The graph-colouring tag has no wiki summary.
6
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2answers
263 views
Decision problem related to coloring
Given a $k$-colorable graph $G$ and vertices $u$ and $v$ of $G$, what is the complexity of deciding if every $k$-coloring of $G$ must assign the same color to both $u$ and $v$?
It does not seem ...
9
votes
0answers
231 views
What is the complexity of this edge coloring problem?
Recently, I have encountered the following variant of edge coloring.
Given a connected undirected graph, find a coloring of the edges that uses the maximum number of colors while also satisfying ...
11
votes
2answers
530 views
Small graph with gap between chromatic and vector chromatic number?
I’m looking for a small graph $G$ whose vector chromatic number is smaller than the chromatic number, $\chi_v(G)<\chi(G)$.
($G$ has vector chromatic number $q$ if there is an assignment $x\colon V ...
12
votes
4answers
484 views
hardness of approximating the chromatic number in graphs with bounded degree
I am looking for hardness results on vertex coloring of graphs with bounded degree.
Given a graph $G(V,E)$, we know that for any $\epsilon>0$, it's hard to approximate $\chi(G)$ within a factor ...
18
votes
5answers
550 views
Reasons for which a graph may be not $k$ colorable?
While reasoning a bit on this question, I've tried to identify all the different reasons for which a graph $G = (V_G,E_G)$ may fail to be $k$ colorable. These are the only 2 reasons that I was able to ...
13
votes
1answer
395 views
Why are perfect graphs called perfect?
Sorry, if this is a naive question, but I could not find the justification in any of the major text books like Bondy-Murty, Diestel or West. Perfect graphs have many beautiful properties, but what is ...
3
votes
1answer
123 views
bounded outdegree bipartite spanners
Given an undirected graph $G = (V,E)$ and an integer $k > 0$, our objective is to find a subgraph $G' = (V ,E')$ where $E' \subseteq E$ such that $G'$ has the three following properties :
$G'$ ...
4
votes
1answer
246 views
Path coloring in general graphs
Path coloring is the problem of coloring a set of paths R in graph G, in such a way that any two paths of R which share an edge in G receive different colors.
We know that coloring a set of paths ...
11
votes
1answer
149 views
An improper planar coloring with monochromatic component size $\leq 2$
Let us relax the coloring a little bit, that is, we allow a small number of adjacent vertices to be assigned the same color. A monochromatic component is defined to be a connected component in the ...
21
votes
1answer
533 views
Coloring complexity of graphs
Suppose $G$ is a graph with coloring number $d = \chi(G)$. Consider the following game between Alice and Bob. At each round, Alice picks a vertex, and Bob answers with a color in $\{1,\ldots,d-1\}$ ...
4
votes
1answer
165 views
Algorithms for Interval Coloring with Capacities and Demands
We are given a graph $G = (V,E)$, where each edge $e \in E$ has a capacity $c_e$. There are a set of $k$ requests $R_1,\ldots,R_k$. Request $R_i$ has a demand $d_i$, and has an interval $I_i = ...
5
votes
1answer
305 views
Razborov's Approximation methods
The approximation mothods is used for deriving lower bounds on the monotone circuit size of k-cliue and perfect matching problem.
People in parameterized complexity theory strongly believe that ...
2
votes
3answers
418 views
Complexity of greedy coloring
I was looking at some heuristics for coloring and found this book on Google books: Graph
Colorings By Marek Kubale
They describe the Greedy algorithm as follows:
...
4
votes
1answer
214 views
Chromatic number of a particular graph
Assume I have a parametrized graph. The parameters are two integers $x$ and $y<x$.
Let $S(x)=\{1, \ldots, x\}$.
The vertices of the graph are all the subsets of $S(x)$ of size $y$. Two vertices ...
10
votes
1answer
228 views
Hardness of approximating fractional chromatic number on bounded degree graphs
Is it apx-hard to approximate fractional chromatic number on bounded degree graphs?
11
votes
2answers
358 views
Approximate graph colouring with a promised upper bound on maximum independent set
In my job the following problem arises:
Is there a known algorithm, that approximates the chromatic number of a graph without an independent set of order 65? (So alpha(G)<=64 is known and |V|/64 ...
1
vote
0answers
76 views
Resources to get started on fractional graph coloring algorithms
I'm interested in using fractional graph coloring algorithms/solvers to solve a problem, where is a good place to start? I'm looking to find basic/introductory to state-of-the-art algorithms more ...
13
votes
0answers
365 views
Question on Products of Graphs
Let $S_{n}(G)$, $C_{n}(G)$, $T_{n}(G)$ be the $n$-fold Strong Product, Cartesian Product and Tensor Product of a graph $G$ on $|V|$ vertices.
