Chromatic numbers from edge ideals: Graph classes with vanishing syzygies are polynomially $χ$-bounded
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911671399743488 |
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| author | Engström, Alexander |
| author_facet | Engström, Alexander |
| contents | The chromatic number $χ$ of a graph is bounded from below by its clique number $ω,$ but it can be arbitrary large. Perfect graphs are defined by $χ=ω$ for all induced subgraphs. An interesting relaxation are $χ$-bounded graph classes, where $χ\leq f(ω).$ 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 $β_{i,j}(I)$ are central algebraic geometric invariants. We study the graph classes where for some fixed $i,j$ that syzygy vanishes, that is, $β_{i,j}(I_G)=0.$
We prove that $χ\leq f(ω),$ where $f$ is a polynomial of degree $2j-2i-4.$ For the elementary special case $β_{i,2i+2}(I_G)=0,$ this amounts to that $(i+1)K_2$-free graphs are ${ω-1+2i \choose 2i}$-colorable, improving on an old combinatorial result by Wagon. We also show that triangle-free graphs with $β_{i,j}(I_G)=0$ 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 $χ.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21800 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chromatic numbers from edge ideals: Graph classes with vanishing syzygies are polynomially $χ$-bounded Engström, Alexander Combinatorics Commutative Algebra The chromatic number $χ$ of a graph is bounded from below by its clique number $ω,$ but it can be arbitrary large. Perfect graphs are defined by $χ=ω$ for all induced subgraphs. An interesting relaxation are $χ$-bounded graph classes, where $χ\leq f(ω).$ 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 $β_{i,j}(I)$ are central algebraic geometric invariants. We study the graph classes where for some fixed $i,j$ that syzygy vanishes, that is, $β_{i,j}(I_G)=0.$ We prove that $χ\leq f(ω),$ where $f$ is a polynomial of degree $2j-2i-4.$ For the elementary special case $β_{i,2i+2}(I_G)=0,$ this amounts to that $(i+1)K_2$-free graphs are ${ω-1+2i \choose 2i}$-colorable, improving on an old combinatorial result by Wagon. We also show that triangle-free graphs with $β_{i,j}(I_G)=0$ 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 $χ.$ |
| title | Chromatic numbers from edge ideals: Graph classes with vanishing syzygies are polynomially $χ$-bounded |
| topic | Combinatorics Commutative Algebra |
| url | https://arxiv.org/abs/2512.21800 |