Questions tagged [graph-colouring]

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Hardness of deciding fractional chromatic number at most $k$

I want to find a reference for the following statement. Here, $\chi_f(G)$ denotes the fractional chromatic number of a graph $G$. For every fixed $r>2$, deciding $\chi_f(G) \leq r$ for a given ...
Minsoo Kim's user avatar
2 votes
0 answers
37 views

Hardness of Coloring based on a Black-Box Hardness of Independent Set

It is well known that both vertex coloring and maximum independent set are very hard to approximate in polynomial time under standard complexity assumptions. Given a black-box hardness of independent ...
John's user avatar
  • 392
1 vote
1 answer
115 views

Balanced set coloring

Let $\{S_1, S_2, ..., S_m\}$ be a collection of subsets of some universe $U$, where each $S_i$ has even size (so does $U$). We want to color the elements of $U$, either red or blue, such that each $...
Arnaud Casteigts's user avatar
0 votes
0 answers
84 views

5-color graph and minor

We have a 5-color graph G without 5-clique. The question is: is there a minor H of G that is a 5-clique? Here the minor definition. With "5-color graph G" I mean $\chi (G)=5$.
Mario Giambarioli's user avatar
4 votes
1 answer
71 views

Complexity of maximum k-edge-colorable subgraph of a bipartite graph

Can the maximum $k$-edge-colorable subgraph of a bipartite graph be found in polynomial time? Equivalently, can the maximum $k$-colorable subgraph of the line graph of a bipartite graph be found in ...
Timothy Chow's user avatar
  • 7,550
3 votes
0 answers
68 views

Property testing algorithm for isomorphism to a balanced 3-sided complete graph

I am looking for testing algorithm in the dense graph model, that checks for a graph with $3n$ vertices whether it's isomorphic to a balanced 3-sided complete graph with $n$ vertices in each set. The ...
Z.L's user avatar
  • 31
3 votes
1 answer
307 views

How do you achieve linear time complexity of greedy graph coloring?

In most resources I could find, greedy algorithm is described as follows: for every vertex $v$, assign the minimal color not used by its neighbors. The above could be implemented as: ...
Sebastian Szczepański's user avatar
11 votes
0 answers
170 views

Complexity of $(\Delta-1)$-coloring graphs of maximum degree $\Delta$

Suppose we have a graph $G$ with maximum degree $\Delta$. Deciding if $G$ can be colored with $\Delta$ colors is easy: Brooks' theorem says that this is always possible, unless $G$ is a clique or an ...
Michael Lampis's user avatar
3 votes
1 answer
196 views

Hardness of Maximum Independent Set in 3-Colorable Graphs

Let $G = (V,E)$ be an undirected graph such that there is a proper coloring of the vertices of $G$ in three colors. Question: In such graphs, are there known results for the hardness of finding a ...
John's user avatar
  • 392
12 votes
1 answer
2k views

Is the 3-coloring problem NP-hard on graphs of maximal degree 3?

Consider the 3-coloring problem: given an undirected graph $G = (V, E)$, decide if there is a 3-coloring of $G$, i.e., a function $f$ from $G$ to $\{1, 2, 3\}$ such that there is no edge $\{u, v\}$ in ...
a3nm's user avatar
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1 vote
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Coloring the $k$-deletion graph “constructively”

For $n,k\ge 1$, we define the graph $D_{n,k}$ to have vertex set $\{0,1\}^n$, with distinct $x,y$ being adjacent if $LCS(x,y)\ge n-k$. My question is: fixing $k>1$, does there exist some $C=C_k$ ...
Zach Hunter's user avatar
5 votes
1 answer
242 views

Graph coloring with limit on number of times a color is used

Are there any results on coloring a graph using a limited number of each color. In other words, the decision problem would be: given a list of colors $C = (c_1, \dots, c_k)$ where each color $c_i$ is ...
user167622's user avatar
2 votes
1 answer
69 views

Upper Bound for distance-two chromatic number in terms of maximum degree

Let us consider simple,finite, undirected graphs. A distance-two colouring of a graph $G$ is a function $f:V(G)\to\{1,2,\dots\}$ such that $f(u)\neq f(v)$ whenever $dist_G(u,v)\leq 2$. A distance-two ...
Cyriac Antony's user avatar
1 vote
1 answer
175 views

Graph associated to a mathematical statement (for the purpose of zero-knowledge proofs)

I'll preface this question by saying I have very little (zero!) knowledge of theoretical computer science, and this post is a genuine attempt to understand something, even if at an intuitive level, ...
Emilio Ferrucci's user avatar
2 votes
2 answers
274 views

When should one start looking at existing results in theoretical CS?

I'm currently a PhD student in theoretical computer science. I've been working on this problem daily for almost a month that has been well studied and was assigned to me by my advisor. The problem is ...
user3508551's user avatar
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2 votes
1 answer
80 views

Is every 4-colourful Eulerian Orientation of a planar 4-regular graph good?

