Separation Choosability in Graph Coloring

• Separation choosability is a refinement of list coloring that restricts adjacent vertex list overlaps to control coloring thresholds in graphs.
• It connects to adaptable choosability and employs probabilistic and design-theoretic methods to derive asymptotic bounds for complete and multipartite graphs.
• Applications extend to planar, sparse, and outerplanar graphs, with ongoing challenges in proving (3,1) and (4,2)-choosability under specific structural constraints.

Separation choosability is a refinement of list coloring in which the adversarial power of a list assignment is restricted by a local overlap bound on adjacent vertices. In its standard form, a graph $G$ is $(k,s)$-choosable if every list assignment $L$ with $|L(v)| \ge k$ for all $v\in V(G)$ and $|L(u)\cap L(v)|\le s$ for every edge $uv\in E(G)$ admits a proper $L$-coloring; equivalently, one may define the separation choosability parameter $ch_s(G)$ as the minimum such $k$ (Vasudevan et al., 30 Jun 2026). The subject lies at the intersection of list coloring, structural graph theory, and probabilistic combinatorics. It has been studied in several closely related notational frameworks, including $(k,s)$0 for the case $(k,s)$1 (Casselgren et al., 17 Sep 2025), $(k,s)$2 on complete graphs (Furedi et al., 2013), and $(k,s)$3-fold variants in which each vertex receives a $(k,s)$4-subset of its list rather than a single color (Godin et al., 2020). Across these formulations, the recurring theme is that bounding adjacent list intersections can sharply alter choosability thresholds, especially for planar graphs, complete graphs, complete multipartite graphs, and sparse graph classes (Casselgren et al., 2020).

1. Definitions, notation, and basic monotonicity

In standard list coloring, a list assignment on a graph $(k,s)$5 is a function $(k,s)$6 assigning to each vertex $(k,s)$7 a finite set $(k,s)$8 of admissible colors, and a proper $(k,s)$9-coloring is a map $L$0 such that $L$1 for all $L$2 and $L$3 whenever $L$4 (Vasudevan et al., 30 Jun 2026). Separation choosability imposes the additional condition that adjacent lists have bounded overlap. One standard formulation is: a list assignment has separation at most $L$5 if $L$6 for every edge $L$7, and $L$8 is $L$9-choosable if every such assignment with $|L(v)| \ge k$0 admits a proper coloring (Vasudevan et al., 30 Jun 2026). The corresponding invariant is

$|L(v)| \ge k$1

The case $|L(v)| \ge k$2 is especially prominent and is often denoted $|L(v)| \ge k$3 (Casselgren et al., 17 Sep 2025).

Several equivalent or near-equivalent notational conventions coexist in the literature. For complete graphs one often writes $|L(v)| \ge k$4 for the least $|L(v)| \ge k$5 such that every $|L(v)| \ge k$6-list assignment with $|L(v)| \ge k$7 on edges is colorable (Furedi et al., 2013). In work on cycles and outerplanar graphs, one fixes $|L(v)| \ge k$8 and $|L(v)| \ge k$9 and asks for the largest $v\in V(G)$0 such that every $v\in V(G)$1-separating $v\in V(G)$2-list assignment admits a $v\in V(G)$3-fold coloring; this yields the separation number $v\in V(G)$4 and the free-separation number $v\in V(G)$5 (Godin et al., 2020). In the ordinary one-color-per-vertex setting, standard list coloring is recovered once $v\in V(G)$6 is large enough: when $v\in V(G)$7, every $v\in V(G)$8-list assignment automatically satisfies the separation condition, so $v\in V(G)$9-choosability becomes equivalent to $|L(u)\cap L(v)|\le s$0-choosability and $|L(u)\cap L(v)|\le s$1 for all $|L(u)\cap L(v)|\le s$2 (Vasudevan et al., 30 Jun 2026).

