The graph-colorings tag has no wiki summary.

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### How to prove a certain connection between the list-chromatic number of a bipartite graph and a cardinal?

Let $G=(L,R,E)$ be a complete bipartite graph, such that $|L|=\aleph_0$ and $|R|=\kappa$. I'd like to show that if $\kappa<2^{\aleph_0}$ then the list-chromatic number of $G$ can't be more than ...

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### Coloring of subgraphs of G^n

Let $G=(L,R,E)$ be a finite bipartite graph, such that for each $v\in L\cup R: deg(v)>0$. Define $E^{(n)}=\{(\overline{l},\overline{r}) | \overline{l}=(l_1,...,l_n)\in L^n , ...

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### Size of b-matching constructed from b maximal matchings

The Question: Let $G$ be a (simple) graph and let $b\in\mathbb{N}$. Suppose that we have $b$ disjoint edge-subsets $M_1,\ldots , M_b$ satisfying the following condition: The set $M_1$ is a maximal ...

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### What is the status of this strong form of Hedetniemi's conjecture?

In this question, a graph is a finite, undirected graph without loops or multiple edges, and a colouring of a graph is a proper vertex colouring. The product $G \times H$ of graphs $G$ and $H$ is the ...

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334 views

### Coloring $K_n$ via edge-weight sums

This is a question inspired by
and tangential to "A Question on 1, 2 ,3 Conjecture"—and certainly
much easier!
Suppose one assigns a random edge weight among $\{1,2,3,\ldots,k\}$ to each
edge ...

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**1**answer

905 views

### A Question on 1, 2 ,3 Conjecture

The 1, 2, 3 conjecture is well-known:
If $G$ is a simple graph which is not $K_2$ then one can assign a number among $1, 2, 3$ to every edge such that if we label each vertex with the sum of the ...

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### coloring in lattice

This is a mathematical question raised from engineering and physics:
Is there some established mathematical approach in filling a physical lattice with some colored basis (black and white here)? For ...

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### Achieving the largest possible minimum spacing between vertices of the same color in an integer lattice

Consider an infinite integer lattice, or an infinite hexagonal lattice with unit length edges. Provided a set of $k$ possible vertex colors, is there a known largest possible minimum spacing that can ...

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245 views

### Crystal structure, lattice, Graph and coloring

I am working across mathematics, physics and engineering. And I am looking for whether there exists already formally established knowledge in the field.
Given a periodic graph (actually a physical ...

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290 views

### When does graph Laplacian have eigenvalue -1?

Consider an undirected graph $G$ with (symmetric) adjacency matrix $A \in \{0,1\}^{n \times n}$ and degree sequence $d = (d_i)$ where $d_i = \sum_{j} A_{ij}$. Assume that every node has degree at ...

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### Colouring a graph whose edge set is a special union of cliques

I am trying to show that a certain family of graphs can always be properly coloured with at most $6$ colours (where "properly coloured" means that each vertex gets a colour and no edge has both ends ...

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### Can one measure the infeasibility of four color proofs?

Terms like "impractical" and "unfeasible" are used to say the Robertson, Sanders, Seymour, and Thomas proof of the four color theorem needs computer assistance. Obviously no precise measure is ...

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409 views

### Minimal graphs of prescribed girth and chromatic number

The well known result of Erdős, states that
Given integers $g > 2$ and $k > 1$ there exist a graph $G$ with $\chi(G) \geq k$ and girth at least $g.$
What I am wondering is
When can we ...

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221 views

### Probability that a random edge coloring of the complete graph is proper

This is a repost of this math.se question that I am posting here since it received no attention there.
What is the probability that a random edge coloring of $K_n$ with
$m \geq n$ colors ...

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### Is There a Graph that is Ramsey for $P_{2n}$ but is $C_{2n+1}-$free

Write $F\to G$ to mean that for every two coloring of the edges of $F$, there exists a monochromatic copy of $G$. Nesetril and Rodl proved that for a graph $G$, there exists a graph $F\to G$ with ...

