Adapted List Coloring in Graph Theory

• Adapted List Coloring is a variant where allowable color lists are customized based on structural features like surface embeddings or vertex degrees.
• The framework uses global counting and local extension techniques, employing Euler’s formula and Heawood numbers to set optimal list-size thresholds.
• This approach extends to equitable, supermodular, distributed, and reconfiguration variants, impacting practical algorithms and complexity in graph coloring.

Searching arXiv for recent and foundational papers on adapted list coloring and closely related variants. Adapted list coloring denotes a family of refinements of list coloring in which permissible lists are not treated as a uniform global resource, but are adjusted to structural features of the underlying object. In one prominent surface-theoretic sense, it refers to Heawood-type choosability for graphs embedded on a surface with one distinguished face whose vertices receive shorter lists than the ambient Heawood bound. In a broader sense, the same principle appears in equitable, supermodular, distributed, online, additive, correspondence, reconfiguration, and optimization settings, where the adaptation is to a face, a degree parameter, a supermodular demand, a defect budget, or a dynamic recoloring constraint (Hutchinson, 2013, Yokoi, 2017, Fuchs et al., 2024).

1. Core framework

For a graph $G$, a list assignment $L$ gives each vertex $v$ a set $L(v)$ of allowable colors. An $L$-coloring is a proper vertex-coloring $c$ such that $c(v)\in L(v)$ for every vertex. A graph is $k$-choosable if it admits an $L$-coloring for every list assignment with $|L(v)|\ge k$ for all $L$0. This is the common substrate on which the various adapted forms are built (Hutchinson, 2013, Kaul et al., 2018).

In the surface setting, surfaces are parameterized by Euler genus $L$1: orientable genus $L$2 corresponds to $L$3, and a nonorientable surface with $L$4 crosscaps has $L$5. The Heawood number is

$L$6

It gives the optimal chromatic bound on a surface of Euler genus $L$7 and also governs choosability. If $L$8 is embedded on such a surface and $L$9 is a distinguished face, the adapted surface problem asks whether one may lower the list sizes on $v$0 while retaining Heawood-sized lists elsewhere (Hutchinson, 2013).

This local-to-global adjustment recurs in other settings. Equitable list coloring keeps properness but requires each color to be used at most $v$1 times (Kaul et al., 2018). In list supermodular coloring, the required list size at an element $v$2 is adapted to local supermodular demand functions $v$3 and $v$4 rather than to a single global maximum parameter (Yokoi, 2017). In distributed defective list coloring, each node $v$5 receives a list $v$6 together with per-color defect bounds $v$7, and feasibility depends on aggregate local slack such as $v$8 relative to the outdegree $v$9 (Fuchs et al., 2024).

2. The surface-theoretic model: Heawood adaptation on a distinguished face

The central theorem for surfaces is a Thomassen-style adaptation of Heawood choosability. Let $L(v)$0 with $L(v)$1, let $L(v)$2 be embedded on a surface of Euler genus $L(v)$3, and let $L(v)$4 be a distinguished face. If $L(v)$5 for every $L(v)$6 and $L(v)$7 for every $L(v)$8, then $L(v)$9 is $L$0-colorable unless the induced subgraph $L$1 contains $L$2 (Hutchinson, 2013).

This theorem is explicitly modeled on Thomassen’s planar theorem for a plane graph with a designated face $L$3, where vertices on $L$4 have 3-lists and all other vertices have 5-lists. The surface result replaces the planar $L$5-list baseline by the Heawood number and allows a reduction by exactly $L$6 on the designated face. In that sense it is a Heawood-parameterized analogue of Thomassen’s $L$7 phenomenon (Hutchinson, 2013).

Concrete small-genus instances make the statement explicit. For the projective plane, $L$8 and $L$9, so 4-lists on $c$0 and 6-lists elsewhere suffice unless $c$1 contains $c$2. For the torus, $c$3 and $c$4, giving the pair $c$5 with obstruction $c$6 on $c$7. For Euler genus $c$8, $c$9, so the adapted threshold is $c(v)\in L(v)$0 with obstruction $c(v)\in L(v)$1 on the distinguished face (Hutchinson, 2013).

The theorem is formulated uniformly in orientable and nonorientable genus. The only notable geometric caveat in the background is the Klein bottle exception: although $c(v)\in L(v)$2, the largest complete graph on the Klein bottle is $c(v)\in L(v)$3 rather than $c(v)\in L(v)$4 (Hutchinson, 2013).

3. Obstructions, tightness, and the exceptional case $c(v)\in L(v)$5

The obstruction $c(v)\in L(v)$6 is necessary. If every vertex of the designated face has only $c(v)\in L(v)$7 available colors, then an induced clique of size $c(v)\in L(v)$8 on that face cannot be $c(v)\in L(v)$9-colored, because the clique requires $k$0 distinct colors while every vertex list has size only $k$1 (Hutchinson, 2013).