Two basic monotonicity principles are explicit in the survey literature. First, $|L(u)\cap L(v)|\le s$3, since restricting list overlaps can only make coloring easier (Vasudevan et al., 30 Jun 2026). Second, $|L(u)\cap L(v)|\le s$4 is nondecreasing in $|L(u)\cap L(v)|\le s$5: if $|L(u)\cap L(v)|\le s$6, then $|L(u)\cap L(v)|\le s$7 (Vasudevan et al., 30 Jun 2026). At the opposite extreme, $|L(u)\cap L(v)|\le s$8 forces adjacent lists to be disjoint, and in particular any graph is $|L(u)\cap L(v)|\le s$9-choosable (Vasudevan et al., 30 Jun 2026). This sharp contrast between $uv\in E(G)$0, small fixed $uv\in E(G)$1, and unrestricted overlap is one of the structural signatures of the area.

2. Relation to ordinary choosability, adaptable choosability, and single-conflict coloring

A central structural relation links separation choosability to adaptable choosability. Given an edge-coloring $uv\in E(G)$2, a vertex coloring $uv\in E(G)$3 is adapted to $uv\in E(G)$4 if no color appears simultaneously on an edge and on both its endpoints; a graph is adaptably $uv\in E(G)$5-choosable if this can be done from every list assignment of size at least $uv\in E(G)$6 (Casselgren et al., 2020). Esperet–Kang–Thomassé observed that if $uv\in E(G)$7 is adaptably $uv\in E(G)$8-choosable, then $uv\in E(G)$9 is $L$0-choosable (Casselgren et al., 2020). The proof is direct: for each edge $L$1 with $L$2, the separation-1 condition makes this intersection a singleton, so one colors $L$3 by that unique common color and then uses adaptedness to forbid equal endpoint colors (Casselgren et al., 2020). This implication underlies several planar results where one first proves adaptable $L$4-choosability and then deduces $L$5-choosability.

Orientation methods provide another bridge. If $L$6 admits an orientation with maximum out-degree at most $L$7, then $L$8 is separation $L$9-choosable; in the notation of the later unified treatment, $ch_s(G)$0 (Casselgren et al., 17 Sep 2025). A related formulation uses edge arboricity: if $ch_s(G)$1, then $ch_s(G)$2 is adaptably $ch_s(G)$3-choosable, and thus $ch_s(G)$4-choosable (Casselgren et al., 2020). Immediate consequences recorded in the literature include that triangle-free planar graphs are adaptably $ch_s(G)$5-choosable and hence $ch_s(G)$6-choosable, and that planar graphs are adaptably $ch_s(G)$7-choosable and hence $ch_s(G)$8-choosable (Casselgren et al., 2020).

The more recent unified perspective places separation choosability together with adaptable choosability $ch_s(G)$9 and the single conflict chromatic number $k$0. The general inequality chain is

$k$1

and also $k$2 (Casselgren et al., 17 Sep 2025). This framework makes it possible to separate the parameters explicitly. For example, cactus graphs comprised solely of triangles with at least two disjoint cycles satisfy

$k$3

showing that separation $k$4-choosability can hold even when the analogous adaptable and single-conflict invariants are $k$5 (Casselgren et al., 17 Sep 2025). Conversely, there are infinite families with

$k$6

under explicit cycle conditions and maximum-degree bounds (Casselgren et al., 17 Sep 2025).

This comparison also clarifies a common misconception. Separation choosability at $k$7 is not the same as adaptable $k$8-colorability or its DP-coloring analogue. In particular, planar graphs without $k$9-cycles are known to be $(k,s)$00-choosable, yet there exist planar graphs without $(k,s)$01-cycles that are not adaptably $(k,s)$02-colorable (Casselgren et al., 2020). Likewise, the correspondence-coloring analogue of $(k,s)$03-choosability fails in general even though positive list-coloring results are available under triangle restrictions (Smith-Roberge, 2022).

3. Extremal and asymptotic behavior on complete and multipartite graphs

For complete graphs, separation choosability has a markedly different scale from ordinary choosability. Kratochvíl, Tuza, and Voigt proved that for fixed $(k,s)$04, $(k,s)$05 is on the order of $(k,s)$06, and later work determined the exact asymptotic constant: $(k,s)$07 for every fixed $(k,s)$08 (Furedi et al., 2013). The proof is mediated by abundant packings. A $(k,s)$09-packing is a family of $(k,s)$10-subsets of a $(k,s)$11-element set in which every $(k,s)$12-set is contained in at most one block; letting $(k,s)$13 denote the smallest $(k,s)$14 admitting more than $(k,s)$15 blocks, the paper proves

$(k,s)$16

and derives the complete-graph asymptotic from the equivalence between non-colorable separated list assignments on $(k,s)$17 and abundant packings with $(k,s)$18 (Furedi et al., 2013). For infinitely many $(k,s)$19, exact values are obtained by finite-field constructions (Furedi et al., 2013).

