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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13
votes
2answers
372 views

Has there been a computer search for a 5-chromatic unit distance graph?

The existence of a 4-chromatic unit distance graph (e.g., the Moser spindle) establishes a lower bound of 4 for the chromatic number of the plane (see the Nelson-Hadwiger problem). Obviously, it ...
3
votes
1answer
89 views

A modified bipartite assignment problem

Consider the following optimization problem. I have $n$ advisors and $dn$ students. I want to assign each student an advisor so that each advisor has exactly $d$ students. Each advisor/student pair ...
5
votes
2answers
130 views

Finite graph colorings without symmetries

Let $G$ be a connected finite simple graph with vertex set $V$, $F$ a finite set and let $\Delta(G)$ denote the degree of $G$, i.e. $\Delta(G)= \max_{v\in V} \deg(v)$. We say that a coloring $\phi\...
4
votes
3answers
184 views

Relation between diametral path and regularity of a graph

Let $G(V,E)$ be a graph. A path whose length is equal to the diameter of a graph is called a diametral path. In a cycle graph every vertex has $2$ diametral paths. Now I need to prove that this: ...
1
vote
1answer
153 views

What does the higher coefficients of ihara zeta function reveal?

Assume we have a graph $G=(V,E)$. The ihara zeta function $Z(G,u)$ is of form $$\frac1{\displaystyle\sum_{i=0}^{2|E|}c_iu^i}$$ A graph which has $|E|$ edges cannot have a simple cycle of length ...
1
vote
0answers
48 views

How to count the number of shortest paths in a 2x2 grid? [closed]

Say that we have a 2x2 regular grid or network. We label the nodes 0 to 3 row-wise. Then, for each node, we want to compute the number of shortest paths that pass through them. I have a Python code ...
1
vote
0answers
36 views

Infinite graphs with number of common neighbors given for each pair of vertices

This is a follow-up to this question. For any set $X$ we set $[X]^2 = \big\{\{x,y\}: x\neq y \in X\big\}$. If $G=(V,E)$ is a simple undirected graph, and $v\in V$, we set $N(v) = \{w\in V:\{v,w\}\in ...
1
vote
0answers
62 views

Number of rooted spanning forests

Let $G$ be a connected simple graph, and identify two vertices $s$ and $t$. Let $\tau(G)$ be the number of spanning trees of $G$, and let $f(G)$ be the number of spanning forests of $G$ with $2$ ...
1
vote
1answer
76 views
7
votes
1answer
134 views

Can the graph removal lemma be proved directly from the triangle removal lemma?

The Triangle Removal Lemma states that any graph with $o(n^3)$ triangles can be made triangle-free by removing only $o(n^2)$ edges. More generally, the Graph Removal Lemma states that for any graph $...
0
votes
1answer
130 views

Stable marriages for infinite bipartite graphs

Short and informal version: Does the stable marriage problem have a solution if there are $\kappa$ men and $\kappa$ women for any cardinal $\kappa \geq \aleph_0$? Long and formal version: Let $\kappa$...
-2
votes
1answer
61 views

Splitting the vertices of undirected graphs into 2 sparse sets

(A version of this question for undirected graphs.) Let $G=(V,E)$ be a finite, simple, undirected graph. For $v\in V$ set $$ N(v) := \{x\in V: \{x,v\}\in E\}. $$ Is it possible to find a ...
15
votes
5answers
479 views

Cayley graphs of $A_n.$

Consider the Cayley graphs of $A_n,$ with respect to the generating set of all $3$-cycles. Their properties must be quite well-known, but sadly not to me. For example: what is its diameter? Is it an ...
1
vote
1answer
78 views

many 5-list colorings

If a graph is 4-list-colorable, then it is easy to see that it has exponentially many 5-list colorings. This is the first sentence in [Exponentially many 5-list-colorings of planar graphs,JCTB,97 (...
1
vote
1answer
72 views

Partitioning finite directed graphs into 3 “incoming-sparse” sets

Let $G=(V,E)$ be a directed graph. For $v\in V$ set $\text{In}(v)=\{x\in V: (x,v)\in E\}$. Is it possible to find a partition $P_1,P_2,P_3$ of $V$ such that for every $P_i$ and every vertex $v\in ...
7
votes
3answers
299 views

A simplified Art Gallery Problem in a matrix

Let's take a $m \times n$ matrix as an area with $m \times n$ blocks (likes a 2D-version of the world in Minecraft). We have to put some lamps in this matrix to illuminate the whole matrix. Here is ...
1
vote
0answers
42 views

How to estimate the size of balanced biclique in random bipartite graph?

