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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7
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0answers
244 views

A maximum discrepancy hypergraph 2-colouring problem

This is sort of a hypergraph-ish question that I feel should be easy to prove or disprove but I can't see it right now. The setup is as follows. We have a vertex set partitioned in to sets ...
7
votes
0answers
698 views

Decomposition of graphs as symmetric differences of copies of $K_{a,b}$

I was wondering if the following decomposition of graphs has been studied, whether it has a name, and what the literature might be on it. Given a labelled graph G, we decompose its edge-set as a ...
6
votes
0answers
101 views

Has anyone seen these binary trees (Catalan-type related to the Gegenbauer polynomials and Motzkin paths)?

The OEIS entry A121448 enumerates binary trees with $n$ edges and $k$ vertices with outdegree 1. Has anyone seen these trees? The o.g.f. for this entry, $G(x,t)$, is essentially a discriminant ...
6
votes
0answers
70 views

Contradicting claims about complexity of directed path graphs isomorphism

Thesis and a paper give conflicting claims about the complexity of graph isomorphism for directed path graphs. Since this means GI is polynomial likely I am missing something or there is something ...
6
votes
0answers
233 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
0answers
273 views

Have topographs been studied before?

This is my first post on MO so I hope this question is suitable. I have quite a few definitions which I will need to state before my questions at the end of this post. Please let me know if anything ...
6
votes
0answers
160 views

Does the weak Hadwiger conjecture imply the Hadwiger conjecture?

For any cardinal $\kappa$, let $K_\kappa$ denote the complete graph on $\kappa$. We consider the following statements: (H) If $G$ is a graph and $\chi(G) = \kappa$ then $K_\kappa$ is a minor of $G$. ...
6
votes
0answers
299 views

the length of paths in a specific graph

Let $n$ be a positive integer and $K$ be the set of all the $2$-elements subsets of $\{1,2,...,n\}$,then $|K|= \binom{n}{2}$. Define $$S=\{P\subseteq K:\bigcup_{I\in P}I=\{1,2,...,n\}\}.$$ For any ...
6
votes
0answers
146 views

Uniformly sampling from the set of all simplicial maps

Let $K$ and $L$ be finite simplicial complexes that remain fixed throughout. How does one efficiently sample (according to the uniform distribution) elements from the finite set of simplicial ...
6
votes
0answers
389 views

Graphs with graphic imbalance sequences

Let $G$ be simple undirected graph and $e=uv\in E(G)$. The imbalance of the edge $e$ is the value $imb(e)=|d(u)-d(v)|$. Let $M_{G}$ denotes the imbalance sequence (or more correctly, multiset of ...
6
votes
0answers
308 views

Maximum fractional chromatic number of a 4-regular triangle-free graph (updated)

Let $G$ be a graph with maximum degree 4 and clique number 2. The fractional version of Reed's Conjecture tells us that $\chi_f(G) \leq 7/2$. But how high can $\chi_f(G)$ be? The Chvátal graph has ...
6
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0answers
196 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
681 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
347 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
1k 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
votes
0answers
667 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
votes
0answers
771 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
85 views

Is there a Ramsey theory for Kneser graphs?

Ramsey theory for graphs usually studies colorings of the edges of complete graphs. I'm interested whether there are any results about edge-colorings of Kneser graphs. More specifically, I'm most ...
5
votes
0answers
65 views

Chromatic numbers for coloring-constrained graphs

I am interested in any and all articles about chromatic numbers applying to constrained colorings of a graph. For example, if a graph must be (properly) colored so that there is a 2-color path ...
5
votes
0answers
97 views

“Edge Density” of Infinite Planar Graphs

Given an infinite planar graph $G$, let's denote by $\{H_1,H_2,\dots,H_m\}$ all the labeled graphs on $n$ vertices that appear as subgraphs of $G$. Also let $$d_n=\frac{\sum_{i=1}^m \#E(H_i)}{nm}$$ ...
5
votes
0answers
85 views

Diameter of the modified bubble-sort graph

The modified bubble-sort graph is the Cayley graph $Cay(S_n,S)$ of $S_n$ generated by $n$ cyclically adjacent transpositions. Thus $S = \{ (1,2),(2,3),\ldots,(n,1)\}$. I was wondering whether the ...
5
votes
0answers
105 views

Does squaring a directed random graph more than double its out-degree?

As far as I know, it is an unsolved question whether or not this is true: If $G$ is a directed an oriented graph, $G^2$ always has some node whose outdegree is at least double that of its ...
5
votes
0answers
92 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
79 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
55 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
74 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
votes
0answers
121 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
209 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
110 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
114 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
115 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
187 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
102 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
174 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
94 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
129 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
177 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
223 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
votes
0answers
168 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
99 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
167 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
300 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
254 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
371 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
600 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 ...
4
votes
0answers
82 views

Graph spectra and topology

This is a somewhat vague question, but I'm wondering if there has been any research into connections between the spectrum of a graph and some notion of the "topology" of that graph. To give an ...
4
votes
0answers
71 views

A relaxation of proper coloring

I am wondering if the following relaxation of proper coloring appears somewhere. I have tried some searching and have found a few relaxations of proper coloring, but none the coincides with what I ...
4
votes
0answers
78 views

Finding closest set of K disjoint hyperspheres to a point in $\mathbb{R}^n$ with uniform radius

I am interested in the following problem: in $\mathbb{R}^n$, we have $N$ overlapping hyperspheres all with the same radius. Given a point $p$ in $\mathbb{R}^n$, the objective is to find the $K$ non ...
4
votes
0answers
85 views

Relationship of Weisfeiler-Lehman algorithm to weak isomorphism of coherent algebras

A coherent algebra is a matrix algebra (over $\mathbb{C}$) closed under conjugate transpose and Schur (entrywise) product, and that contains the identity matrix $I$ and all ones matrix $J$. Given ...