[Paper Review] Coloring graphs with forbidden bipartite subgraphs

James Anderson, Anton Bernshteyn+1 more|arXiv (Cornell University)|Jul 12, 2021

Limits and Structures in Graph Theory11 references4 citations

TL;DR

This paper proves that for any bipartite graph $F$, the chromatic number $\chi(G)$ of any $F$-free graph $G$ of maximum degree $\Delta$ is at most $(1+o(1))\Delta/\log\Delta$, improving prior bounds that depended on $F$'s structure. The result extends to DP-coloring, establishing a uniform constant factor independent of the forbidden bipartite subgraph.

ABSTRACT

A conjecture of Alon, Krivelevich, and Sudakov states that, for any graph $F$, there is a constant $c_F > 0$ such that if $G$ is an $F$-free graph of maximum degree $Δ$, then $χ(G) \leq c_F Δ/ \logΔ$. Alon, Krivelevich, and Sudakov verified this conjecture for a class of graphs $F$ that includes all bipartite graphs. Moreover, it follows from recent work by Davies, Kang, Pirot, and Sereni that if $G$ is $K_{t,t}$-free, then $χ(G) \leq (t + o(1)) Δ/ \logΔ$ as $Δ o \infty$. We improve this bound to $(1+o(1)) Δ/\log Δ$, making the constant factor independent of $t$. We further extend our result to the DP-coloring setting (also known as correspondence coloring), introduced by Dvořák and Postle.

Motivation & Objective

• To resolve a conjecture by Alon, Krivelevich, and Sudakov on the chromatic number of $F$-free graphs with maximum degree $\Delta$.
• To close the gap between known upper bounds and the conjectured optimal bound $c_F \Delta / \log \Delta$ for bipartite forbidden subgraphs $F$.
• To unify the dependence on $F$ in the constant factor, showing it can be reduced to $1+o(1)$, independent of $F$.
• To extend the result to the more general DP-coloring (correspondence coloring) framework, which generalizes list coloring.

Proposed method

• Uses a randomized coloring procedure with iterative color reduction, modeling the survival of available colors through a branching process.
• Applies a modified version of the entropy method and probabilistic analysis to bound the expected number of available colors per vertex.
• Employs a DP-cover formalism to model correspondence colorings, where color matches between adjacent vertices are defined by edge-specific matchings.
• Introduces a sequence of color sets $\ell_i$ and $d_i$ representing available colors and uncolored vertices, tracking their evolution over iterations.
• Uses concentration bounds and exponential decay estimates to show that the number of available colors grows faster than the number of uncolored vertices.
• Applies a key lemma to establish the existence of a stage $i^\star$ where the number of available colors exceeds eight times the number of uncolored vertices, enabling completion of the coloring.

Experimental results

Research questions

• RQ1 Can the constant factor in the chromatic number bound for $F$-free graphs be made independent of the forbidden bipartite graph $F$?
• RQ2 Does the conjectured bound $\chi(G) \leq (1+o(1))\Delta / \log \Delta$ hold for all bipartite $F$, even when $F = K_{t,t}$ for large $t$?
• RQ3 Can the result be extended beyond ordinary coloring to the more general DP-coloring model?
• RQ4 What is the minimal constant factor achievable in the asymptotic chromatic number bound for $F$-free graphs with large maximum degree?
• RQ5 Is the bound $\chi(G) \leq (1+o(1))\Delta / \log \Delta$ tight for $F$-free graphs, especially in the context of random regular graphs or high-girth graphs?

Key findings

• For any bipartite graph $F$, every $F$-free graph $G$ of maximum degree $\Delta$ satisfies $\chi(G) \leq (1+\varepsilon)\Delta / \log \Delta$ for sufficiently large $\Delta$, with the constant factor independent of $F$.
• The bound $\chi(G) \leq (1+o(1))\Delta / \log \Delta$ is asymptotically optimal up to a factor of 2, matching the lower bound from random $\Delta$-regular graphs.
• The result extends to DP-coloring: every $F$-free graph admits a DP-coloring with at most $(1+o(1))\Delta / \log \Delta$ colors.
• The proof establishes that the number of available colors grows sufficiently fast during iterative coloring steps, ensuring that at some stage $i^\star$, $\ell_{i^\star} \geq 8d_{i^\star}$, enabling completion of the coloring.
• The analysis shows that the survival probability of colors decays exponentially, but the initial number of colors is large enough to overcome this decay and maintain a superlinear growth in available colors.
• The final bound matches the shattering threshold for random graphs and the factor of i.i.d. independent set threshold in $\Delta$-regular trees, suggesting that further improvement would be extremely difficult.