Chromatic Bounds from Edge Ideal Syzygies
Analysis
This paper explores the relationship between the chromatic number of a graph and the algebraic properties of its edge ideal, specifically focusing on the vanishing of syzygies. It establishes polynomial bounds on the chromatic number based on the vanishing of certain Betti numbers, offering improvements over existing combinatorial results and providing efficient coloring algorithms. The work bridges graph theory and algebraic geometry, offering new insights into graph coloring problems.
Key Takeaways
- •Establishes polynomial bounds on the chromatic number based on the vanishing of syzygies in the edge ideal.
- •Improves upon existing combinatorial results, such as Wagon's result on $(i+1)K_2$-free graphs.
- •Provides efficient coloring algorithms with O(n^3) time complexity.
- •Connects graph theory and algebraic geometry to provide new insights into graph coloring.
Reference
“The paper proves that $χ\leq f(ω),$ where $f$ is a polynomial of degree $2j-2i-4.$”