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### 3-edge-coloring of 3-regular multigraphs

Given that a 3-regular multigraph is 3-edge-colorable, is there an expression for how many 3-edge-colorings exist?
(For example, if a 2-regular multigraph is 2-edge-colorable, there are $2^k$ ...

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284 views

### Fractional chromatic number, find reference to a particular alternate definition for

I'm searching for a reference to a particular alternate definition of the fractional chromatic number of graphs.
Let me review the most common definition and basic properties first.
Let $ G $ be ...

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223 views

### Strategic vertex labeling

We are given a graph $G=(V,E)$ with positive edge weights $w_{i}$ and numerical {0,1,-1} labels $l$ for all vertices . We know that $G$ has a subset $G'$ with all vertices labeled 0(all vertices with ...

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403 views

### bipartite graph coloring

Hi, I don't know much about graph theory so I would need to know if the following problem has a positive answer or a reference. There is a bipartite graph G with the two vertex sets V1, V2. Each ...

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### Clique number of a $k$th power of a graph in terms of maximum degree?

Let $G$ be a graph of maximum degree $d$. Are there any known/easy bounds on the clique number of $G^k$, in terms of $d$ and $k$? I would like something at least epsilon better than the bound on the ...

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181 views

### A yes no question concerning induced group

We are given a permutation group G acting on a finite set X. Finite set $\Omega$ contains all mappings $\omega$ from the set X to finite set K. We define mapping $\hat{g}$ on the set $\Omega$ in the ...

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286 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 ...

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518 views

### Vector chromatic number and Lovasz theta

For $\alpha \ge 2$, an $\alpha$-vector coloring of a graph $X$ is an assignment of unit vectors to the vertices of $X$ such that vectors assigned to adjacent vertices have inner product less than or ...

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279 views

### coincidence between minimal triangulation numbers and chromatic numbers

A friend and I were reading up on the classification of compact surfaces when we realized that the minimal number of triangles in a triangulation of the sphere, torus, and projective plane are 4 ...

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### Edge-coloring of the complete graph without any rainbow paths

For a given $2k-1$ edge coloring of the complete graph $K_{2k}$,
say a Hamiltonian path $P$ is a rainbow path if every color appears exactly once in $P$.
My question is
"For each $2k (k \geq 2)$, is ...

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629 views

### Coloring a graph by Maximum Independent Set extraction

recursively extracting a MIS from an undirected simple graph $G$ does produce a minimal coloring for $G$ ?
I searched extensively the internet and found a paper [1] which answer partially to this ...

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**1**answer

243 views

### Isomorphism of connected, rigid, N-regular graphs with chromatic index N?

Background/Motivation
I'm working on algorithms for canonical labeling of a certain class of graphs (motivated by biology). The "difficult" instances of this problem can be reduced to graphs of the ...

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### 2-Coloring a planar hypergraph

Consider a hypergraph (of rank 3) $H = (V, E)$ (where the rank of $H$ is the maximum cardinality of a hyperedge). $H$ is said to be planar if we can construct a planar graph $G = (V, A)$, and a ...

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### A copy of the Vizing's classic article about List Coloring.

Does anyone know where I get a copy of the Vizing's classic article about List Coloring?
"V. G. Vizing, Coloring the vertices of a graph in prescribed colors, Metody Diskret. Anal. v Teorii Kodov i ...

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172 views

### Hypergraph coloring

I am investigating whether the following hypergraph is $2$-colorable.
Let $0\le c < d < e$ be fixed natural numbers and consider a graph on $2e$ vertices, with the vertices labelled as ...

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### Chromatic number of the power set

Let $X$ be a non-empty set. Consider $\mathcal{P}(X)$, the power-set of $X$. We say that $a,b \in \mathcal{P}(X)$ form an edge if and only if their symmetric difference is a singleton, i.e. ...

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304 views

### Graph Theory Terminology Question

Given a (non-multi)graph $G$ let $N_G$ be the least number of nodes that must be colored (by a single color) such that every other node in $G$ shares an edge with at least one colored node. (I am only ...

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### Coloring tensor products of graphs

Let $ G,H $ are simple finite graphs and $A = G \times H$. Here $ G \times H $ is the tensor product (also called the direct or categorical product) of $ G $ and $ H $.
Let $G$ has smaller chromatic ...

