Stationary List Colorability

• Stationary list colorability is defined as the minimum list size (list chromatic number) ensuring a proper vertex coloring in graphs with clique restrictions.
• The approach leverages entropy-compression and local correction methods to derive nearly optimal bounds in triangle-free and Kₙ-free graphs.
• Probabilistic techniques and independent set estimations are used to keep flaw probabilities extremely low, ensuring efficient coloring even for large maximum degrees.

Stationary list colorability, commonly formalized as the list chromatic number $\chi_\ell(G)$ of a graph $G$, concerns the minimum integer $q$ such that for any assignment of color lists $L(v)$ of size $q$ to each vertex $v$, there exists a proper coloring $c$ with $c(v)\in L(v)$ for all $v$. For graphs with forbidden substructures—most notably constraints on the clique number—stationary list colorability exhibits strong asymptotic improvements compared to general graphs. Molloy (Molloy, 2017) establishes nearly optimal upper bounds for $\chi_\ell(G)$ in graphs with small clique number, specifically triangle-free graphs and more generally $K_r$-free graphs.

1. Definitions and Role of Clique Restrictions

The list chromatic number $\chi_\ell(G)$ is the smallest $q$ such that for every assignment of color lists $L(v)$ of size $q$, the vertices of $G$ admit a proper coloring $c$ with $c(v)\in L(v)$. Clique restrictions, such as forbidding triangles (triangle-free) or more generally forbidding $K_r$ subgraphs, impose local sparsity constraints. In triangle-free graphs, each neighborhood forms an independent set; in $K_r$-free graphs, the neighborhoods lack an $(r-1)$-clique. These structural restrictions yield combinatorial advantages, particularly in bounding the list chromatic number by exploiting properties of independent sets and local coloring flexibility.

2. Main Bounds for List Chromatic Number

Triangle-Free Graphs

Molloy establishes the following bound: for every $\epsilon>0$ there is $\Delta_0(\epsilon)$ such that every triangle-free graph $G$ with maximum degree $\Delta \ge \Delta_0$ satisfies:

$$\chi_\ell(G) \le (1+\epsilon) \frac{\Delta}{\ln \Delta}.$$

As $\Delta\to\infty$, this sharpens to

$\chi_\ell(G) = (1+o(1)) \frac{\Delta}{\ln \Delta}.$

$K_r$-Free Graphs

If $G$ is $K_r$-free, $r\ge 4$, and $\Delta$ is the maximum degree, Molloy proves:

$$\chi_\ell(G) \le 200\,r\,\Delta\,\frac{\ln\ln\Delta}{\ln\Delta}.$$

These results match or explicitly quantify the leading constants in previous asymptotic formulations.

3. Methodological Framework and Local Correction

The proof strategies revolve around a blend of entropy-compression (inspired by Moser–Tardos) and combinatorial sampling. The approach for both triangle-free and $K_r$-free cases starts with an all-blank partial coloring. For each vertex $v$, the list of usable colors is

$$L_v = \{\text{colors in}\ L(v)\ \text{not used on}\ N(v)\}\cup\{\mathsf{Blank}\}.$$

Two bad events (“flaws”) are monitored:

• $B_v$: $|L_v|<L$, for an auxiliary parameter $L\ll q$,
• $Z_v$: $|\{u\in N(v)\mid u \text{ is Blank}\}|<L$.

Upon detecting a flaw, a local correction (FIX) recolors the neighborhood $N(v)$ stochastically, with recursive calls to correct newly arising flaws within bounded distance. For triangle-free graphs this recursion is within distance 2 for $B$ flaws and distance 3 for $Z$ flaws. In $K_r$-free graphs, the roles are interchanged.

Probability bounds for flaws, established via Chernoff-type (negatively correlated) inequalities and independent-set counting, confirm that each flaw has occurrence probability $<\Delta^{-4}$. With these bounds, entropy-compression ensures that the correction process halts with high probability in $O(n)$ steps, and the resultant coloring extends to a full solution via the Local Lemma.

4. Estimation of Independent Sets and Local Sampling

For $K_r$-free graphs, key lemmas provide independent set counts. If $H$ is $K_r$-free,

$2^{|V(H)|/(r-1)} \le I(H) \le 2^{|V(H)|},$

where $I(H)$ is the number of independent sets. At least half of the independent sets in $H$ have size at least

$\frac{\log_2 I(H)}{\log_2\log_2 I(H)}.$

These counts enable random recoloring procedures based on tuples of independent sets, with iterative sampling producing distributions equivalent to uniform partial assignments. Such largescale enumeration of independent sets is central for probabilistic bounds required in the correction arguments.

5. Asymptotic Behavior and Parameter Choices

In the triangle-free setting, $\epsilon$ may vanish as $\Delta\to\infty$, with auxiliary parameters (e.g., $L=\Delta^{\epsilon/2}$) scaling to ensure error terms in Chernoff bounds converge to $o(1)\cdot(\Delta/\ln\Delta)$, yielding the precise asymptotic

$\chi_\ell(G)\le (1+o(1))\,\Delta/\ln\Delta.$

For $K_r$-free graphs, the factor $200\,r$ derives from constants in the independent-set and flaw probability lemmas, and the $(\ln\ln\Delta)/(\ln\Delta)$ factor is enforced by selecting $$q\sim \text{const}\cdot\Delta(\ln\ln\Delta)/\ln\Delta$$ so that the flaw probabilities remain sufficiently low.

6. Historical Perspective and Comparison with Prior Work

Johansson previously established $\chi_\ell(G)=O(\Delta/\ln\Delta)$ for triangle-free graphs with a constant approximately 4. Kim showed the same for graphs of girth at least 5. Molloy’s result improves this to the exact leading constant $1+o(1)$, and his argument streamlines the original methods through entropy-compression. For $K_r$-free graphs, Johansson’s bound of $O(\Delta\ln\ln\Delta/\ln\Delta)$ is recovered and extended, though Molloy obtains an explicit constant $200\,r$ and a shortened proof structure. These advances deliver the first simple, nearly optimal choice-number bounds for both triangle-free and $K_r$-free graphs, unifying and refining the landscape of stationary list colorability in sparse settings (Molloy, 2017).