The graph-colorings tag has no usage guidance.

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### Prime labelling of graphs

A prime labeling of a graph is an injective function $f: V(G) \to \{1, 2, ..., |V(G)|\}$
such that for every pair of adjacent vertices $u$ and $v$, $\text{gcd}(f(u), f(v)) = 1$ (labels of any two ...

**2**

votes

**1**answer

134 views

### How to construct a graph with arbitrarily large girth and large chromatic number? [closed]

Erdos theorem says it is possible and it is not so easy. What is the general procedure to construct graphs like Grötzsch graph?

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### Rate of convergence of graph-theoretic quantity to fractional graph-theoretic counterpart

Let $G^n$ denote the OR product of a graph with itself $n$ times, i.e. the graph which has an edge between distinct vertices $(v_1,v_2,\ldots,v_n)$ and $(u_1,u_2,\ldots,u_n)$ if there exists some $i$ ...

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### Generalizations of the Four-Color theorem

The four color theorem asserts that every planar graph can be properly colored by four colors.
The purpose of this question is to collect generalizations, variations, and strengthenings of the four ...

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157 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 $G$.
...

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**1**answer

324 views

### Coloring the edges of a torus graph

Question:Consider the $2k \times 2k$ grid graph on a torus. Is it true that for every $2$-coloring of the edges, there is an antipodal pair of vertices connected by a path that changes colors at most ...

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136 views

### What is known about the chromatic number for minimum-distance graphs in higher dimensions?

For a set of points in $\mathbb{R}^d$ with minimum distance $a$, the minimum-distance graph connect two points iff they are at distance $a$. We can also view it as the tangency graph for a set of ...

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votes

**2**answers

136 views

### Coloring of a normal map

can the following proposition be proved? If so please suggest a method. Can Kempe’s Argument be used for proof ?
Proposition: A normal map has a colouring of countries by 4 colours iff the edges of ...

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votes

**1**answer

370 views

### Coloring of the plane

I would like to know the minimum number k such that the plane R^2 can be coloured with k colors such that no colour contain all the possible distances. In other words, a colouring such that each color ...

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votes

**2**answers

72 views

### Coloring maximal independent sets with 1 color

Let $G$ be a graph and $M\subseteq V(G)$ be a maximal independent set. Is there a coloring $c:V(G)\to\chi(G)$ such that $c$ is constant on $M$?
(The answer is positive for graphs with infinite ...

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vote

**2**answers

113 views

### Regular graphs with strongly regular edge colorings

Given a positive integer $c>1$, for what parameters $(v,k,\lambda,\mu)$ does there exist a $c k$ regular graph on $v$ vertices that can be given an edge coloring with $c$ colors, such that the ...

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votes

**1**answer

173 views

### Coloring algorithm maximising color difference between neighbors

Consider a graph and a set of ordered colors ${\cal C} = \{1,2,\cdots,C\}$. I want to color each node $i$ with a color $c_i\in{\cal C}$ so as to maximize the minimum color difference between two ...

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**1**answer

213 views

### Coin graph is 4-colorable

How can we prove that a coin graph is 4-colorable???Also, can we find any example of an non-3-colorable coin graph.

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**1**answer

236 views

### On the maximum number of $t$-subset of $\{1,\ldots, n\}$ having pairwise singleton or empty intersections

Let $n$, $t \in \mathbb N$ two natural numbers such that $0<t<n$, and let $A$ be a set of $n$ elements.
We call a quasi-partition or q-p of $A$ a subset $W \subset \mathcal P(A)$ such that we ...

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**1**answer

91 views

### Coloring graph such that the coloring classes are not maximal independent sets

Let $G$ be a (finite or infinite) simple graph. We let $\mathrm{Ind}(G)$ be the collection of independent sets. For any cardinal $\kappa$ and coloring map $\chi: G\to \kappa$ we have ...

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**0**answers

264 views

### Kempe chain color swaps in a partially colored map

Crossposted from math.stackexchange.com:
http://math.stackexchange.com/questions/904932/kempe-chain-color-swaps-in-a-partially-colored-map
Question: In this partially Tait's colored map, using ...

