# Tagged Questions

Vertex colouring, Edge Colouring, List Colouring, Fractional Chromatic Number and other variants of graph colouring problems are all on topic.

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### SDP based heuristics for graph coloring

This is a question about semidefinite programming heuristics for graph vertex coloring based on the Lovasz theta number such as "Approximate Graph Coloring" by Karger, Motwani and Sudan or "A ...
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### many 5-list colorings

If a graph is 4-list-colorable, then it is easy to see that it has exponentially many 5-list colorings. This is the first sentence in [Exponentially many 5-list-colorings of planar graphs,JCTB,97 (...
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### How many uniquely colored degree two vertices in 3-coloring of subcubic graph?

Is there a graph with maximum degree three that has 3 degree two vertices that must get the same (resp. different) color in every 3-coloring of the graph? I'm interested in any similar results as ...
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### A property of minimal prime ideals in rings with finite chromatic number

Let $R$ be a commutative ring with identity. There are so many ways to associate a graph to $R$. Consider this: take the elements of $R$ (All elements including zero) as vertices an two distinct ...
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### 2-edge colorable graph approximation

A 2 edge-colorable graph is a graph in which we can color the edges with two colors, in a way such that no edges of the same color share a vertex. Given a graph G = (V,E) I want to find a 2 edge-...
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### What is known about the chromatic number for minimum-distance graphs in higher dimensions?

For a set of points in $\mathbb{R}^d$ with minimum distance $a$, the minimum-distance graph connect two points iff they are at distance $a$. We can also view it as the tangency graph for a set of ...
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### Colorings of triangulations as a generalization of the four-color problem

My question can be viewed as a generalization of the four-color problem. Instead of a planar graph, consider a triangulation of a d-sphere. One wants to color vertices with N colors so that no two ...
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### Graphs with a coloring that majorizes all other colorings

By a coloring of a graph $G = (V,E)$ I mean a map $\kappa:V\to\mathbb{N}$ such that $\kappa(u)\ne \kappa(v)$ whenever $u$ and $v$ are adjacent. (Sometimes this is called a proper coloring but I am ...
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### Coloring a Ferrers diagram

I've shopped the problem below around a bit and it seems like it might be known, or not that hard to resolve, but so far I've come up empty-handed. Say that a coloring of the dots of a Ferrers ...
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### Can one make Erdős's Ramsey lower bound explicit?

Erdős's 1947 probabilistic trick provided a lower exponential bound for the Ramsey number $R(k)$. Is it possible to explicitly construct 2-colourings on exponentially sized graphs without large ...
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### Performance guarantee of RLF [closed]

I cannot manage to find the performance guarantee of the Recursive Largest First (RLF) algorithm for approximating the chromatic number of a graph. I know DSATUR has a $\mathcal{O}(n)$ guarantee, ...
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### Generalizations of the Four-Color theorem

The four color theorem asserts that every planar graph can be properly colored by four colors. The purpose of this question is to collect generalizations, variations, and strengthenings of the four ...
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### Why is “P vs. NP” necessarily relevant?

I want to start out by giving two examples: 1) Graham's problem is to decide whether a given edge-coloring (with two colors) of the complete graph on vertices $\lbrace-1,+1\rbrace^n$ contains a ...
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### A relaxation of proper coloring

I am wondering if the following relaxation of proper coloring appears somewhere. I have tried some searching and have found a few relaxations of proper coloring, but none the coincides with what I ...
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### Bounds on chromatic number of $k$-planar graphs

A $1$-planar graph can be drawn in the plane so that each arc is crossed at most once by another arc. A $k$-planar graph can be drawn so that each arc is crossed at most $k$ times. Planar graphs are ...
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### The numbers of edge colorings and partitions into vertex covers

Let $G = \langle X \cup Y, E\rangle$ be a $d$-regular bipartite graph such that $|U|=|V|=n$, and assume that $d$ divides $n$. Let $F(G)$ denote the number of proper edge colorings of $G$ using $d$ ...
The reason for the gorgeous simplicity of the classic proofs of $R(3,3)$, $R(4,4)$, $R(3,4)$ and $R(3,5)$ is that essentially all you need is the trivial bound and a picture. But for bigger Ramsey ...
Let $G=(V,E)$ be a simple, undirected graph with the following properties: Contracting any edge increases the chromatic number by $1$; For each minor $M$ of $G$ we have $\chi(M) \leq \chi(G) + 1$. ...