Let $G$ be a simple undirected plane 4-regular graph. Basic definitions are given at the bottom of this question. An Eulerian orientation of $G$ is good if for each vertex $v$ of $G$, the edges around ...
Cyriac Antony's user avatar
2 votes
1 answer
207 views

Does abundance of max cliques make it easy to solve COLORABILITY?

Let $q\geq 3$. We know that $q$-COLORABILITY is an NP-complete problem. Suppose that $G$ is a graph such that each vertex of $G$ is part of a $q$-clique (i.e. $K_q$). Since we may assume that $G$ does ...
Cyriac Antony's user avatar
1 vote
1 answer
52 views

Graph classes where giving a q-clique edge cover makes testing for q-colouring easy

A $q$-clique of a graph is a complete subgraph on $q$ vertices. A $q$-clique edge cover of $G$ is a set of subgraphs of $G$ such that each subgraph is a $q$-clique and each edge of G is contained in ...
Cyriac Antony's user avatar
2 votes
1 answer
180 views

Coloring intersection graph of squares

It is known that the coloring intersection graph of axis-parallel rectangles is NP-Hard. What about squares and more specific case "unit squares"? Thanks.
R.hatam's user avatar
  • 21
2 votes
1 answer
211 views

Is the difference between the acyclic chromatic number and the star chromatic number unbounded?

Is $\chi_s(G)-\chi_a(G)$ unbounded in general graphs? I thought $\chi_s(G)-\chi_a(G)$, the difference between the star chromatic number and the acyclic chromatic number, is unbounded for general ...
Cyriac Antony's user avatar
1 vote
0 answers
111 views

Has this notion of connectivity in edge-colored graphs been studied?

Consider a simple graph $G$ where each edge is either red or blue. I'm interested in the following notion of connectivity: Two vertices $u$ and $v$ are said to be connected if there is a path ...
Tassle's user avatar
  • 881
1 vote
0 answers
121 views

Complexity of counting 3-colourings of planar bounded degree graphs

The following are known: It is #P-complete to count the number of 3-colourings of a planar graph (with respect to randomized reductions) [1]. For all $d\geq 3$, it is #P-complete to count the number ...
Cyriac Antony's user avatar
4 votes
1 answer
237 views

Complexity of relaxed edge colouring

A (proper) $k$-edge colouring of a graph $G(V,E)$ is a function $f:E\to\{1,2,\dots,k\}$ such that adjacent vertices are mapped to different colours; that is, $f(e)\neq f(e')$ if $e$ and $e'$ are ...
Cyriac Antony's user avatar
1 vote
1 answer
75 views

Weak incidence colouring

Let $G(V,E)$ be a simple undirected graph, let $k\in \mathbb{N}$, and let $I(G)=\{(v,e)\ :\ e\in E, v\in e \}$ (here, $v\in e$ means $v$ is incident on $e$). A $k$-incidence colouring of $G$ is a ...
Cyriac Antony's user avatar
3 votes
1 answer
352 views

ETH based lower bound for $k$-COLORING of bounded degree graph

It is known that there is no $2^{o(n)}$-time algorithm for 3-COLORABILITY of graphs of maximum degree four, unless ETH fails [1]. Is a there a similar result for $k$-COLORABILITY assuming only ETH (...
Cyriac Antony's user avatar
3 votes
2 answers
131 views

Graph labelling where vertices with a common neighbour get different labels

Is the following notion of graph labeling (let me call it special labeling) ever studied in the literature? A special labelling of a (simple undirected) graph $G$ is a function $f:V(G)\to\mathbb{N}$ ...
Cyriac Antony's user avatar
1 vote
2 answers
129 views

Name of this graph partitioning problem? (related to coloring)

Given a graph $G=(V,E)$ and an integer $k$, find a partition $P_1, P_2, \dots, P_k$ of $V$ into $k$ parts that minimize the total number of edges between two vertices in the same part, i.e. $\sum_i |(...
Matthieu Latapy's user avatar
9 votes
0 answers
151 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. ...
William Gasarch's user avatar
0 votes
1 answer
92 views

How many more colours do you need if you add to $G$ a maximum matching from $G^c$?

The title says it all. Let $G$ be a (simple undirected finite) graph, and let $G^c$ be the complement of $G$ (i.e. contains edges not in $G$ and no more). Let $M$ be a perfect matching in $G^c$. How ...
Cyriac Antony's user avatar
3 votes
2 answers
319 views

Coloring where all colors are present in closed neighborhood of every vertex

I am interested in (proper) vertex colorings of graphs with the following condition: for every vertex $v$ in the graph, all colors should be present in the closed neighborhood of $v$. Is this studied ...
Cyriac Antony's user avatar
2 votes
1 answer
127 views

