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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2
votes
0answers
45 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 ...
1
vote
0answers
25 views

About the small set expansion conjecture.

Given a graph $G=(V,E)$ and a $\delta > 0$ one wants to calculate $h(G,\delta)=min_{\vert S\vert \leq \delta \vert V \vert } \phi(S)$. ($\phi(S) = \frac{ E(S,\bar{S}) }{d min \{\vert S \vert , n - ...
3
votes
0answers
56 views

Aperiodic graphs

The concepts of being non-periodic and aperiodic for tilings have obvious versions for connected graphs with a countable set of vertices and a finite number of edges meeting at each vertex. A graph ...
1
vote
1answer
28 views

Linear intersection number and maximum degree

This question is inspired by a Andrew D. King's comment in Linear intersection number and coloring (not chromatic) number A linear hypergraph is a pair $\pi=(X, L)$ where $X\neq \emptyset$ is a set ...
3
votes
0answers
42 views

Linear intersection number and coloring (not chromatic) number

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 ...
1
vote
2answers
243 views

Independence Number of Graphs

Suppose $\alpha(G)\leq\alpha(H)$ where $G$ and $H$ are graphs, and $\alpha(.)$ is the independence number of graph. Is the following statement true? $\alpha(G\boxtimes G) \leq \alpha(H\boxtimes H)$ ...
4
votes
1answer
81 views

Edges of $K_n$ are colored, connected few-colored subgraph is needed

Assume that all edges of a complete graph $K_n$ are colored in $k$ colors. We want to choose $m$ colors so that the graph formed by edges of chosen colors is connected. It is always possible if $m\geq ...
-1
votes
0answers
16 views

which graph srtuctures can help you calculate the pair-wise shortest paths efficiently? [on hold]

Is BFS structure good enough for this task? Any comments would be greatly appreciated.
2
votes
1answer
78 views

Relationship of clique, independence, and chromatic numbers

For any graph $G=(V,E)$ let $\bar{G}$ be the complement graph. Is $$\text{inf}\big\{\frac{\omega(G)+\omega(\bar{G})}{\chi(G)} : G \text{ is a finite graph}\big\}$$ known? If not, what lower bounds are ...
3
votes
2answers
95 views

regular polyhedra (and polytopes) in hyperbolic geometry, and generalisations

While there exist regular tesselations of the hyperbolic plane with arbitrary regular polygons, there are no new regular polyhedra in hyperbolic (3D) space. This being quite trivial, it is probably ...
1
vote
1answer
44 views

Graph of bounded continous functions with distance 1

Let $V = \{f:[0,1]\to \mathbb{R}: f \text{ is continuous}\}$ and consider the metric that is defined for $f,g\in V$ by $$d(f,g) = \max\{|f(t)-g(t)|: t\in [0,1]\}.$$ We set $E = \{\{f,g\}: f,g \in ...
2
votes
0answers
67 views

the choosing of an independent set in a specific $k$-partite graph

Let $k\geq2$ be an integer, a graph $G=(V,E)$ is called $k$-partite if $V$ admits a partition into $k$ classes such that every edge of $G$ has its ends in different classes: vertices in the same class ...
-3
votes
0answers
58 views

What does this graph notation mean? E\S [closed]

I am studying graph theory right now but I am confused what E\S means where both E and S are sets of edges. What does the "\" indicate?
7
votes
1answer
135 views

When is the tensor product of two graphs planar?

Given two graphs $G=(V_1,E_1)$ and $H=(V_2,E_2)$, the tensor product of $G$ and $H$ is the graph $G \times H = (V,E)$, where $V=V_1 \times V_2$ is the Cartesian product of the $V_i$ and $ (u,v) \ E \ ...
0
votes
0answers
37 views

Weighted eigenvector-entry sums of bipartite graphs

Let $\lambda_m$ denote the $m$-th eigenvalue of the adjacency matrix $A$ of a bipartite graph $G$ of order $N$ and ${\bf v}_m$ the normalized eigenvector corresponding to $\lambda_m$. I am looking for ...
4
votes
1answer
117 views

Hadwiger's conjecture for coloring number instead of chromatic number

For any graph $G=(V,E)$, the coloring number $\text{Col}(G)$ is defined to be the smallest cardinal $\kappa$ such that there is a well-ordering $\leq$ on $V$ such that for every vertex $v\in V$ we ...
3
votes
0answers
28 views