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364 views

### Matching number and chromatic number

If $G$ is a (finite) graph, denote with $\mu(G)$ the size of any maximum matching in $G$ (this number is also called the "matching number" of $G$).
For odd integers $n$ we have $n=\chi(K_n) = ...

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79 views

### asymptotic notation with graph colouring

This is my first ever post so I hope this is an appropriate question.
Basically I am looking at the paper here: http://homepages.math.uic.edu/~mubayi/papers/biclique.pdf
Namely theorem 5.
Now, feel ...

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**0**answers

228 views

### Computing the chromatic polynomial of graph modulo $x-3$

The chromatic polynomial of graph $P(G,x)$ is univariate
polynomial which counts the number of colorings of $G$
with $x$ colors for natural $x$.
Graph is not $k$ colorable iff $P(G,k)=0$.
The ...

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votes

**2**answers

248 views

### Conjecture: for perfect graphs the fractional chromatic index rounded up equals the chromatic index

Let $\chi'_f(G)$ be the fractional chromatic index.
Based on limited experiments (up to 8 vertices and few larger graphs),
I suspect:
Conjecture For perfect graphs $\lceil \chi'_f(G) \rceil = ...

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votes

**1**answer

369 views

### Postnikov's approach to perfect matchings of graphs

Over a decade ago Alexander Postnikov developed his own way of looking at perfect matchings of bipartite plane graphs. As I recall, he starts with a 2-coloring of the square grid and creates a new ...

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257 views

### The Universal Labeling of graph

The universal labeling of a graph $G$ is a labeling of the edge set in $G$ such that in every orientation $\ell$ of $G$ for every two adjacent vertices $v$ and $u$, the sum of incoming edges of $v$ ...

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**6**answers

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### Is it easy to produce hard-to-color graphs?

This question arises from my recent visit to my daughter's second-grade class, where I led some discussion and activities on graph coloring (see Math for seven-year-olds). In one such activity, each ...

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182 views

### Bounds on chromatic number of $k$-planar graphs

A $1$-planar graph can be drawn in the
plane so that each arc is crossed at most once by another arc.
A $k$-planar graph can be drawn so that each arc is crossed at most $k$ times.
Planar graphs are ...

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votes

**1**answer

67 views

### Graph construction to double coloring & Hadwiger number

For any graph $G$ let $\eta(G)$ be the Hadwiger number of $G$.
Is there for every graph $G$ a graph $2G$ such that
-- $\chi(2G) = 2\chi(G)$, and
-- $\eta(2G) = 2\eta(G)$?
For each one of the above ...

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### Independence Number of K4-free planar graphs

The independence ratio is defined as the size of the maximum independent set divided by $n$. By the 4-color theorem, every planar graph with $n$ vertices has independence ratio at least $1/4$. ...

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**2**answers

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### Big Mono-Chromatic Subgraphs of Vertex 2-Colourings

I'm not a graph theorist, but the following quantity came up in my work and I'm curious if it has been studied. Given a connected finite graph $\Gamma = (V,E)$ define: $$ c(\Gamma) = \min_{f : V ...

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### Bipartite Graphs arising from two k-partitions of a given Graph

Let $G$ be an $n$-chromatic connected graph. Let $(V_1, V_2, \cdots, V_n)$ and $(U_1, U_2, \cdots, U_n)$ be two partitions of $V(G)$ corresponding to proper n-colorings of $G$.
Consider the bipartite ...

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### Are infinite planar graphs still 4-colorable?

Imagine you have a finite number of "sites" $S$ in the positive quadrant
of the integer lattice $\mathbb{Z}^2$,
and from each site $s \in S$, one connects $s$ to every lattice point to which it
has a ...

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votes

**1**answer

76 views

### 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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**1**answer

194 views

### 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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338 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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votes

**1**answer

954 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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**1**answer

348 views

### 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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votes

**1**answer

153 views

### 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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**3**answers

278 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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**1**answer

335 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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### 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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**1**answer

225 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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**2**answers

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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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### 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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226 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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### 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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**1**answer

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 ...