--- title: Separation Choosability in Graph Coloring url: https://www.emergentmind.com/topics/separation-choosability type: topic --- # Separation Choosability in Graph Coloring 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 \(k\) [2606.31702]. 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 \(ch_{\mathrm{sep}}(G)\) for the case \(s=1\) [2509.13913], \(\chi_\ell(G,c)\) on complete graphs [1303.4030], and \(b\)-fold variants in which each vertex receives a \(b\)-subset of its list rather than a single color [2009.00287]. 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 [2010.16190]. ## 1. Definitions, notation, and basic monotonicity In standard list coloring, a list assignment on a graph \(G\) is a function \(L\) assigning to each vertex \(v\) a finite set \(L(v)\) of admissible colors, and a proper \(L\)-coloring is a map \(c:V(G)\to \bigcup_v L(v)\) such that \(c(v)\in L(v)\) for all \(v\) and \(c(u)\neq c(v)\) whenever \(uv\in E(G)\) [2606.31702]. Separation choosability imposes the additional condition that adjacent lists have bounded overlap. One standard formulation is: a list assignment has separation at most \(s\) if \(|L(u)\cap L(v)|\le s\) for every edge \(uv\), and \(G\) is \((k,s)\)-choosable if every such assignment with \(|L(v)|\ge k\) admits a proper coloring [2606.31702]. The corresponding invariant is \[ ch_s(G)=\min\{k:\; G \text{ is }(k,s)\text{-choosable}\}. \] The case \(s=1\) is especially prominent and is often denoted \(ch_{\mathrm{sep}}(G)\) [2509.13913]. Several equivalent or near-equivalent notational conventions coexist in the literature. For complete graphs one often writes \(\chi_\ell(G,c)\) for the least \(k\) such that every \(k\)-list assignment with \(|L(u)\cap L(v)|\le c\) on edges is colorable [1303.4030]. In work on cycles and outerplanar graphs, one fixes \(a\) and \(b\) and asks for the largest \(c\) such that every \(c\)-separating \(a\)-list assignment admits a \(b\)-fold coloring; this yields the separation number \(s(G;a,b)\) and the free-separation number \(s_{\mathrm{free}}(G;a,b)\) [2009.00287]. In the ordinary one-color-per-vertex setting, standard list coloring is recovered once \(s\) is large enough: when \(s\ge k\), every \(k\)-list assignment automatically satisfies the separation condition, so \((k,s)\)-choosability becomes equivalent to \(k\)-choosability and \(ch_s(G)=\chi_\ell(G)\) for all \(s\ge \chi_\ell(G)\) [2606.31702]. Two basic monotonicity principles are explicit in the survey literature. First, \(ch_s(G)\le \chi_\ell(G)\), since restricting list overlaps can only make coloring easier [2606.31702]. Second, \(ch_s(G)\) is nondecreasing in \(s\): if \(s_1\le s_2\), then \(ch_{s_1}(G)\le ch_{s_2}(G)\) [2606.31702]. At the opposite extreme, \(s=0\) forces adjacent lists to be disjoint, and in particular any graph is \((1,0)\)-choosable [2606.31702]. This sharp contrast between \(s=0\), small fixed \(s\), 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 \(F:E(G)\to \mathbb{N}\), a vertex coloring \(c\) is adapted to \(F\) if no color appears simultaneously on an edge and on both its endpoints; a graph is adaptably \(k\)-choosable if this can be done from every list assignment of size at least \(k\) [2010.16190]. Esperet–Kang–Thomassé observed that if \(G\) is adaptably \(k\)-choosable, then \(G\) is \((k,1)\)-choosable [2010.16190]. The proof is direct: for each edge \(uv\) with \(L(u)\cap L(v)\neq \emptyset\), the separation-1 condition makes this intersection a singleton, so one colors \(uv\) by that unique common color and then uses adaptedness to forbid equal endpoint colors [2010.16190]. This implication underlies several planar results where one first proves adaptable \(3\)-choosability and then deduces \((3,1)\)-choosability. Orientation methods provide another bridge. If \(G\) admits an orientation with maximum out-degree at most \(k\), then \(G\) is separation \((k+1)\)-choosable; in the notation of the later unified treatment, \(ch_{\mathrm{sep}}(G)\le k+1\) [2509.13913]. A related formulation uses edge arboricity: if \(a(G)\le k\), then \(G\) is adaptably \((k+1)\)-choosable, and thus \((k+1,1)\)-choosable [2010.16190]. Immediate consequences recorded in the literature include that triangle-free planar graphs are adaptably \(3\)-choosable and hence \((3,1)\)-choosable, and that planar graphs are adaptably \(4\)-choosable and hence \((4,1)\)-choosable [2010.16190]. The more recent unified perspective places separation choosability together with adaptable choosability \(ch_{ad}\) and the single conflict chromatic number \(\chi_{\nleftrightarrow}\). The general inequality chain is \[ \chi_{\nleftrightarrow}(G)\ge ch_{ad}(G)\ge ch_{\mathrm{sep}}(G), \] and also \(ch(G)\ge ch_{ad}(G)\ge ch_{\mathrm{sep}}(G)\) [2509.13913]. 