We have a random bipartite graph $G=(V,U,E)$ and $|V|=|U|=n$, in which any vertex pair $<v,u>$ ($v\in V$,$u\in U$) exists an edge with probability $p$. A balanced bipartite complete graph is a ...
1
vote
0answers
59 views

Equivalence between bipartite undirected graph and arbitrary directed graph

Let the adjacency matrix of an undirected bipartite graph be $A = \begin{pmatrix} 0 & B \\ B^T & 0 \end{pmatrix}$ where B is called the biadjacency matrix. Now, by instead interpreting B as ...
-2
votes
1answer
62 views

Degrees and common neighbors

For any simple, finite, undirected graph $G=(V,E)$ and $v\in V$ let $N(v) = \{w\in V:\{v,w\}\in E\}$. Suppose $G, H$ are finite, simple, undirected graphs and there is a bijection between the vertex ...
-6
votes
1answer
96 views

Do degrees determine the chromatic number?

Suppose $G, H$ are finite, simple, undirected graphs and there is a bijection between the vertex sets $\varphi:V(G) \to V(H)$ such that for all $v\in V$ we have $$\text{deg}_G(v) = \deg_H(\varphi(v)).$...
5
votes
0answers
119 views

Complexity of graph isomorphism

Last year, Laszlo Babai proved that the graph isomorphism problem can be solved in time: $$ \exp(O(\log^c n)) $$ where $n$ is the number of vertices. What is the best bound we have for $c$? (The ...
4
votes
2answers
207 views

Algorithms for finding graph isomorphisms

I was wondering if anybody knows where I can find some information about the current (practical) algorithms for finding graph isomorphisms. I've joined the bandwagon and wrote my own which I would ...
3
votes
1answer
74 views

Let $S$ be the nonempty set of strongly regular graphs with given parameters. Must $S$ contain vertex transitive graph?

As the title says, let $S$ be the nonempty set of strongly regular graphs with given parameters. Must $S$ contain vertex transitive graph? I suspect the most likely counterexample would be $|S|=1$.
5
votes
2answers
130 views

“Common-neighbor-regular” graphs

Let $G=(V,E)$ be a finite, simple, undirected graph. For $v\in V$ we set $N(v)=\{w\in V:\{v,w\}\in E\}$. We say that $G$ is $k$-common-neighbor-regular if for all $v\neq w\in V$ we have $|N(v)\cap N(...
3
votes
1answer
84 views

Tournament whose large subtournaments contain no automorphism

For sufficiently large $n$, it is known that most tournaments of size $n$ contains no nontrivial automorphism, though I forgot the reference. For sufficiently large $n$, does there always exist a ...
4
votes
1answer
125 views

Representing a graph's vertices as linear combinations of paths

I have a (directed graded) graph $G=(V,E)$ with two distinguished subsets of $V$: sources and destinations. I am to show that the set of indicator functions of paths starting in a source and ending in ...
5
votes
1answer
193 views

Construction of a graph

To construct a specific kind of undirected graph $G=(V,E)$, which $|V|=n>2$. For convenience, label the vertices with $v_1,v_2,\dots ,v_n\in V$, and $(v_i,v_j)\in E$ means there is a edge between ...
0
votes
0answers
59 views

Tight bound of Turan number for K_{1,t,t}?

I'm looking for a tight bound for Turan number $ex_2(n,K_{1,t,t})$, where $K_{1,t,t}$ is the complete 3-partite graph with parts of size 1, t, and t. The motivation is that we now $ex_2(n,K_{t,t})=O(n^...
0
votes
0answers
36 views

Question about number of faces for regular graphs

I came upon the following problem. Consider a regular 3-graph with faces of degree $2, 3, 4, 5$, 4-graph and faces of degree $2, 3$ and 5-graph with faces of degree $2, 3$. Are there any such graphs ...
3
votes
1answer
96 views

Menger's Theorem for planar triangulations

I was reading the paper "Planar separators" by Alon, Seymour and Thomas (available on the first author's webpage). They consider a planar triangulation, that is, a maximally planar graph $G$ drawn in ...
4
votes
1answer
82 views

The minimum number of Hamiltonian paths in a strongly connected tournament of order $n$

For $n\ge3$ let $a(n)$ be the minimum number of Hamiltonian paths in a strong (i.e., strongly connected) tournament of order $n.$ Where is $a(n)$ discussed in the literature? Is the exact value ...
2
votes
1answer
128 views

Removing cycles in a directed graph by swapping edges orientation

I have the following problem: let $G$ be a finite directed graph with $V$ vertices $v_i$ and $E$ (directed) edges $e_j$. I know that if an edge $e_k = (v_i, v_j)$ is in the graph, then the opposite ...
0
votes
0answers
25 views

A measurement for vertex- and edge failure sensitivity

I want to have a metric that helps me judge how independent different paths in one graph are. So here are my assumptions: A graph consists of a set of vertices ($V$) and edges ($E$). A path ...
1
vote
0answers
103 views

Extending continuous functions from $\partial X$ to $X\cup \partial X$

Consider a proper geodesic hyperbolic space $X$ (in the sense of Gromov). Let $\partial X$ be its Gromov boundary. Consider a complex-valued continuous function on the boundary $f\colon\partial X\to\...
0
votes
1answer
92 views

Topology on $\omega\times\omega$ such that topologically connected equals graph-connected

For any set $X$ we define $[X]^2 =\big\{\{a,b\}: a, b \in X\text{ and }a\neq b\big\}$. Let $$E = \big\{\{(a_1, a_2), (b_1, b_2)\}\in[\omega\times\omega]^2: |a_i-b_i| = 1\text{ for some } i\in\{1,2\}\...
0
votes
0answers
36 views

For which matrices deciding permutation similarity is polynomial?

