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Coloring graphs with forbidden almost bipartite subgraphs

James Anderson, Anton Bernshteyn, Abhishek Dhawan

TL;DR

This work addresses the AKS conjecture, seeking universal upper bounds on the chromatic number of F-free graphs with maximum degree Δ. It proves that for every almost bipartite graph F, the constant c(F) can be taken uniformly as c(F) ≤ 4, yielding χ(G) ≤ (4+o(1)) Δ/ log Δ in the relevant regime, and extends the result to DP-coloring (corresponding χ_{DP}(G) ≤ (4+o(1)) Δ/ log Δ). The authors develop a Rödl nibble–style coloring procedure that advances by carefully controlling average color-degrees rather than worst-case degrees, leveraging the K_{1,s,t}-free structure to tame dependencies, and employing advanced probabilistic tools (LLL, Chernoff, Exceptional Talagrand, Harris–Kleitman). They further connect these bounds to DP-coloring palettes and derive algorithmic consequences, including efficient randomized coloring algorithms and palette sparsification results that hold in the DP-coloring setting. Collectively, the results provide a uniform bound in all currently known AKS cases and open pathways toward a deeper understanding of uniform bounds in graph coloring under forbidden subgraph constraints. The methods combine structural graph theory with sophisticated probabilistic analysis, promising impact on independence-number bounds and related coloring problems.

Abstract

Alon, Krivelevich, and Sudakov conjectured in 1999 that for every finite graph $F$, there exists a quantity $c(F)$ such that $χ(G) \leq (c(F) + o(1)) Δ/ \logΔ$ whenever $G$ is an $F$-free graph of maximum degree $Δ$. The largest class of connected graphs $F$ for which this conjecture has been verified so far, by Alon, Krivelevich, and Sudakov themselves, comprises the almost bipartite graphs (i.e., subgraphs of the complete tripartite graph $K_{1,t,t}$ for some $t \in \mathbb{N}$). However, the optimal value for $c(F)$ remains unknown even for such graphs. Bollobás showed, using random regular graphs, that $c(F) \geq 1/2$ when $F$ contains a cycle. On the other hand, Davies, Kang, Pirot, and Sereni recently established an upper bound of $c(K_{1,t,t}) \leq t$. We improve this to a uniform constant, showing $c(F) \leq 4$ for every almost bipartite graph $F$. This surprisingly makes the bound independent of $F$ in all the known cases of the conjecture. We also establish a more general version of our bound in the setting of DP-coloring (also known as correspondence coloring) and consider some algorithmic consequences of our results.

Coloring graphs with forbidden almost bipartite subgraphs

TL;DR

This work addresses the AKS conjecture, seeking universal upper bounds on the chromatic number of F-free graphs with maximum degree Δ. It proves that for every almost bipartite graph F, the constant c(F) can be taken uniformly as c(F) ≤ 4, yielding χ(G) ≤ (4+o(1)) Δ/ log Δ in the relevant regime, and extends the result to DP-coloring (corresponding χ_{DP}(G) ≤ (4+o(1)) Δ/ log Δ). The authors develop a Rödl nibble–style coloring procedure that advances by carefully controlling average color-degrees rather than worst-case degrees, leveraging the K_{1,s,t}-free structure to tame dependencies, and employing advanced probabilistic tools (LLL, Chernoff, Exceptional Talagrand, Harris–Kleitman). They further connect these bounds to DP-coloring palettes and derive algorithmic consequences, including efficient randomized coloring algorithms and palette sparsification results that hold in the DP-coloring setting. Collectively, the results provide a uniform bound in all currently known AKS cases and open pathways toward a deeper understanding of uniform bounds in graph coloring under forbidden subgraph constraints. The methods combine structural graph theory with sophisticated probabilistic analysis, promising impact on independence-number bounds and related coloring problems.

Abstract

Alon, Krivelevich, and Sudakov conjectured in 1999 that for every finite graph , there exists a quantity such that whenever is an -free graph of maximum degree . The largest class of connected graphs for which this conjecture has been verified so far, by Alon, Krivelevich, and Sudakov themselves, comprises the almost bipartite graphs (i.e., subgraphs of the complete tripartite graph for some ). However, the optimal value for remains unknown even for such graphs. Bollobás showed, using random regular graphs, that when contains a cycle. On the other hand, Davies, Kang, Pirot, and Sereni recently established an upper bound of . We improve this to a uniform constant, showing for every almost bipartite graph . This surprisingly makes the bound independent of in all the known cases of the conjecture. We also establish a more general version of our bound in the setting of DP-coloring (also known as correspondence coloring) and consider some algorithmic consequences of our results.
Paper Structure (12 sections, 35 theorems, 158 equations, 1 figure, 1 table)

This paper contains 12 sections, 35 theorems, 158 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Bounds for the Alon--Krivelevich--Sudakov conjecture
  3. DP-coloring
  4. Algorithmic considerations
  5. Proof strategy
  6. Preliminaries
  7. The Coloring Procedure
  8. Proof of Lemma \ref{['iterationTheorem']}
  9. Proof of Lemma \ref{['expectationKeptUncolor']}
  10. Proof of Lemma \ref{['concentrationKeptUncolor']}
  11. Proof of Theorem \ref{['mainTheorem']}
  12. Palette sparsification

Key Result

Theorem 1.3

Conjecture conj:AKS holds for almost bipartite graphs $F$. That is, $c(F) < \infty$ for every almost bipartite graph $F$.

Figures (1)

  • Figure 1: A $C_6$-free graph with a $2$-fold cover containing $C_6$.

Theorems & Definitions (60)

  • Conjecture 1.2: Alon--Krivelevich--Sudakov AKSConjecture
  • Theorem 1.3: Alon--Krivelevich--Sudakov AKSConjecture
  • Theorem 1.4: Alon--Krivelevich--Sudakov AKSConjecture
  • Theorem 1.5: Davies--Kang--Pirot--Sereni DKPS
  • Theorem 1.6
  • Corollary 1.7
  • Definition 1.9
  • Proposition 1.10: anderson2024coloring
  • Conjecture 1.11: DP-version of the Alon--Krivelevich--Sudakov Conjecture
  • Theorem 1.12
  • ...and 50 more