Questions tagged [graph-colouring]

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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. ...
0
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1answer
76 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 ...
3
votes
2answers
137 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 ...
2
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1answer
70 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 ...
2
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0answers
159 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 $\...
4
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0answers
75 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 ...
3
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0answers
136 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 ...
4
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2answers
141 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 ...
2
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1answer
112 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) $...
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 ...
2
votes
1answer
74 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 ...
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0answers
70 views

Hardness of vertex colouring on hypergraphs with $O(\log n)$ edges

I'm interested to know whether there has been any work done on the problem in the title. For the problem to be meaningful, we would naturally need that the hyperedges must have large ($\omega(1)$) ...
2
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2answers
120 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....
1
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0answers
46 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.
3
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2answers
101 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 ...
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 ...
7
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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 ...
-4
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1answer
110 views

conversion to DAG

Can we reverse directions instead?
9
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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 ...
1
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0answers
52 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 ...
2
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0answers
84 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 ...
2
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0answers
37 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 ...
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0answers
169 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?
3
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0answers
269 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-...
1
vote
1answer
93 views

Color shifting in a bipartite graph

Assume that we have a directed bipartite graph $G = \langle L\dot\cup R, E\rangle $. Where $E$ contains directed edges only from $L$ to $R$, that is, $E\subseteq L\times R$. Assume further that the ...
2
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0answers
29 views

Heuristics for exact #3COLORING close to the 3-colorability threshold

What are some fast heuristics for exactly counting 3-colorings of graphs close to or at the 3-colorability threshold? Is there literature on the average-case performance for any of these methods?
4
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0answers
96 views

2-hop distributed coloring in the CONGEST model

Consider a graph $G=(V,E)$ and let $d(u,v)$ denote the distance between $u$ and $v$ in $G$. A 2-hop coloring is a mapping $c:V\to{1,\ldots, C}$ ($C$ is the number of colors) such that $d(u,v)\le 2 \...
3
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0answers
145 views

Is 3-coloring bounded degree graphs subexponential: $O(\exp{(\sqrt{n}\log^2{n})})$? [closed]

We got an argument that 3-coloring bounded degree graphs is subexponential with complexity $O(\exp{(\sqrt{n}\log^2{n})})$. The treewidth of a planar graphs on $n$ vertices is $O(\sqrt{n})$ and 3-...
1
vote
1answer
118 views

How good of an approximate 2-coloring can you get of the halved cube graph?

We say that a 2-coloring $col : V_G \rightarrow \{0, 1\}$ of a graph $G$ is $\epsilon$-approximate if $Pr_{(w, v) \in E_G}(col(w) \neq col(v)) \geq \epsilon$. For every $n$, what is the maximum $\...
0
votes
1answer
271 views

3-coloring graph zero-knowledge proof [closed]

I was researching about zero-knowledge proofs and in this link http://web.mit.edu/~ezyang/Public/graph/svg.html I've seen the exercise question: Currently, you can only select adjacent pairs of nodes ...
0
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0answers
110 views

Path-width and chromatic number

How would I prove that the chromatic number of a graph $G$ is smaller than or equal to the path-width of $G$ + 1? or $\chi(G) \leq pw(G) + 1$
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0answers
73 views

Describe the condition of “non-adjacent 3-cycles” in terms of the cubic adjacency matrix

Oleg Borodin and André Raspaud "A sufficient condition for planar graphs to be 3-colorable" Journal of Combinatorial Theory B88, 2003, 17–27 state the following conjecture: Conjecture 1.2: ...
7
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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?
1
vote
1answer
153 views

Deciding $\omega(G)>k$ when $\alpha(G)$ and $\chi(G)$ have bounds and are known

Given a $k>0$ and a graph $G(V,E)$ with known independence number $\sqrt{|V|}\leq\alpha(G)\leq\alpha\sqrt{|V|}$ and chromatic number $\frac1\beta\sqrt{|V|}\leq\chi(G)\leq\sqrt{|V|}$ for some fixed $...
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 ...
3
votes
1answer
113 views

What is the recognition complexity of k-uniform k-partite hypergraphs? [duplicate]

We can easily recognize bipartite graphs, but I surprisingly couldn't find anything on the recognition complexity of 3-uniform tripartite hypergraphs, though I'm sure this has been studied. It's also ...
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 ...
2
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0answers
64 views

About complexity of recovering or learning Bayesian networks

Are there complexity theoretic results about recoverability or learnability of the marginals (of the source vertices) and the conditionals (along each of the edges) of a Bayesian network from having ...
2
votes
1answer
530 views

Partition into triangles in a 3-partite graphe

Let $G=(X\cup Y\cup Z,E)$ be a 3-partite graph such that: $|X|=|Y|=|Z|=q$. $2 \leq d(v) \leq 6$ for all $v \in X\cup Y\cup Z$, where $d(v)$ is the degree of v. $\sum_{x \in X} d(x)=\sum_{y \in Y}d(y)=...
-5
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1answer
868 views

What is the complexity of the fastest method of k-coloring any graph? [closed]

I heard brute-force is the only method. Is there any other way? Is there a way to prove that the complexity cannot be exponential?
4
votes
1answer
654 views

Canonical way of coloring graphs (individualization) for isomorphism purpose

Stefan Kratsch and Pascal Schweitzer in their paper Graph Isomorphism for Graph Classes Characterized by two Forbidden Induced Subgraphs give a characterization of graphs on which Graph Isomorphism ...
0
votes
1answer
73 views

Number of rounds of iterative one-round distributed color reduction

We are talking about one-round coloring algorithms for distributed graphs. In "On the complexity of distributed graph coloring" (theorem 5.1) Kuhn and Wattenhoffer presented a one-round algorithm to ...
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
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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
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 $u,v$. ...
1
vote
1answer
183 views

Two paper appear to imply collapse via coloring $P_5$-free graphs

Found this from graphclasses.org. Two papers give conflicting results for coloring $P_5$-free graphs which appear to imply $P=NP$. From Polynomial-time algorithm for vertex k-colorability of P_5-...
12
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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. ...
2
votes
0answers
30 views

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 $i$ ...
3
votes
2answers
303 views

Graphs whose maximal clique intersection graph has bounded chromatic number

In general, The chromatic number of the clique intersection graph a graph can again be arbitrarily high. For example, take $n$ maximal cliques of size $n$ such that every pair of maximal cliques has ...
6
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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 ...