Chromatic numbers from edge ideals: Graph classes with vanishing syzygies are polynomially $\chi$-bounded

TL;DR

This paper links algebraic properties of edge ideals to graph coloring, proving polynomial bounds on chromatic number for classes with vanishing syzygies and providing efficient coloring algorithms.

Contribution

It establishes polynomial bounds on chromatic number based on algebraic syzygy vanishing and improves existing combinatorial bounds for specific graph classes.

Findings

Graphs with certain vanishing syzygies are polynomially $oldsymbol{ ext{chi-bounded}}$.

Triangle-free graphs with specific syzygy conditions are $(j-1)$-colorable.

Colorings can be computed in $O(n^3)$ time for graphs with $n$ vertices.

Abstract

The chromatic number $\chi$ of a graph is bounded from below by its clique number $\omega,$ but it can be arbitrary large. Perfect graphs are defined by $\chi=\omega$ for all induced subgraphs. An interesting relaxation are $\chi$-bounded graph classes, where $\chi\leq f(\omega).$ It is not always possible to achieve this with a polynomial $f.$ The edge ideal $I_G$ of a graph $G$ is generated by monomials $x_ux_v$ for each edge $uv$ of $G.$ The bi-graded betti numbers $\beta_{i,j}(I)$ are central algebraic geometric invariants. We study the graph classes where for some fixed $i,j$ that syzygy vanishes, that is, $\beta_{i,j}(I_G)=0.$ We prove that $\chi\leq f(\omega),$ where $f$ is a polynomial of degree $2j-2i-4.$ For the elementary special case $\beta_{i,2i+2}(I_G)=0,$ this amounts to that $(i+1)K_2$-free graphs are ${\omega-1+2i \choose 2i}$-colorable, improving on an old combinatorial result by Wagon. We also show that triangle-free graphs with $\beta_{i,j}(I_G)=0$ are $(j-1)$-colorable. Complexity wise, we show that these colorings can be derived in time $O(n^3)$ for graphs on $n$ vertices. Moreover, we show that for almost all graphs with parabolic $i,j,$ there are better bounds on $\chi.$