The computational complexity of determining the chromatic number () and independence number () for n-fold graph products depends on the product type and the structure of the base graph . Here's a breakdown:

Strong Product ()

• Chromatic Number: . Given , this can be computed in polynomial time in (via exponentiation), s...

Short answer:

• For the Cartesian power Cn(G), the chromatic number is trivial: χ(Cn(G)) = χ(G) for all n (Sabidussi, 1957/58). For certain G (e.g., complete graphs Kq and K2 in particular), α(Cn(G)) also has a closed form and can be computed in time polynomial in n.
• For the strong power Sn(G), computing α(Sn(G)) is exactly the “finite-n” version of the Shannon-capacity problem. Even for very sm...

The short answer is that, at present, no unconditional (i.e. provable in the sense of “P ≠ NP’’-type results) exponential–time lower bounds are known for any of the six numbers you list, and for some of them the value is in fact trivial. What we do know is the following.

1. Cartesian powers – the chromatic number is trivial
   For the Cartesian product we have χ(G □ H)=max{χ(G),χ(H)}. ...

Here is essentially the answer: all of these “large” product‐graphs

Gⁿ (strong, cartesian or tensor) can be viewed as exponentially large but succinctly represented graphs, and it has been known for twenty‐five years that standard NP‐hard problems on succinct graphs jump up one exponent to become NEXP‐complete. In particular, if you present

• a small “base‐graph’’ G (with |V(G)...

Below we write
Sₙ(G) … n-fold strong (a.k.a. AND, or, Shannon) product,
Cₙ(G) … n-fold Cartesian product,
Tₙ(G) … n-fold tensor / categorical / Kronecker product.

The input of the algorithm is

• a graph G with |V(G)| = m vertices, given in the ordinary way,
• the two numbers χ(G) and α(G) (you are allowed to look them up in O(1) time),
• an integer n written in binary....

Excellent question. It touches upon some deep and famous problems in graph theory. The key to answering this is to understand the known (and unknown) properties of the chromatic number and independence number with respect to these graph products.

Let's break down the six problems. The short answer is that three of these are easy (polynomial in log n), while the other three are indeed hard (beli...

Computational Complexity of Graph Products: Chromatic and Independence Numbers

This question asks about the computational complexity of calculating chromatic numbers and independence numbers for different types of graph products, specifically whether exponential time (in terms of n) is required for n-fold products.

Cartesian Product ()

For the chromatic number of the Cartesian prod...

<think> We are given a graph G with known chromatic number χ(G) and independence number α(G), computed by an algorithm that might be exponential time. The question asks whether computing the chromatic number and independence number for the n-fold strong product (S_n(G)), Cartesian product (C_n(G)), and tensor product (T_n(G)) of G requires exponential time in n, even for special graphs.

First, re...

This is an interesting question about the computational complexity of graph parameters on graph products. Let me address what is known about these problems.

Known Results

Chromatic Number of Graph Products

1. Cartesian Product: For the Cartesian product, we have the exact formula: This follows from the fact that the Cartesian product of a graph with i...