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 \[ \bigl(ch_{\mathrm{sep}}(G),\, ch_{ad}(G),\, \chi_{\nleftrightarrow}(G)\bigr)=(2,3,3), \] showing that separation \(2\)-choosability can hold even when the analogous adaptable and single-conflict invariants are \(3\) [2509.13913]. Conversely, there are infinite families with \[ ch_{\mathrm{sep}}(G)=ch_{ad}(G)=\chi_{\nleftrightarrow}(G)=ch(G)=3 \] under explicit cycle conditions and maximum-degree bounds [2509.13913]. This comparison also clarifies a common misconception. Separation choosability at \(s=1\) is not the same as adaptable \(3\)-colorability or its DP-coloring analogue. In particular, planar graphs without \(4\)-cycles are known to be \((3,1)\)-choosable, yet there exist planar graphs without \(4\)-cycles that are not adaptably \(3\)-colorable [2010.16190]. Likewise, the correspondence-coloring analogue of \((4,2)\)-choosability fails in general even though positive list-coloring results are available under triangle restrictions [2203.13348]. ## 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 \(s\), \(\chi_\ell(K_n,s)\) is on the order of \(\sqrt{sn}\), and later work determined the exact asymptotic constant: \[ \lim_{n\to\infty}\frac{\chi_\ell(K_n,c)}{\sqrt{c\,n}}=1 \] for every fixed \(c\ge 1\) [1303.4030]. The proof is mediated by abundant packings. A \((v,k,t)\)-packing is a family of \(k\)-subsets of a \(v\)-element set in which every \(t\)-set is contained in at most one block; letting \(v_0(k,t)\) denote the smallest \(v\) admitting more than \(v\) blocks, the paper proves \[ v_0(k,t)=\left(1+o(1)\right)\frac{k^2}{t-1} \] and derives the complete-graph asymptotic from the equivalence between non-colorable separated list assignments on \(K_n\) and abundant packings with \(t=c+1\) [1303.4030]. For infinitely many \(n\), exact values are obtained by finite-field constructions [1303.4030]. Balanced complete multipartite graphs behave differently. For the complete \(r\)-partite graph \(K_{n,\dots,n}\) with fixed \(r\ge 2\), \[ \chi_\ell(K_{n,\dots,n},1)=(1+o(1))\log_{r/(r-1)} n \] as \(n\to\infty\), and by monotonicity the same leading term holds for every fixed \(s\ge 1\) [1109.2969]. The corresponding result for complete \(k\)-partite \(k\)-uniform hypergraphs is \[ \chi_\ell\!\bigl(K^{(k)}_{n,\dots,n},1\bigr)=(1+o(1))\log_k n \] for fixed \(k\) [1109.2969]. These asymptotics match the classical list chromatic number, so in dense multipartite structures the separation restriction \(s=1\) does not change the leading order [1109.2969]. Lower bounds of the same logarithmic form persist under strong global separation. For complete \(r\)-partite graphs \(K_{m*r}\), there exists an absolute constant \(k\) such that when \(r\ge 2\) and \(m\ge r^k\), one can construct a non-colorable list assignment \(L\) with \[ |L(v)|\ge \frac{7}{750}r\ln m \] for all vertices and \[ |L(u)\cap L(v)|\le \left\lfloor \frac{2r}{r-1}\right\rfloor \] for all distinct pairs \(u\neq v\) [1403.3370]. 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 \(\Theta(r\ln m)\) scale in this family [1403.3370]. For complete bipartite graphs \(K_{k,n}\), the unified 2025 treatment records an exact threshold shared by four invariants. If \(n\le k^k-1\), then \[ ch(K_{k,n})=\chi_{\nleftrightarrow}(K_{k,n})=ch_{ad}(K_{k,n})=ch_{\mathrm{sep}}(K_{k,n})\le k; \] if \(n\ge k^k\), then all four parameters equal \(k+1\) [2509.13913]. 