Questions tagged [graph-colouring]

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39
votes
2answers
944 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 $v$, $c(v)\...
38
votes
17answers
4k 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 computer-...
37
votes
6answers
3k 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? ;...
27
votes
1answer
889 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 $\{1,\ldots,d-1\}$ ...
26
votes
3answers
953 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 ...
23
votes
2answers
829 views

Are shift-chains two-colorable?

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 ${\...
21
votes
5answers
3k 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 ...
21
votes
2answers
1k 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 ...
17
votes
3answers
487 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 ...
17
votes
1answer
940 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 ...
16
votes
1answer
634 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 ...
16
votes
0answers
433 views

Is graph coloring complete for poly-APX?

Is the graph coloring problem complete for poly-APX under C-reductions (alternatively, under AP-reductions)? For the graph coloring problem, speaking of a feasible solution means a proper coloring for ...
15
votes
3answers
2k 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 cubic ...
14
votes
0answers
413 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 ...
13
votes
4answers
2k 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 ...
13
votes
2answers
1k 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
2answers
540 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 ...
12
votes
1answer
401 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 ...
12
votes
1answer
753 views

Have these coloring games been solved?

In the paper "On the complexity of some coloring games", Bodlaender gives some open questions about the complexity of deciding if player 1 or 2 has a winning strategy in some graph coloring games. ...
11
votes
3answers
647 views

Graph coloring minimizing the number of colors in every independent set

Is the following claim known? Claim: For any graph $G$ with $n$ vertices there exists a coloring of $G$ such that every independent set is colored by at most $O(\sqrt{n})$ colors.
11
votes
1answer
253 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 ...
10
votes
2answers
470 views

Reference Request: Asymptotic hardness of $hk$ coloring $k$-colorable graphs

I heard of a result in approximate graph coloring, but cannot find the source. The result is: For every constant $h$ there exists a sufficiently large $k$ such that coloring a $k$-colorable graph ...
10
votes
1answer
163 views

Is there a planar 4-regular graph that is 3-acyclic colourable?

A colouring is said to be an acyclic colouring if there is no bicoloured cycle (i.e each cycle gets at least 3 colours). Burstein proved that 4-regular graphs are 5-acyclic colourable. It seems to me ...
10
votes
1answer
410 views

Hardness of approximating fractional chromatic number on bounded degree graphs

Is it apx-hard to approximate fractional chromatic number on bounded degree graphs?
10
votes
0answers
128 views

Is 4-Coloring restricted to graphs with crossing number 1 NP-complete?

Planar graphs are 4-colorable. Determining if a planar graph is 3-colorable is NP-Complete. A graph with a crossing number 1 (graph such that it can be drawn with $\le 1$ crossing) is 5-colorable. ...
9
votes
1answer
303 views

3-coloring planar graphs in $O\left(3^{n^.5}\right)$?

I was wondering if the task of searching for planar 3-colorings is known to be of complexity $O\left(c^{\sqrt{n}}\right)$ or lower? This feels like it would be an intuitive consequence based from ...
9
votes
1answer
223 views

What is known about the hardness of the chromatic index for restricted graph classes?

There is a nice paper from 1991 that contains three diagrams about different graphclass-families showing what is known about the hardness of determining the chromatic index for them. Are there any ...
8
votes
2answers
355 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 ...
8
votes
1answer
188 views

Complexity of graph isomorphism with properly colored edges (ref. request)

An edge-colored graph $G$ is a graph whose edges are labeled with a color (generally represented by an integer). Such a coloring is proper if all adjacent edges in $G$ have different colors. I ...
8
votes
1answer
1k views

How bad can the greedy coloring (list color) for the c-chromatic number of graph be?

c-chromatic number is defined in the paper Partitions of graphs into cographs. It asks for the minimum number of colors used to color vertices such that each color class is a cograph. Cograph is a P4-...
7
votes
3answers
627 views

Hard problems for bounded vertex cover

We know that list coloring problem is W[1]-hard when parameterized by vertex cover. Are there any other problems which are also W[1]-hard parameterized by vertex cover?
7
votes
2answers
735 views

Anti-chromatic number

What is the maximum number of colors that can be used for coloring the vertices of a given graph, with no isolated vertices, such that each vertex should share its color with at least one of its ...
7
votes
1answer
288 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 ...
7
votes
1answer
337 views

Chromatic number of G+v where G is a cograph

Cograph is a well-know graph that does not have induced $P_4$. My questions are about determining the chromatic number of graphs in the class cograph+v. Notations: Denote by $\chi(G)$ the chromatic ...
7
votes
1answer
518 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 k-...
7
votes
1answer
305 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 ...
7
votes
2answers
293 views

Computing the edge orbits of a graph (and discussing definitions)

A (vertex) automorphism in a graph $G=(V,E)$ is a permutation $\sigma$ of the vertices that preserves adjacency, namely $\sigma(u) \sigma(v) \in E$ if and only if $uv \in E$. The automorphisms of a ...
7
votes
1answer
393 views

The Overfull conjecture in graph theory and $coNP$

I am not good at complexity, but got a possible relation between a plausible conjecture in graph theory and $coNP$. Graph $G$ is Class 1 if it can be edge colored with $\Delta(G)$ colors, otherwise ...
7
votes
1answer
245 views

3-color a cubic graph such that a MIS receives only two colors

According to Brooks'_theorem, a cubic graph (3-regular graph) containing no $K_4$ can be properly colored by three colors. (Such a color can actually be found in linear time, which is not our primary ...
6
votes
1answer
298 views

NP-hardness of coloring uniform hypergraphs

Since a $2$-uniform hypergraphs are just graphs. The problem of deciding if $2$-uniform is $k$-colorable for $k=1,2$ is easy, and NP-hard for $k \geq 3$ colors. This is well know and I have seen ...
6
votes
2answers
334 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 ...
6
votes
1answer
113 views

Existence of certain graph gadget related to coloring odd hole free graph

Crossposted from MO. Wondering about the existence of graph gadget related to coloring (or 3-coloring) odd hole free graphs. Let $G$ be simple $k$-chromatic connected graph with two vertices $u,v$. ...
6
votes
0answers
89 views

How much is known about coloring of planar graphs with degree bounds?

Are there any references that address the following (open?) questions: 1) Is there an algorithm that 4-colors any planar graph of maximal degree at most 5 in linear time? 2) What is the largest ...
6
votes
0answers
122 views

General Results for Complicated Constraint Satisfaction Problem

Consider the following problem: on a finite two-dimensional grid (say the grid points are vertices of a graph), I need to color the vertices in such a way to satisfy the existence and nonexistence of ...
5
votes
3answers
420 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 ...
5
votes
2answers
4k 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 $...
5
votes
1answer
382 views

Worst case ratio between minimum clique cover and maximum independent set

The maximum independent set problem gives a lower bound for the minimum clique cover problem. This is easy to see because given any clique cover together with an independent set, any two vertices ...
5
votes
1answer
541 views

Chromatic number of planar graph with girth at least k

The four-color theorem claims that, any loopless planar graph is 4-colorable. However, it is NP-Complete to determine the chromatic number of planar graphs, even for those 4-regular ones. Girth is ...
4
votes
1answer
241 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 ...
4
votes
1answer
10k views

Flood fill vs depth first search

Is the flood fill algorithm the same as depth first search? If not, how do they differ in complexity?