Q1 For which matrices deciding permutation similarity is polynomial? It is not easier than graph isomorphism (and very likely is equivalent to it). If necessary, assume the entries are nonnegative ...
0
votes
0answers
39 views

Is the Hadwiger number reconstructible?

Let $G, H$ be two finite, simple, undirected graphs. We call them "reduce-by-1"-isomorphic, or $r_1$-isomorphic for short, if there is a bijection $\psi: V(G) \to V(H)$ such that for all $v \in V(G)$ ...
0
votes
1answer
110 views

“Reduce-by-1”-isomorphic graphs

Let $G, H$ be two finite, simple, undirected graphs. We call them "reduce-by-1"-isomorphic, or $r_1$-isomorphic for short, if there is a bijection $\psi: V(G) \to V(H)$ such that for all $v \in V(G)$ ...
0
votes
0answers
45 views

Approximation to colouring for bounded degree graphs

I have already asked one question on colouring, this question is more specific. Given a bounded degree graph $G$ with $\Delta(G)=2d$, is there a well know algorithm to achieve an approximation ratio ...
3
votes
0answers
84 views

What is the complexity of finding a third Hamilton Cycle in cubic graph?

According to Smith Theorem: if a cubic graph has a hamilton circuit then it must have a second one. SMITH : Given a Hamilton circuit in a 3-regular graph, find a second Hamilton circuit. It is known ...
2
votes
0answers
134 views

Is the class of Heyting algebras originating from directed graphs a variety?

The category RefGph of reflexive directed graphs is the functor category $\hat{∆}_1=\mbox{Fun}(∆^◦_1,$Set), where $∆_1$ is the simplex category truncated at level 1. Hence the poset Sub(X) of ...
1
vote
1answer
76 views

Graph colouring for bounded degree graphs

I'm fairly new to colourings on bounded degree graphs i'm interested in the following questions, For planar graphs with bounded degree $4$ is finding the colouring number $NP$-hard? So is ...
1
vote
1answer
69 views

Intersection number of categorical product of graphs

Let $G, H$ be finite, simple, undirected graphs. Suppose $i(G), i(H)$ are their respective intersection numbers. If $G\times H$ denotes the categorical product of graphs, what is $i(G\times H)$ in ...
2
votes
1answer
40 views

Generalizing series-parallel digraphs with feedback

There are common definitions of series-parallel (SP) graphs and digraphs: the basic idea is as follows. A SP graph (or digraph) has two distinguished vertices $s$ ("source") and $t$ ("target"). The ...
1
vote
1answer
157 views

Product and coproduct for bipartite graphs

Consider the category $BiGraph$ of bipartite graphs (vertices are named "places" and "transitions" like in Petri nets) and continous maps. A subgraph is named open iff it is place-bordered, and a map ...
8
votes
1answer
145 views

Non-isomorphic graphs with isomorphic edge vectors

Let $G$ and $H$ be graphs on the vertex set $\{1, \ldots, n\}$ and let $(e_i)$ be the standard basis of $\mathbb{R}^n$. For each edge $\{i,j\}$ define edge vectors $e_i - e_j$ and $e_j - e_i$ in $\...
5
votes
2answers
92 views

Majority coloring for directed graphs

I came up with the following coloring concept when studying neural networks (which are often modelled using directed graphs). No idea whether there is already an established name for it. If $X$ is a ...
3
votes
0answers
47 views

Number of Hamiltonian paths in the Hoffman-Singleton graph/Moore graphs?

Does anyone happen to know how many Hamiltonian paths are in the Hoffman-Singleton graph, or have any idea on how I could count such paths?
0
votes
1answer
76 views

Majority colorings

If $X$ is a non-empty set, we say that $M\subseteq X$ is a majority if $|M| > |X\setminus M|$. Let $G=(V,E)$ be a finite, simple, undirected graph. For $v\in V$ we set $N(v)=\{x\in V: \{x,v\} \in ...
4
votes
3answers
88 views

Name for directed graphs with “balanced cycles”

Does the following class of graphs have a name? I'm interested in directed graphs with the following property: for every cycle (of the underlying undirected graph) half of the edges go in one ...