Isomorphic Hadwiger graphs of connected infinite graphs

Let $G$ be a graph, then we define its Hadwiger graph $\textrm{Hadw}(G)$ in the following way: $V(\textrm{Hadw}(G)) = \{S\subseteq (V(G): S\neq \emptyset\textrm{ and } S \textrm{ is connected}\}$; ...
1
vote
1answer
57 views

Hadwiger-Nelson problem in higher dimensions

Given a positive integer $n\in \mathbb{N}$ we define the Hadwiger-Nelson graph $\text{HN}_n$ by $V(\text{HN}_n) = \mathbb{R}^n$; $E(\text{HN}_n) = \{\{v_1, v_2\}: v_1, v_2 \in \mathbb{R}^n \text{ ...
6
votes
1answer
193 views

A variant to the Hadwiger-Nelson problem

Consider the following graph $G=(V,E)$ where $V=\mathbb{R}^2$ and $E = \{\{x,y\}: x,y \in \mathbb{R}^2 \text{ and } |x-y|\in \mathbb{Q}\}$. What is $\chi(G)$? (This is a variant of the ...
0
votes
0answers
14 views

there exists k such that all elements of A^k are positive iff G is bipartite [migrated]

How we can prove this: Suppose G is a connected graph with adjacent matrix A. prove that there exists k such that all elements of A^k are positive iff G is bipartite. Is A^n >0 if G is not bipartite? ...
4
votes
1answer
92 views

Countable hypo-hamiltonian graph

If $G = (V,E)$ is a graph, then a $\omega$-path is an injective map $p:\omega\to V$ such that $\{p(k),p(k+1)\}\in E$ for all $k\in \omega$. In a similar fashion, we define a $\mathbb{Z}$-path. Is ...
0
votes
1answer
131 views

Ore's theorem for countable graphs

Ore's theorem states that in a finite graph $G$ with $|V(G)|=n$, there is a Hamiltonian path, provided that the sums of the degrees of 2 distinct, non-adjacent vertices is $\geq n$. For countable ...
3
votes
1answer
216 views

How many hamiltonian paths can be removed from a complete directed graph before it becomes disconnected?

The question started from a problem brought home by a friend's 5th grader: "How many ways you can arrange 5 people sitting around a round table so that the people sitting to the left of any person are ...
2
votes
2answers
126 views

Upper bounds on the edge clique cover number on special graph classes

An edge clique cover of an undirected graph $G$ is a set of cliques of $G$ such that every edge of $G$ is an edge in at least one clique in the set. The edge clique cover number $\theta(G)$ is the ...
0
votes
1answer
50 views

Counting the orderings of outward-directed trees where the degree of each vertex is $2$

Let $T$ be a connected directed tree with the following properties: The degree of each vertex of $T$ is at most $2$ (I am sure there is a name for such a graph but I do not know it). $T$ has a ...
-4
votes
1answer
59 views

Resources to learn about hypergraphs [closed]

I am working on a project that is based on hyper graphs. Is there any resource that I could refer to understand the basics properties of hyper graphs ?
2
votes
1answer
94 views

Network flows with shared capacities

Suppose we have a flow network, with capacity constraints on weighted sums of arc flows, such as: $$2 f(1, 2) + 3 f(4, 5) + f(3, 7) \leq 10,$$ where $f(1, 2)$ denotes the flow through arc $(1, ...
0
votes
1answer
71 views

A question about graphs not having non-trivial automorphisms [closed]

Let call a simple graph (not containing neither loops, nor multiple edges) "prime", if it has no non-trivial automorphisms, i.e. graph that has only "identity" automorphic transformation. I cannot ...
5
votes
0answers
194 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 ...
1
vote
0answers
62 views

Discrete p-Laplacian

One of the definitions of the discrete (weighted) $p$-Laplacian is the following: $$\Delta_{p,w}u(x):=\sum_y |u(y)-u(x)|^{p-2}(u(y)-u(x))w(x,y).$$ Consider the one dimensional case. Then the free ...
3
votes
1answer
126 views

Condition(s) for the full autormophism group $\operatorname{Aut}(C(G, S))$ of the Cayley graph of $G$ to be isomorphic to $G$

If $\Gamma = C(G, S)$ is the (undirected) Cayley graph of a finite group $G$ with generating set $S$, then $G \le \operatorname{Aut}(\Gamma)$, the "full" automorphism group of $\Gamma$. When is ...
2
votes
2answers
86 views

Matching polynomials and Ramanujan graphs

Is it purely coincidental that the same number $2\sqrt{d-1}$ appears in these two following apparently disparate concepts? A $d-$regular graph is said to be called Ramanujan if its adjacency ...
2
votes
2answers
137 views