Balanced complete multipartite graphs behave differently. For the complete $(k,s)$20-partite graph $(k,s)$21 with fixed $(k,s)$22,

$(k,s)$23

as $(k,s)$24, and by monotonicity the same leading term holds for every fixed $(k,s)$25 (Füredi et al., 2011). The corresponding result for complete $(k,s)$26-partite $(k,s)$27-uniform hypergraphs is

$(k,s)$28

for fixed $(k,s)$29 (Füredi et al., 2011). These asymptotics match the classical list chromatic number, so in dense multipartite structures the separation restriction $(k,s)$30 does not change the leading order (Füredi et al., 2011).

Lower bounds of the same logarithmic form persist under strong global separation. For complete $(k,s)$31-partite graphs $(k,s)$32, there exists an absolute constant $(k,s)$33 such that when $(k,s)$34 and $(k,s)$35, one can construct a non-colorable list assignment $(k,s)$36 with

$(k,s)$37

for all vertices and

$(k,s)$38

for all distinct pairs $(k,s)$39 (Puleo, 2014). Since this intersection bound is global rather than merely edge-local, it yields valid lower bounds for ordinary separation choosability and shows that small pairwise overlap does not collapse the choice number below the $(k,s)$40 scale in this family (Puleo, 2014).

For complete bipartite graphs $(k,s)$41, the unified 2025 treatment records an exact threshold shared by four invariants. If $(k,s)$42, then

$(k,s)$43

if $(k,s)$44, then all four parameters equal $(k,s)$45 (Casselgren et al., 17 Sep 2025). This gives a rare exact formula for separation choosability in a nontrivial infinite family.

4. Planar graphs and the central conjectures

Planar graphs are the main testing ground for small-separation list coloring. Kratochvíl–Tuza–Voigt showed that every planar graph is $(k,s)$46-choosable, paralleling Thomassen’s $(k,s)$47-choosability theorem (Casselgren et al., 2020). At the same time, Voigt constructed planar graphs that are not $(k,s)$48-choosable, and the question whether all planar graphs are $(k,s)$49-choosable remains open (Berikkyzy et al., 2015). For $(k,s)$50, the central conjecture is that every planar graph is $(k,s)$51-choosable (Casselgren et al., 2020). This conjecture holds for triangle-free planar graphs (Casselgren et al., 2020), for planar graphs without $(k,s)$52-cycles (Choi et al., 2013), and for planar graphs without $(k,s)$53- and $(k,s)$54-cycles (Choi et al., 2013).

A substantial line of work proves $(k,s)$55-choosability under short-cycle restrictions by way of adaptable choosability and maximum average degree. Montassier–Raspaud–Zhu established

$(k,s)$56

which yields that every graph $(k,s)$57 is $(k,s)$58-choosable (Casselgren et al., 2020). Thus, showing $(k,s)$59 suffices for adaptable $(k,s)$60-choosability and hence for $(k,s)$61-choosability (Casselgren et al., 2020). Using planar discharging, one obtains that a planar graph with no intersecting triangles and with every triangle adjacent to at most one $(k,s)$62-cycle has $(k,s)$63, so it is adaptably $(k,s)$64-choosable and therefore $(k,s)$65-choosable (Casselgren et al., 2020). The same paper records a second family: if no triangle is adjacent to a triangle or a $(k,s)$66-cycle, and each $(k,s)$67-cycle is adjacent to at most three triangles, then again $(k,s)$68 and the graph is adaptably $(k,s)$69-choosable (Casselgren et al., 2020).