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204 views

### maps with a large number of 4-colorings

I brought this up in January, but now know more and can be more precise.
I will have two questions.
How much of this is known?
If you don't know the answer to 1, then who does?
I invite all ...

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523 views

### Regular graph colorings

[Since I didn't get any feedback at MSE, I dare to post this question here, too.]
Call a coloring $C:V(G) \rightarrow \lbrace 1,\dots,n \rbrace$ of a graph $G$ regular when every vertex with color ...

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185 views

### A Ramsey-like lower bound?

Does there exist a graph $G$ which cannot be properly vertex-coloured with 3 colours (i.e. $G$ has chromatic number at least 4), such that for every graph $H$, if $H$ contains a triangle but there ...

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183 views

### What is the definition of a discharge rule?

This question is in the reverse direction of a common MO question. Instead of being faced with a formal definition and asking for some intuition for the definition, I have a concept with I understand ...

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266 views

### Edge Colorings of Directed Graphs which Respect an Involution

Let G be a graph and let C be a set of coloring. Suppose that there is an involution $\phi$ from C to C. We can think about the element of C as the nonzero elements of some Abelian group and ...

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240 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 ...

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1k views

### Highly symmetric 6-regular graph with 20 vertices

I'm interested in (node/edge-)symmetric 6-regular graphs on 20 vertices and 60 edges, especially ones with a A5/icosahedral/dodecahedral symmetry group and especially their chromatic number. So far I ...

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### The Problem about 2-coloring finite plane

Suppose we color a $X \times X$ finite plane by red and blue arbitrarily. How large does X need to be to guarantee a monochromatic combinatorial square $k \times k$
1 0 1 0 1 1
1 1 1 1 1 1
1 ...

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### What is the chromatic number of the graph whose vertices dimension $n-2$ subsimplicies of $\Delta^n$ and an edge between two vertices is given if the two associated $n-2$ vertices are contained in the same $n-1$ subsimplex?

Let us first remark that all of this takes place on the boundary of $\Delta^n$. The question that I wanted to solve that led to the question in the title is as follows: Let ...

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509 views

### Graph Coloring - searching for some interesting problems

Hi,
I'm a high school student and writing a paper about graph coloring. Can you tell me something about some interesting problems in graph theory connected with graph coloring? Such as full triangle ...

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772 views

### chromatic number of the hyperbolic plane

A notorious problem in combinatorics is the following:
If we color $\mathbb{R}^2$ so that no pair of points at unit distance get the same color, what is the fewest number of colors required?
This ...

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### a new lower bound for the chromatic number of a graph?

Let S+(G) denote the sum of the squares of the positive eigenvalues of the adjacency matrix of a graph G. Let S-(G) denote the sum of the squares of the negative eigenvalues and q the chromatic ...

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184 views

### Maximum number of 4-colorings of planar graphs (precise version)

This is a redo of my earlier question.
I'm trying again with more precision. I ignored one of the dicta in the "how to ask" page.
For a fixed $n$, what is known (references preferred) about the ...

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365 views

### Maximum number of different 4-colorings of planar graphs of a given size

I understand that it is computationally hard to count the 4-colorings of a given graph. See answers to
this question
In a given class of planar graphs (will leave choice of class open) is there a ...

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557 views

### Graphical representation of chromatic polynomial

Suppose $c(G,u)$ is chromatic polynomial of connected simple graph $G$. We know that
$|c(G,-1)|$, as Stanley proved, is the total number of directed graph on $G$, without any cycle.
Also, we know ...

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106 views

### Counting walks on proper colorings of odd cycles

Let $G$ be an undirected odd cycle. Let $f$ be a proper 3-coloring of $G$. If $w=v_1v_2...v_k$ is a walk on $k$ vertices of $G$, let $f(w)=f(v_1)f(v_2)...f(v_k)$. Let $W_k=\{f(w)|w$ is a walk on ...

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931 views

### Has anyone seen this graph?

I recently constructed the graph shown below in the process of investigating some problems regarding line graphs and homomorphisms, and then happened to see it on wikipedia. I was wondering if anyone ...