with the same number of vertices $N=O(10)$. The graphs have a significant automorphism group (these are disconnected ...