Let the chromatic number ($\chi(G)$) and the independence ...
5
votes
1answer
181 views
Parallel algorithms to color interval graphs
Several NP-hard graph problems get easy if we consider interval graphs. There is a greedy algorithm to color optimally an interval graph. Just sort the intervals according their left endpoints and ...
4
votes
1answer
450 views
Any relation between the size of maximum independent set and the chromatic number on graph of bounded degree?
Consider an connected undirected graph $G$ with $n$ vertices and maximum degree $\Delta$. Assume $G$ contains a maximum independent set of size $k$. Is there any relation between the chromatic number ...
4
votes
1answer
239 views
3 colorable graphs
I was trying to understand the underlying difficulty of coloring 3 colorable graphs with as least number of colors as possible. Though i am aware of hardness result of coloring it with 4 colors, i ...
6
votes
2answers
231 views
Hardness of additive approximation to Graph Coloring problem.
In paper Approximation Algorithms for the Chromatic Sum, page 18, authors state that based on the fact that the Graph Coloring problem is hard to approximate with a ratio less than 2 (under the ...
34
votes
15answers
3k views
Conjectures implying Four Color Theorem
Four Color Theorem (4CT) states that every planar graph is four colorable. There are two proofs given by [Appel,Haken 1976] and [Robertson,Sanders,Seymour,Thomas 1997]. Both these proofs are ...
1
vote
1answer
382 views
graph coloring with 3 colors
I'm searching for an algorithm that can calculate a suboptimal solution for:
color a graph with 3 colors
some vertices already have a color and can't be changed
the edges have values and the ...
6
votes
1answer
231 views
An image coloring problem
I have a large collection of microscopy images of cell cultures. Each image consists of $10^{10} \times 10^{10}$ pixels. These images have been "segmented", meaning that their pixels have been ...
4
votes
3answers
269 views
Hardness of finding a graph coloring given the optimal number of colors
We know that graph coloring is NP-complete even in some special graph classes. On the other hand if someone tells you the exact value of the chromatic number of the input graph, is this problem ...
17
votes
2answers
852 views
Coloring Planar Graphs
Consider the set of planar graphs where all the internal faces are triangles. If there is an interior point of odd degree the graph cannot be three colored. If every interior point has even degree can ...
12
votes
3answers
263 views
Is there a constant factor approximation algorithm for 2D rectangle coloring problem?
The problem we consider here is the extension of the well-known interval coloring problem. Instead of intervals we consider rectangles having sides parallel to axes. The objective is to color the ...
11
votes
2answers
418 views
Do Shift-chains have Property B?
For $A\subset [n]$ denote by $a_i$ the $i^{th}$ smallest element of $A$.
For two $k$-element sets, $A,B\subset [n]$, we say that $A\le B$ if $a_i\le b_i$ for every $i$.
A $k$-uniform hypergraph ...
33
votes
1answer
609 views
How many distinct colors are needed to lower-bound the choosability of a graph?
A graph is $k$-choosable (also known as $k$-list-colorable) if, for every function $f$ that maps vertices to sets of $k$ colors, there is a color assignment $c$ such that, for all vertices $v$, ...
11
votes
3answers
924 views
Complexity of edge coloring in planar graphs
3-edge coloring of cubic graphs is $NP$-complete. Four Color Theorem is equivalent to "Every cubic planar bridgeless graphs is 3-edge colorable".
What is the complexity of 3-edge coloring of ...
23
votes
3answers
801 views
When is relaxed counting hard?
Suppose we relax the problem of counting proper colorings by counting weighted colorings as follows: every proper coloring gets weight 1 and every improper coloring gets weight $c^v$ where $c$ is some ...
0
votes
3answers
626 views
Is it possible to have a 4-coloring for a non-planar graph ? [closed]
I have been working on this thread http://cstheory.stackexchange.com/questions/791/grid-k-coloring-without-monochromatic-rectangles, and I am aware that the four color theorem implies that all planar ...
11
votes
1answer
291 views
Efficient algorithm for near-optimal edge-colourings of hypergraphs
Graph colouring problems are, already, hard enough for most people. Even so, I'm going to have to be difficult and ask a problem about hypergraph colouring.
Question.
What efficient algorithms are ...
35
votes
6answers
2k views
Grid $k$-coloring without monochromatic rectangles
Update: The obstruction set (i.e. the NxM "barrier" between colorable and uncolorable grid sizes) for all monochromatic-rectangle-free 4-colorings is now known.
Anyone feel up to trying 5-colorings? ...
