Restricted List Chromatic Number

• The restricted list chromatic number is a graph invariant that extends traditional list coloring by imposing cardinality constraints on vertex color lists.
• It unifies finite and infinite combinatorics by formalizing robust proper colorings and introducing variants for forbidden induced subgraphs.
• Key research employs probabilistic methods and cardinal invariant analyses to establish extremal bounds and unveil complexities in graph structures.

The restricted list chromatic number generalizes the conventional list chromatic number in both finite and infinite combinatorics, quantifying the minimum order of “robust” proper list coloring under cardinal constraints on the structure of lists. It appears in several variants, most notably for infinite graphs, where it is defined via functions $L:V_X \to [\kappa]^\kappa$ with the demand that every such assignment of $\kappa$-sized lists from a cardinal $\kappa$ permits a proper coloring selecting one color per list so that adjacent vertices receive distinct colors. This invariant, and its analogs in the context of forbidden induced subgraphs (“$\mathcal{G}$-free colorings”), has been the subject of extensive recent work, addressing both foundational properties and extremal bounds in finite and infinite cases.

1. Definitions and Basic Properties

For a graph $X = (V_X, E_X)$ and infinite regular cardinal $\kappa$, $X$ is said to be $\kappa$-restricted-list-colorable, denoted $RList(X, \kappa)$, if for every function $L: V_X \to [\kappa]^\kappa$ there exists a choice function $c: V_X \to \kappa$, with $c(v) \in L(v)$ and $c(v) \neq c(w)$ for every edge $\{v, w\} \in E_X$ (Hayashi, 7 Dec 2025). The restricted list chromatic number is

$$\chi^R_\ell(X) = \min\{\,\kappa : RList(X, \kappa)\text{ holds}\,\}.$$

It follows the hierarchy

$$List(X,\kappa) \Rightarrow RList(X,\kappa) \Rightarrow SList(X,\kappa) \Rightarrow Chr(X,\kappa),$$

where $List(\cdot)$ refers to standard list coloring, $SList(\cdot)$ to stationary set list coloring, and $Chr(\cdot)$ to ordinary chromaticity. In the context of finite graphs and forbidden substructures, a parallel generalization appears: for a graph $H$ and family of forbidden graphs $\mathcal{G}$, a coloring is $\mathcal{G}$-free if no color class induces a copy of any member of $\mathcal{G}$, and the restricted list $\mathcal{G}$-free chromatic number $\chi^L_{\mathcal{G}}(H)$ is defined as the minimum $k$ such that every $k$-list assignment admits a $\mathcal{G}$-free $L$-coloring (Rowshan, 2022).

2. Structural Results and Characterizations

The restricted list chromatic number interacts delicately with classical cardinal invariants. For complete bipartite graphs $K_{\lambda,\lambda}$ and infinite regular $\kappa \ge \lambda$, $RList(K_{\lambda,\lambda}, \kappa)$ holds if and only if $\lambda < \mathfrak{r}_\kappa$, the $\kappa$-reaping number (Hayashi, 7 Dec 2025). Under Generalized Continuum Hypothesis (GCH), if $X$ is infinite and $\chi^R_\ell(X) = \kappa$ is infinite, then the coloring number satisfies $\mathrm{Col}(X) \leq \kappa^{++}$.

In the setting of restricted $\mathcal{G}$-free choosability for finite graphs, an essential tool is a Hall-type lemma: if $H$ fails to admit a $\mathcal{G}$-free $L$-coloring for some $k$-list assignment, then there exists $S \subset V(H)$ with $|S| > \delta \cdot |\bigcup_{v \in S}L(v)|$, where $\delta$ is the minimum degree of any forbidden graph in $\mathcal{G}$ (Rowshan, 2022).

3. Extremal Bounds and Lower-Bound Constructions

Boundedness of the restricted list chromatic number is closely tied to Hadwiger-type and Reed-type conjectures. For ordinary list coloring, Hadwiger's conjecture (that $K_t$-minor-free graphs are $(t-1)$-colorable) does not extend: Steiner proved that for all sufficiently large $t$, there exist $K_t$-minor-free graphs $G_t$ with $\chi_\ell(G_t) \geq (2 - o(1))t$, refuting the Kawarabayashi-Mohar conjecture at $c=3/2$ and establishing that $c \geq 2$ is necessary (Steiner, 2021). This construction employs probabilistic methods, notably pasting copies of the dense complement of a random bipartite graph across a shared clique, exploiting local expansion properties to both avoid $K_t$ minors and force high list chromaticity.