The face reduction by $k$2 is also best possible in a stronger sense. Proposition 1.2 constructs infinitely many surfaces and embeddings with all vertices on one face $k$3, no $k$4 on that face, yet the graph is not $k$5-list-colorable. In special cases where $k$6 embeds, one obtains an embedded graph $k$7 with all vertices on one face that is $k$8-critical, hence not $k$9-choosable (Hutchinson, 2013).

This sharpness has a second aspect: planar-style reductions cannot be transferred wholesale to higher-genus settings. There are 2-connected outerplanar near-triangulations modified by identifying two boundary edges that produce embeddings on any surface with all vertices on one face and that are not 3-choosable. A plausible implication is that the planar face bounds are genuinely exceptional and that higher-genus analogues require Heawood-scaled parameters rather than fixed planar constants (Hutchinson, 2013).

The excluded case $L$0 remains unresolved. The counting argument used for $L$1 is too weak there, and the paper notes that multiple copies of $L$2 can be arranged compatibly with Euler genus $L$3. The question whether a “5-list on $L$4, 7-list elsewhere” theorem holds for $L$5, or whether some modified obstruction is unavoidable, is left open (Hutchinson, 2013).

4. Proof methods and constructive consequences

The proof of the adapted Heawood theorem combines global counting with local extension arguments around the designated face. Euler’s formula and edge bounds for triangulations are used to control the number of vertices of degree $L$6 and of degree at least $L$7 on $L$8. Outside special cases, these estimates force favorable degree regimes: vertices on $L$9 have degree at most $|L(v)|\ge k$0, while vertices off $|L(v)|\ge k$1 have degree at least $|L(v)|\ge k$2 (Hutchinson, 2013).

Several structural ingredients are then used. Vizing and Erdős–Rubin–Taylor’s list-Brooks theorem is applied whenever the maximum degree does not exceed the available list size, except for complete graphs and odd cycles. Kostochka–Stiebitz list-critical density is used to rule out hypothetical $|L(v)|\ge k$3-list-critical subgraphs in the all-on-$|L(v)|\ge k$4 regime. A block-cutvertex analysis excludes chains of $|L(v)|\ge k$5-blocks on the face boundary, and induction on $|L(v)|\ge k$6 handles the remaining base cases (Hutchinson, 2013).

The special cases in which $|L(v)|\ge k$7 may embed require separate counting. There the refined inequality

$|L(v)|\ge k$8

controls high-degree vertices on the designated face, and the proof splits according to $|L(v)|\ge k$9 (Hutchinson, 2013).

No discharging is used. The methods are counting, structural decomposition, and precoloring extension. The argument is constructive in the sense that it yields a polynomial-time procedure: compute $L$00, verify the list-size conditions, test whether $L$01 contains $L$02, color $L$03 by the degree-based lemmas, and extend to $L$04 sequentially (Hutchinson, 2013).

A related algorithmic development appears in later work on fixed surfaces. For type 345 list assignments—$L$05 with no girth assumption, $L$06 with girth at least $L$07, or $L$08 with girth at least $L$09—there are linear-time algorithms for planar graphs and for graphs embedded on a fixed surface, and deterministic distributed algorithms running in $L$10 rounds in the LOCAL model (Postle, 2019). This suggests that adapted list coloring on surfaces is not only existential but also algorithmically tractable in structurally sparse regimes.

5. Other major adaptations of list coloring

A substantial branch of the literature adapts list coloring by balancing color usage. In equitable list coloring, an equitable $L$11-coloring is a proper list-coloring in which each color is used on at most $L$12 vertices. For total graphs, the conjectural threshold is $L$13, and this is proved for all graphs with $L$14. In the same paper, powers of paths satisfy equitable $L$15-choosability for $L$16, and powers of cycles satisfy equitable $L$17-choosability for $L$18 when $L$19 and $L$20 (Kaul et al., 2018). A sharper balancing notion, strongly equitable list coloring, requires that at most $L$21 color classes are full; every $L$22-sparse graph with minimum degree at least $L$23 is equitably $L$24-choosable, and every $L$25-sparse graph with minimum degree at least $L$26 is equitably $L$27-choosable, with the proofs established at the strongly equitable level (Kierstead et al., 2024).

Another adaptation is local-demand coloring. In the supermodular framework, the global requirement $L$28 is replaced by the local bound $L$29, where $L$30 is defined from effective sets of an intersecting-supermodular function. In bipartite multigraphs this recovers both Galvin’s global $L$31-bound and the Borodin–Kostochka–Woodall local bound $L$32 (Yokoi, 2017).

Distributed variants adapt lists to orientation, outdegree, and defect budgets. In one direction, an $L$33-coloring on a directed graph of maximum outdegree $L$34 can be converted in two LOCAL rounds into a proper list-coloring provided each list has size

$L$35

and the message complexity is essentially just the transmission of the lists themselves (Maus et al., 2020). In another, recursive partitioning of the global color space preserves the ratio $L$36 up to a constant factor per recursion level and yields deterministic distributed $L$37-list-coloring algorithms in $L$38 rounds, with faster bounds under bounded neighborhood independence (Rzadkowski et al., 2019). A simpler two-sweep framework later extended this to oriented list defective coloring, proving that if $L$39 and $L$40, then each node can choose a color $L$41 with at most $L$42 same-colored outneighbors (Fuchs et al., 2024).