The 2013 work of Choi–Lidický–Stolee gives two further landmark positive results: every planar graph without $(k,s)$70-cycles is $(k,s)$71-choosable, and every planar graph without $(k,s)$72- and $(k,s)$73-cycles is $(k,s)$74-choosable (Choi et al., 2013). These are established in strengthened precoloring-extension forms. For the $(k,s)$75-free case, the paper proves an outer-face extension theorem for a path $(k,s)$76 of at most three vertices under a $(k,s)$77-list assignment with boundary list-size constraints, no two $(k,s)$78-vertices adjacent, and no $(k,s)$79-vertex adjacent to two vertices of $(k,s)$80 (Choi et al., 2013). For the $(k,s)$81-free case, it proves colorability from $(k,s)$82-lists with one precolored vertex (Choi et al., 2013). A related strengthening by Kierstead and Lidický shows that in the planar $(k,s)$83-theorem one may allow an independent set of vertices to have lists of size $(k,s)$84 instead of $(k,s)$85 (Kierstead et al., 2013).

The $(k,s)$86-problem occupies a parallel track. Berikkyzy, Drilleau, Kozik, and Micek proved that if a planar graph contains no chorded $(k,s)$87-cycle for any fixed $(k,s)$88, then it is $(k,s)$89-choosable (Berikkyzy et al., 2015). Later, a Thomassen-style inductive extension argument showed that every planar graph with a $(k,s)$90-list assignment $(k,s)$91 and no triangle $(k,s)$92 satisfying

$(k,s)$93

is $(k,s)$94-colorable (Smith-Roberge, 2022). More generally, if there is a clique $(k,s)$95 meeting every such offensive triangle, then the graph is $(k,s)$96-colorable (Smith-Roberge, 2022). The 2026 survey also records Zhu’s theorem that every planar graph is $(k,s)$97-choosable in the weaker sense that every $(k,s)$98-list assignment satisfying $(k,s)$99 on edges is colorable (Vasudevan et al., 30 Jun 2026). This demonstrates concretely that separation can bridge the gap between planar graphs being $L$00-choosable and not uniformly $L$01-choosable (Vasudevan et al., 30 Jun 2026).

5. Sparse graphs, cycles, outerplanar graphs, and bipartite minimum-degree bounds

Cycles and outerplanar graphs admit exact formulas in the $L$02-fold setting. For even cycles $L$03,

$L$04

while for odd cycles $L$05,

$L$06

These formulas are obtained by combining greedy orientation arguments, explicit extremal list assignments, and a parameter-shift proposition stating that if $L$07 is $L$08-choosable, then it is also $L$09-choosable for any $L$10 (Godin et al., 2020).

The same paper determines the free-separation number for cycles, where one arbitrary vertex is precolored. For $L$11,

$L$12

where $L$13 is a piecewise linear function of $L$14, $L$15, and $L$16, and for $L$17 there is a separate three-case exact formula (Godin et al., 2020). These cycle formulas extend to cactus graphs: if $L$18 is a cactus of finite girth $L$19 and either $L$20 or all cycles are triangles, then

$L$21

while in mixed cases involving both triangles and longer cycles the controlling cycle length can shift to the shortest cycle of length at least $L$22 in a narrow parameter band (Godin et al., 2020). For outerplanar graphs of girth $L$23, one has the bounds

$L$24

derived by breadth-first coloring of facial cycles in the weak dual tree (Godin et al., 2020).

Halin graphs provide another constructive planar family. Using that cycles are $L$25-choosable and trees are trivially $L$26-choosable, it is proved that every Halin graph is $L$27-choosable (Casselgren et al., 2020). The construction colors the internal tree first, then removes used colors from the leaf lists to obtain a $L$28-list assignment on the outer cycle (Casselgren et al., 2020).

Beyond planar and outerplanar settings, the strongest general lower bound currently recorded in the supplied literature concerns bipartite minimum degree. There exists an absolute constant $L$29 such that for any bipartite graph $L$30 with minimum degree $L$31,

$L$32

(Esperet et al., 2018). This extends the complete-bipartite asymptotic and implies, via the inequality $L$33, that adaptable choosability also satisfies a $L$34 lower bound in graphs of minimum degree $L$35 (Esperet et al., 2018). The proof combines nearly disjoint hypergraph constructions with a two-stage randomized list assignment on the two sides of the bipartition (Esperet et al., 2018).

