2
votes
0answers
158 views

Computing the chromatic polynomial of graph modulo $x-3$

The chromatic polynomial of graph $P(G,x)$ is univariate polynomial which counts the number of colorings of $G$ with $x$ colors for natural $x$. Graph is not $k$ colorable iff $P(G,k)=0$. The ...
20
votes
3answers
1k views

Can one measure the infeasibility of four color proofs?

Terms like "impractical" and "unfeasible" are used to say the Robertson, Sanders, Seymour, and Thomas proof of the four color theorem needs computer assistance. Obviously no precise measure is ...
1
vote
3answers
213 views

Strategic vertex labeling

We are given a graph $G=(V,E)$ with positive edge weights $w_{i}$ and numerical {0,1,-1} labels $l$ for all vertices . We know that $G$ has a subset $G'$ with all vertices labeled 0(all vertices with ...
26
votes
4answers
2k views

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 ...
12
votes
6answers
2k views

A decision problem in graph coloring

It'll be great to get a pointer or answer to the following question: What is the complexity of the following problem? Given an unweighted and undirected graph, can we have a proper (not necessarily ...
9
votes
1answer
610 views

Finding Two Rainbow Spanning Trees

Suppose we have a graph whose edges are coloured. It's not necessarily a proper colouring: a given node may have 0, 1, or several incident edges of a given colour. Is the following problem ...
4
votes
7answers
2k views

Suggest effective heuristic (not precise) graph colouring algorithm

Can you suggest a good rough graph colouring algorithm? What are the best such algorithms nowadays?