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 \((4,1)\)-choosable, paralleling Thomassen’s \(5\)-choosability theorem [2010.16190]. At the same time, Voigt constructed planar graphs that are not \((4,3)\)-choosable, and the question whether all planar graphs are \((4,2)\)-choosable remains open [1512.03787]. For \(s=1\), the central conjecture is that every planar graph is \((3,1)\)-choosable [2010.16190]. This conjecture holds for triangle-free planar graphs [2010.16190], for planar graphs without \(4\)-cycles [1303.2753], and for planar graphs without \(5\)- and \(6\)-cycles [1303.2753]. A substantial line of work proves \((3,1)\)-choosability under short-cycle restrictions by way of adaptable choosability and maximum average degree. Montassier–Raspaud–Zhu established \[ ch_{ad}(G)\le \left\lfloor \frac{\mathrm{Mad}(G)}{2}\right\rfloor +1, \] which yields that every graph \(G\) is \(\bigl(\lfloor \mathrm{Mad}(G)/2\rfloor +1,1\bigr)\)-choosable [2010.16190]. Thus, showing \(\mathrm{Mad}(G)<4\) suffices for adaptable \(3\)-choosability and hence for \((3,1)\)-choosability [2010.16190]. Using planar discharging, one obtains that a planar graph with no intersecting triangles and with every triangle adjacent to at most one \(4\)-cycle has \(\mathrm{Mad}(G)<4\), so it is adaptably \(3\)-choosable and therefore \((3,1)\)-choosable [2010.16190]. The same paper records a second family: if no triangle is adjacent to a triangle or a \(4\)-cycle, and each \(5\)-cycle is adjacent to at most three triangles, then again \(\mathrm{Mad}(G)<4\) and the graph is adaptably \(3\)-choosable [2010.16190]. The 2013 work of Choi–Lidický–Stolee gives two further landmark positive results: every planar graph without \(4\)-cycles is \((3,1)\)-choosable, and every planar graph without \(5\)- and \(6\)-cycles is \((3,1)\)-choosable [1303.2753]. These are established in strengthened precoloring-extension forms. For the \(C_4\)-free case, the paper proves an outer-face extension theorem for a path \(P\) of at most three vertices under a \((*,1)\)-list assignment with boundary list-size constraints, no two \(L_2\)-vertices adjacent, and no \(L_2\)-vertex adjacent to two vertices of \(P\) [1303.2753]. For the \(\{C_5,C_6\}\)-free case, it proves colorability from \((*,1)\)-lists with one precolored vertex [1303.2753]. A related strengthening by Kierstead and Lidický shows that in the planar \((4,1)\)-theorem one may allow an independent set of vertices to have lists of size \(3\) instead of \(4\) [1306.5283]. The \((4,2)\)-problem occupies a parallel track. Berikkyzy, Drilleau, Kozik, and Micek proved that if a planar graph contains no chorded \(\ell\)-cycle for any fixed \(\ell\in\{5,6,7\}\), then it is \((4,2)\)-choosable [1512.03787]. Later, a Thomassen-style inductive extension argument showed that every planar graph with a \((4,2)\)-list assignment \(L\) and no triangle \(T\) satisfying \[ \left|\bigcap_{v\in V(T)}L(v)\right|=2 \] is \(L\)-colorable [2203.13348]. More generally, if there is a clique \(H\) meeting every such offensive triangle, then the graph is \(L\)-colorable [2203.13348]. The 2026 survey also records Zhu’s theorem that every planar graph is \((4,2)\)-choosable in the weaker sense that every \(4\)-list assignment satisfying \(|L(x)\cap L(y)|\le 2\) on edges is colorable [2606.31702]. This demonstrates concretely that separation can bridge the gap between planar graphs being \(5\)-choosable and not uniformly \(4\)-choosable [2606.31702]. ## 5. Sparse graphs, cycles, outerplanar graphs, and bipartite minimum-degree bounds Cycles and outerplanar graphs admit exact formulas in the \(b\)-fold setting. For even cycles \(C_{2p+2}\), \[ s(C_{2p+2};a,b)= \begin{cases} a-b,& b\le a<2b,\\ a,& a\ge 2b, \end{cases} \] while for odd cycles \(C_{2p+1}\), \[ s(C_{2p+1};a,b)= \begin{cases} a-b,& b\le a<2b,\\ b+(p+1)(a-2b),& 2b\le a\le \left\lfloor \left(2+\frac1p\right)b\right\rfloor,\\ a,& a\ge \left\lceil \left(2+\frac1p\right)b\right\rceil. \end{cases} \] These formulas are obtained by combining greedy orientation arguments, explicit extremal list assignments, and a parameter-shift proposition stating that if \(C_n\) is \((a,b,c)\)-choosable, then it is also \((a+2k,b+k,c+k)\)-choosable for any \(k\ge 1\) [2009.00287]. The same paper determines the free-separation number for cycles, where one arbitrary vertex is precolored. For \(n\ge 4\), \[ s_{\mathrm{free}}(C_n;a,b)=\lfloor c(n,a,b)\rfloor, \] where \(c(n,a,b)\) is a piecewise linear function of \(a\), \(b\), and \(n\), and for \(C_3\) there is a separate three-case exact formula [2009.00287]. These cycle formulas extend to cactus graphs: if \(G\) is a cactus of finite girth \(g\) and either \(g\ge 4\) or all cycles are triangles, then \[ s_{\mathrm{free}}(G;a,b)=s_{\mathrm{free}}(C_g;a,b), \] while in