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7
votes
1answer
151 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 ...
2
votes
1answer
70 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 ...
0
votes
0answers
47 views

Individualization of two Subgraphs

Notation: $H$ is the adjacency matrix of graph $H'$ respectively. $H_k$ is the block or sub-matrix of matrix $H$. The adjacency matrix of graph $H_k \cup H_e$ (subgraphs of $H'$) is $M_{(k,e)}$ ...
8
votes
1answer
153 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 ...
1
vote
0answers
53 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
118 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
votes
1answer
90 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?
5
votes
1answer
207 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
54 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
256 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 ...
7
votes
0answers
58 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 ...
5
votes
1answer
96 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
175 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
votes
1answer
409 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. ...
1
vote
0answers
26 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
195 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
votes
0answers
104 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 ...
14
votes
0answers
328 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 ...
7
votes
1answer
283 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 ...
2
votes
1answer
108 views

For which graph classes the fractional chromatic index rounded up equals the chromatic index?

Let $\chi'_f(G)$ be the fractional chromatic index. For which graph classes $\lceil \chi'_f(G) \rceil = \chi'(G)$? Since $\chi'_f(G)$ is computable in polynomial time, solving this gives tractable ...
0
votes
0answers
131 views

An interesting class of colored graphs?

Let $G$ be a complete graph edge-colored with $k$ colors. We say that $G$ is Gallai-colored if no triangle is colored with three distinct colors. Fix a tuple of integers $c = (c_1,\ldots,c_k)$. We may ...
1
vote
0answers
99 views

Chromatic number of a graph where neighbourhood of each vertex is isomorphic

Consider a graph $G(V,E)$. For any two vertices $v_1 , v_2 \in V$, the subgraph induced by the neigbourhood of vertex $v_1$, denoted by $G_{v_1}$ be isomorphic to the subgraph induced by the ...
6
votes
2answers
700 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 ...
5
votes
1answer
343 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 ...
7
votes
1answer
208 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 ...
11
votes
3answers
550 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.
8
votes
1answer
511 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-...
2
votes
1answer
104 views

Complexity of Haemers' minimum rank

In 1978 Willem H. Haemmers published "An upper bound on the Shannon capacity of a graph". Tims has a survey of more recent results his thesis. What is the computational complexity of computing ...
5
votes
1answer
137 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 ...
10
votes
2answers
430 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 ...
1
vote
1answer
3k 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?
6
votes
2answers
339 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 ...
17
votes
1answer
836 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 ...
12
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
4answers
938 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 ...
20
votes
5answers
1k 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
544 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
162 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
494 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
200 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 ...
26
votes
1answer
804 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
201 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 = [s_i,...
5
votes
1answer
367 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-...
2
votes
3answers
1k 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
229 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
286 views

Hardness of approximating fractional chromatic number on bounded degree graphs

Is it apx-hard to approximate fractional chromatic number on bounded degree graphs?
12
votes
2answers
500 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
82 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 ...
14
votes
0answers
398 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 ...
7
votes
1answer
259 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 ...