6. Variants, proof techniques, and open directions

Three proof paradigms recur throughout the theory. The first is discharging, especially in planar graph results. In the adaptable-$L$36-choosability approach, one assumes a subgraph $L$37 of average degree at least $L$38, uses Euler’s formula to write

$L$39

and redistributes face charges according to carefully tuned rules until every face has nonnegative final charge, contradicting the assumption (Casselgren et al., 2020). Closely related discharging schemes underlie the $L$40-choosability results for planar graphs with forbidden chorded cycles (Berikkyzy et al., 2015) and the union-separation analogues $L$41, $L$42, and $L$43 under additional structural hypotheses (Kumbhat et al., 2015).

The second paradigm is Thomassen-style boundary induction. It is central in the proofs that planar graphs without $L$44-cycles or without $L$45- and $L$46-cycles are $L$47-choosable (Choi et al., 2013), in the mixed-list-size strengthening of planar $L$48-choosability (Kierstead et al., 2013), and in the offensive-triangle criterion for planar $L$49-choosability (Smith-Roberge, 2022). Typical ingredients are minimal counterexamples, 2-connectivity of the outer face, chord restrictions, and local reductions on a boundary path whose colored vertices have singleton or short lists.

The third paradigm is probabilistic or design-theoretic construction. It dominates the asymptotic theory for complete graphs, complete multipartite graphs, and bipartite minimum-degree lower bounds. Abundant packings settle the complete-graph asymptotic $L$50 (Furedi et al., 2013). Nearly disjoint hypergraphs with small independence number yield lower bounds for complete multipartite graphs and complete $L$51-partite $L$52-uniform hypergraphs (Füredi et al., 2011). Randomized list assignments combined with hitting-set heuristics and alteration produce lower bounds of order $L$53 for complete $L$54-partite graphs under strong separation (Puleo, 2014) and order $L$55 for bipartite graphs of minimum degree $L$56 (Esperet et al., 2018).

Several important variants broaden the scope of the subject. One is choosability with union separation, where the constraint is $L$57 rather than $L$58. The identity

$L$59

connects the two models when list sizes are fixed, but the union framework is asymmetric because trimming lists can destroy the union lower bound (Kumbhat et al., 2015). In that setting, all planar graphs are $L$60-choosable and $L$61-choosable, and planar graphs with no chorded $L$62-cycle are $L$63-choosable (Kumbhat et al., 2015). Another variant is $L$64-fold choosability with separation, where one asks for disjoint $L$65-subsets chosen from adjacent lists; the exact cycle and cactus formulas belong to this framework (Godin et al., 2020). A further generalization is the correspondence-coloring analogue, but the planar $L$66-statement fails there: there exist planar graphs that are not $L$67-correspondence choosable (Smith-Roberge, 2022).

The open problems explicitly recorded in the cited literature remain concentrated around a small set of thresholds. The major planar question is whether every planar graph is $L$68-choosable (Casselgren et al., 2020). The parallel planar question is whether every planar graph is $L$69-choosable (Berikkyzy et al., 2015). On the structural side, the unified 2025 paper asks whether for each $L$70 there exists a graph $L$71 with $L$72, whether there exists any graph with

$L$73

and whether there exists a simple planar graph with

$L$74

(Casselgren et al., 17 Sep 2025). For bipartite induced subgraph methods, several Ramsey-type conjectures were posed in connection with extending the $L$75 lower bound beyond bipartite graphs (Esperet et al., 2018). A plausible implication is that the logarithmic scale in degree should remain fundamental in sparse classes without large cliques, but the precise general statement is not settled in the supplied sources.

Taken together, these developments show that separation choosability is neither a minor perturbation of list coloring nor merely a technical restriction on list assignments. In complete graphs it lowers the scale from linear to square-root growth (Furedi et al., 2013); in complete multipartite graphs it preserves logarithmic asymptotics (Füredi et al., 2011); in planar graph theory it produces a hierarchy of sharp small-list phenomena centered on the unresolved $L$76 and $L$77 thresholds (Casselgren et al., 2020). The subject has thus become a meeting point for discharging, precoloring extension, orientation methods, hypergraph packings, and randomized constructions, with its most delicate questions still concentrated at the interface between local overlap constraints and global graph structure.