mixed cases involving both triangles and longer cycles the controlling cycle length can shift to the shortest cycle of length at least \(4\) in a narrow parameter band [2009.00287]. For outerplanar graphs of girth \(g\ge 5\), one has the bounds \[ s_{\mathrm{free}}(C_{g-1};a,b)\le s_{\mathrm{free}}(G;a,b)\le s_{\mathrm{free}}(C_g;a,b), \] derived by breadth-first coloring of facial cycles in the weak dual tree [2009.00287]. Halin graphs provide another constructive planar family. Using that cycles are \((2,1)\)-choosable and trees are trivially \((2,1)\)-choosable, it is proved that every Halin graph is \((3,1)\)-choosable [2010.16190]. The construction colors the internal tree first, then removes used colors from the leaf lists to obtain a \((2,1)\)-list assignment on the outer cycle [2010.16190]. 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 \(C>0\) such that for any bipartite graph \(G\) with minimum degree \(d\), \[ ch_{\mathrm{sep}}(G)\ge C\log d \] [1802.03727]. This extends the complete-bipartite asymptotic and implies, via the inequality \(ch_a(G)\ge ch_{\mathrm{sep}}(G)\), that adaptable choosability also satisfies a \(C\log d\) lower bound in graphs of minimum degree \(d\) [1802.03727]. The proof combines nearly disjoint hypergraph constructions with a two-stage randomized list assignment on the two sides of the bipartition [1802.03727]. ## 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-\(3\)-choosability approach, one assumes a subgraph \(H\) of average degree at least \(4\), uses Euler’s formula to write \[ \sum_{v\in V(H)}(d(v)-4)+\sum_{f\in F(H)}(r(f)-4)=-8, \] and redistributes face charges according to carefully tuned rules until every face has nonnegative final charge, contradicting the assumption [2010.16190]. Closely related discharging schemes underlie the \((4,2)\)-choosability results for planar graphs with forbidden chorded cycles [1512.03787] and the union-separation analogues \((4,9)\), \((3,11)\), and \((4,7)\) under additional structural hypotheses [1512.07847]. The second paradigm is Thomassen-style boundary induction. It is central in the proofs that planar graphs without \(4\)-cycles or without \(5\)- and \(6\)-cycles are \((3,1)\)-choosable [1303.2753], in the mixed-list-size strengthening of planar \((4,1)\)-choosability [1306.5283], and in the offensive-triangle criterion for planar \((4,2)\)-choosability [2203.13348]. 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 \(\chi_\ell(K_n,c)\sim \sqrt{cn}\) [1303.4030]. Nearly disjoint hypergraphs with small independence number yield lower bounds for complete multipartite graphs and complete \(k\)-partite \(k\)-uniform hypergraphs [1109.2969]. Randomized list assignments combined with hitting-set heuristics and alteration produce lower bounds of order \(r\ln m\) for complete \(r\)-partite graphs under strong separation [1403.3370] and order \(\log d\) for bipartite graphs of minimum degree \(d\) [1802.03727]. Several important variants broaden the scope of the subject. One is choosability with union separation, where the constraint is \(|L(u)\cup L(v)|\ge t\) rather than \(|L(u)\cap L(v)|\le s\). The identity \[ |L(u)\cup L(v)|=|L(u)|+|L(v)|-|L(u)\cap L(v)| \] connects the two models when list sizes are fixed, but the union framework is asymmetric because trimming lists can destroy the union lower bound [1512.07847]. In that setting, all planar graphs are \((4,9)\)-choosable and \((3,11)\)-choosable, and planar graphs with no chorded \(4\)-cycle are \((4,7)\)-choosable [1512.07847]. Another variant is \(b\)-fold choosability with separation, where one asks for disjoint \(b\)-subsets chosen from adjacent lists; the exact cycle and cactus formulas belong to this framework [2009.00287]. A further generalization is the correspondence-coloring analogue, but the planar \((4,2)\)-statement fails there: there exist planar graphs that are not \((4,2)\)-correspondence choosable [2203.13348]. 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 \((3,1)\)-choosable [2010.16190]. The parallel planar question is whether every planar graph is \((4,2)\)-choosable [1512.03787]. On the structural side, the unified 2025 paper asks whether for each \(k\ge 0\) there exists a graph \(G_k\) with \(ch_{ad}(G_k)-ch_{\mathrm{sep}}(G_k)\ge k\), whether there exists any graph with \[ ch_{\mathrm{sep}}(G)