6. Algebraic, matroidal, correspondence, dynamic, and optimization variants

Adaptation also occurs in settings where the color constraints are not purely graph-theoretic. For matroids, a proper coloring is one whose color classes are independent sets, and the online version reveals colors adversarially over time. The central theorem states that the online list-chromatic number equals the chromatic number for every matroid, and more generally that on-line $L$43-colorability is equivalent to colorability from the canonical lists $L$44 (Lasoń et al., 2013).

Polynomial-method variants adapt the Alon–Tarsi theorem to nonstandard coloring constraints. For additive colorings, a new digraph $L$45 is constructed from an orientation $L$46 so that if the number of even Eulerian subdigraphs of $L$47 differs from the number of odd ones, then lists of size $L$48 suffice for an additive list coloring. This yields additive list-colorability for tripartite graphs in which one color class consists of simplicial vertices of outdegree $L$49 (Gossett, 2023). A related computational line develops efficient implementations of the Alon–Tarsi method and shows that when the principal coefficient vanishes, further coefficients impose linear and quadratic constraints on bad list assignments, often allowing one to certify choosability or reduce the search to a few assignments (Dvořák, 2023).

In correspondence coloring, DP-coloring replaces equality of colors across edges by edgewise matchings. Many list-coloring upper bounds persist, but some do not: there exists a planar bipartite graph with DP-chromatic number $L$50, despite the list-coloring fact that every planar bipartite graph is $L$51-list-colorable, and the edge-DP-chromatic number of every $L$52-regular graph with $L$53 is at least $L$54 (Bernshteyn et al., 2017). A further synthesis with variable degeneracy leads to DPG-$L$55-coloring, which simultaneously extends list coloring, DP-coloring, and list-forested coloring. Within that framework, every planar graph is DPG-$L$56-colorable, and several planar DP-coloring theorems under cycle restrictions admit analogous DPG generalizations (Nakprasit et al., 2019).

Some adaptations are dynamic rather than static. In reconfiguration, the vertices of the state space are proper list-colorings and edges correspond to valid single-vertex recolorings. If $L$57 is connected with $L$58 and $L$59 for all vertices, then the reconfiguration graph induced on unfrozen colorings is connected and has diameter $L$60; if one vertex has $L$61, the full reconfiguration graph is connected with diameter at most $L$62 (Cambie et al., 12 May 2025). The same paper shows a sharp phase transition: reducing the list size of a single vertex to $L$63 can shatter the state space into exponentially many components (Cambie et al., 12 May 2025).

Finally, exact optimization adapts list coloring to weighted objectives. In the weighted list coloring problem, each color has a nonnegative weight and one seeks a proper list-coloring minimizing the total weight of active colors. Branch-and-price algorithms model color classes as color-specific stable sets, decompose pricing into maximum-weight stable set problems on the induced color graphs $L$64, and exploit indistinguishable colors to reduce symmetry. Computational studies report strong performance on instances with up to roughly seventy vertices, while also showing that hardness depends not only on graph density and list sizes but also on the distribution of colors across lists (Lucci et al., 2018, Lucci et al., 2023).

7. Conceptual scope and open directions

Taken together, these results suggest a common organizing principle: list coloring is often most effective when the list requirement is calibrated to a structural bottleneck rather than imposed uniformly. On surfaces, the bottleneck is a designated face and the sharp reduction is by $L$65 from the Heawood number (Hutchinson, 2013). In equitable variants it is the permissible load per color (Kaul et al., 2018, Kierstead et al., 2024). In supermodular coloring it is the local effective demand $L$66 (Yokoi, 2017). In distributed algorithms it is the ratio of list size to outdegree or defect budget (Maus et al., 2020, Rzadkowski et al., 2019, Fuchs et al., 2024). In reconfiguration it is the extra local slack $L$67 that determines whether the state space remains connected (Cambie et al., 12 May 2025).

Several open problems remain central. In the original surface setting, the Euler genus $L$68 case is unresolved, as are sharper local reductions on multiple faces or on special low-genus embedding classes (Hutchinson, 2013). In equitable total list coloring, the conjectured threshold beyond $L$69 is still open (Kaul et al., 2018). Distributed work continues to seek smaller list thresholds, fewer rounds, and tighter CONGEST implementations (Maus et al., 2020, Rzadkowski et al., 2019, Fuchs et al., 2024). In DP-coloring, the gap between correspondence coloring and ordinary list coloring remains a recurring obstruction to direct transfer of classical theorems (Bernshteyn et al., 2017). In reconfiguration, rapid mixing of the Glauber dynamics under local slack $L$70 is conjectural (Cambie et al., 12 May 2025).

Adapted list coloring is therefore not a single theorem but a research program. Its unifying theme is the replacement of global, uniform choosability thresholds by list conditions that encode geometry, sparsity, balance, local demand, or dynamics. The surface theorem of Heawood type remains the archetypal instance, but the surrounding literature shows that the same idea has become a general method for refining list coloring across graph theory and adjacent combinatorial frameworks.