A conjecture on 4-coloring maximal planar graphs

The question/task is to prove/disprove the conjecture below. Let $G$ be a maximal planar graph with a 4-coloring $f$. Let $(a,b,c,d)$ be a cycle in $G$. Let $S$ be the collection of all $a,c$-paths in ...
Cyriac Antony's user avatar
2 votes
0 answers
169 views

Complexity of Edge Coloring Regular Graphs With Large Degrees

There is an interesting series of papers on edge colorability / $1$-factorization of regular graphs with large degrees, which over the years have shown better and better lower bounds for the degree $\...
Tacocat's user avatar
  • 21
3 votes
0 answers
91 views

Computing chromatic number of subcubic graphs

According to graphclasses.org, the chromatic number of a subcubic graph can be computed in linear time (because the decision problem COLORABILITY can be solved in linear time). The reference given the ...
Cyriac Antony's user avatar
4 votes
0 answers
168 views

Uniquely 4-colorable Planar Graph Conjecture?

My question is on Uniquely 4-colorable Planar Graph Conjecture mentioned in On purely tree-colorable planar graphs (and other papers of same(?) team such as "Theory on Structure and Coloring of ...
Cyriac Antony's user avatar
4 votes
2 answers
786 views

3-colourability of Eulerian maximal planar graph

The following paragraph is from this answer by David Eppstein (emphasis mine). A maximal planar graph is 3-colorable iff it is Eulerian (if it is not Eulerian, then the odd wheel surrounding a single ...
Cyriac Antony's user avatar
2 votes
1 answer
232 views

Who proved that a triangulation is 3-colourable implies its dual is bipartite

Let $G$ be a maximal planar graph (also called a triangulation); i.e, $G$ is a planar graph every face of which is a triangle. It is well known that the following three statements are equivalent: (i) $...
Cyriac Antony's user avatar
10 votes
1 answer
295 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 ...
Cyriac Antony's user avatar
2 votes
1 answer
102 views

Producing colouring of maximal planar graphs G from colouring of dual of G

A (simple) graph is said to be maximal planar if it is planar and adding one more edge loses planarity. In other words, the graph is planar and every face is a triangle. Tait gave a one-one ...
Cyriac Antony's user avatar
2 votes
2 answers
141 views

On the coloring number of small graphs with small cliques

Given a parameter $k$, and a graph $G$ with $O(k^2)$ vertices that has a maximum clique with $\le k$ vertices, I want to investigate the maximum number of colors $C(k)$ needed to properly color $G$, i....
Mathieu Mari's user avatar
1 vote
0 answers
49 views

K-Uniform Hypergraph Strong Coloring [closed]

I want to ask if strong coloring of a k-uniform hypergraph using only k colors is NP-Hard or NP-Complete? If you can add a reference this will be helpful.
Salwa's user avatar
  • 11
2 votes
2 answers
115 views

Does distance-2 coloring fit in Telle and Proskurowski 's algorithm for partial-k trees?

This question is on "Vertex Partitioning Problems" framework of Telle and Proskurowski. For solving problems in parital $k$-trees (i.e., graphs of bounded treewidth), the "Vertex ...
Cyriac Antony's user avatar
8 votes
1 answer
264 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 ...
Arnaud Casteigts's user avatar
10 votes
2 answers
2k 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 ...
Arnaud Casteigts's user avatar
-4 votes
1 answer
134 views

conversion to DAG

Can we reverse directions instead?
HHH's user avatar
  • 7
10 votes
1 answer
823 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 ...
Zach Hunter's user avatar
1 vote
0 answers
65 views

Directed Acyclic Graph partition into minimum subgraphs with a constraint

I have this problem, not sure there is a name for it, wherein a Directed Acyclic Graph has different colored nodes. The idea is to partition it into minimum number of subgraphs with the following 2 ...
Zakir's user avatar
  • 111
2 votes
0 answers
94 views

Which computational framework lies behind the Chinese “Social Credit System”?

BACKGROUND The Social Credit System is a data-driven reputation system which draws on several sources to label various entities, namely businesses and individual citizens, with a trustworthiness ...
Tfovid's user avatar
  • 149
2 votes
0 answers
47 views

"Planar graph coloring is not self-reducible" is this about all $p$-relations encoding that problem?

I have a question about the paper "Planar graph coloring is not self-reducible" by Samir Khuller and Vijay Vazirani. The final theorem in that paper states that "Planar Graph k-coloring is not self ...
Elle Najt's user avatar
  • 1,439
1 vote
0 answers
181 views

On planar $4$ regular graphs

It is $NP$-hard to decide if a $4$-regular planar graph can be $3$-colored. Is an exact algorithm possible that under uniform distribution is in average polynomial time?
Turbo's user avatar
  • 12.9k
3 votes
0 answers
315 views

Difficulty of graph coloring and independent set?

Given a graph on $n$ vertices it is strongly $NP$-complete to decide it is $3$-colorable while it is easy to decide it is $n$-colorable. Is there a parsimonious reduction from SUBSET-SUM to GRAPH-3-...
VS.'s user avatar
  • 539