Term for vertex connected to every other vertex in a graph

Do you know a good common term for the operation of connecting a new vertex v to every vertex in a graph G (or a term for such vertex v)? The ones I know give me poor search results: a nice word ...
0
votes
0answers
30 views

Least square problems with binary variables

I want to solve the heat equation $T_t(x,t) = - L_x . T(x,t) + F(x,t)$ in an edge-weighted graph where $L_x = \sum_i x_i e_{ij}$ is weighted Laplacian matrix of the graph. Then I conclude to the ...
0
votes
0answers
37 views

Discrete bi-Laplacian

I was wondering whether there exists any kind of literature on the the powers of the discrete Laplacian, in particular the the discrete bi-Laplacian, possibly with weights on the edges. In particular ...
8
votes
1answer
157 views

Is the Cayley graph of Thompson's group isolated in the space of vertex-transitive graphs?

Consider Thompson's group (the one commonly referred to as $T$), which is finitely presentable. Consider the Cayley graph, but then forget the coloring and direction on edges. So now we just have an ...
7
votes
0answers
60 views

Approximation of the effective resistance on Cayley graph

Let $\Gamma$ be a finitely generated group, and denote by $G$ the Cayley graph of $\Gamma$. Denote by $d_R$ the resistance distance metric on this graph. The resistance distance metric between the ...
0
votes
0answers
32 views

Confusion about reduction counting vertex covers to counting cycle covers

Cross-posted from cstheory This confuses me. One easy case of counting is when the decision problem is in $P$ and there are no solutions. A lecture show that the problem of counting the number of ...
5
votes
0answers
100 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 ...
3
votes
1answer
110 views

Directed Hypercube Minimal Cuts

If $[n]:=\{1,2,\ldots, n\}$ for some $n\in\mathbb{N}$, then the hypercube digraph of dimension $n$, denoted $H_n$, is the graph whose set of vertices is the power-set $\wp([n])$ where two vertices ...
6
votes
2answers
156 views

Geometric dominating set: NP-complete?

Let $G=(V,E)$ be a geometric graph, a graph embedded in the plane whose edge lengths are the Euclidean distance between its endpoint vertices. Say that a set of vertices $D \subseteq V$ is a geometric ...
4
votes
1answer
103 views

Statistics of strongly connected components in random directed graphs

I'm interested in the statistics of strongly connected components in random directed graphs. However, I'm unable to find any results on this, partly because I don't know the terminology to search for. ...
4
votes
1answer
74 views

Are there 2-connected regular graphs whose maximum matching leaves 3 vertices uncovered?

I'd like to use Corollary 5 of a paper by Hell & Kirkpatrick on graph packings to obtain an NP-hardness result. They want a 2-vertex-connected graph $F$ such that every matching in $F$ leaves at ...
2
votes
1answer
88 views

When the vertex covering number is smaller than the chromatic number

For any graph $G=(V,E)$ let $\tau(G)$ be the minimum cardinality of a vertex cover of $G$. As noted here, we have $\tau(G) \geq \chi(G) - 1$ for all finite graphs $G$. I'm interested in graphs $G$ ...
1
vote
1answer
87 views

Proof of closed walk generating function identity

In 'Spectral Conditions for the Reconstructibility of a Graph' Godsil and McKay give a short proof of an identity (Lemma 2.1) that relates the generating function for the number of closed walks ...
7
votes
2answers
803 views

Surveys of the items of Erdős' “toolbox”

Could you point out some survey papers and monographs that highlight the kernel of tricks, techniques, and tools that Paul Erdős employed the most in his research work (in particular in graph theory, ...
4
votes
3answers
98 views

Linear intersection number and vertex covering number

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 ...
11
votes
1answer
230 views

Papers about decentralized search and cluster

I just start an independent study about small world network and clusters and I try to find papers about decentralized search and clusters. Can anyone give me some references? Thanks! EDIT (David ...
0
votes
0answers
40 views

About adjacency matrices of $k-$shift lifts of graphs

I am finding the notation of cyclic lifts of graphs to be very confusing. Lets say one is looking at a cyclic $k-$lift of a $\vert V \vert$ sized graph. I would like to understand what is the ...
15
votes
2answers
388 views

Can all unit-distance graphs have their vertices at algebraic integers?

A graph $G$ is described as a unit-distance graph if there exists a function $f:G \rightarrow \mathbb{C}$ such that for every edge $(u,v) \in E(G)$, we have $|f(u) - f(v)| = 1$. Obviously, we can ...