In the forbidden subgraph ($\mathcal{G}$-free) list coloring context, precise thresholds aligning list and ordinary coloring can be established for small graphs and certain structural conditions (e.g., for regular-join operations or in Ohba-type regimes) (Rowshan, 2022).

4. Connections to List Colorings with Forbidden Subgraphs

The restricted list chromatic number unifies with generalized list $\mathcal{G}$-free chromatic numbers. When $\mathcal{G} = \{K_2\}$, the invariant reduces to the classic list chromatic number. For other forbidden families, $k$-$\mathcal{G}$-free-choosable is defined as the property that every $k$-list assignment admits a coloring with each color class inducing a $\mathcal{G}$-free subgraph (Rowshan, 2022).

Key results include:

• Ohba-type equalities: If $$|V(H)| \leq \delta\,\chi_{\mathcal{G}}(H) + \sqrt{\delta\,\chi_{\mathcal{G}}(H)} - (\delta-1)$$, then $\chi^L_{\mathcal{G}}(H) = \chi_{\mathcal{G}}(H)$.
• Subadditivity under graph join: $$\chi^L_G(H \oplus H') \leq \chi^L_G(H) + \chi^L_G(H')$$ under explicit Hall-type size/degrees hypotheses.

A plausible implication, based on these, is that the gap between ordinary $\mathcal{G}$-free chromatic and list $\mathcal{G}$-free chromatic numbers is often limited for graphs with certain structural conditions or when large joins are involved.

5. Infinite-Graph Phenomena and Set-theoretic Pathologies

In the infinite case, restricted list chromaticity is controlled by cardinal invariants and may demonstrate nonmonotonic behavior under cardinal arithmetic—specifically, there are ZFC-consistent models in which $RList(X, \kappa)$ holds but $RList(X, \lambda)$ fails for $\kappa < \lambda$ (Hayashi, 7 Dec 2025). For instance, forcing with $\operatorname{Fn}_{<\kappa}(\mu,2)$ can guarantee $RList(K_{\mu,\mu},\kappa)$ but not $RList(K_{\mu,\mu},\lambda)$. δ

Comparison with stationary list coloring shows these notions are distinct: $SList(K_{2^\kappa,\kappa},\kappa)$ may hold where $RList(K_{2^\kappa,\kappa},\kappa)$ fails.

6. Special Cases, Methodological Innovations, and Open Problems

Notable extremal, structural, and methodological results include:

• For triangle-free graphs with maximum degree $\Delta$, the list chromatic number satisfies $\chi_\ell(G) \le (1+o(1))\Delta / \ln\Delta$ via probabilistic methods and entropy-compression frameworks (Molloy, 2017).
• For $K_r$-free graphs, $$\chi_\ell(G)\le 200r\frac{\Delta\ln\ln\Delta}{\ln\Delta}$$ using Shearer-type bounds on the number of independent sets and local random recoloring plus entropy compression (Molloy, 2017).

Open questions persist regarding tight Brooks-type bounds in the restricted list setting, precise thresholds tying forbidden substructures to list choosability, and further understanding the behavior of these invariants under cardinal arithmetic and forcing. There is continued interest in refining Ohba-type bounds for generalized forbidden subgraph families and extending subadditivity and equality results to more complex hereditary properties.

7. Summary Table: Key Notions and Relationships

Invariant	Definition Summary	Key References
$\chi^R_\ell(X)$	Least $\kappa$: $\forall L: V_X\to [\kappa]^\kappa$, proper $L$-coloring exists	(Hayashi, 7 Dec 2025)
$\chi^L_{\mathcal{G}}(H)$	Least $k$: every $k$-list assignment admits $\mathcal{G}$-free $L$-coloring	(Rowshan, 2022)
List chromatic number $\chi_\ell$	Least $k$: every $k$-list assignment admits proper coloring	(Molloy, 2017)

These results collectively show that the restricted list chromatic number provides a unified technical lens for both finite and infinite combinatorics, linking coloring, cardinal invariants, forbidden substructures, and structural extremal graph theory.