--- title: Adaptable Choosability in Graph Coloring url: https://www.emergentmind.com/topics/adaptable-choosability type: topic --- # Adaptable Choosability in Graph Coloring Adaptable choosability is a list-coloring parameter in which the local obstruction is specified by an edge coloring or edge labelling rather than by adjacency alone. For a graph \(G\), an edge coloring \(F\) (or labelling \(\tau\)), and a list assignment \(L\), one asks for a vertex coloring \(c\) with \(c(v)\in L(v)\) such that no edge \(uv\) satisfies \(c(u)=c(v)=F(uv)\). The least integer \(k\) for which this is possible for every edge coloring and every list assignment with \(|L(v)|\ge k\) is the adaptable choosability of \(G\), denoted \(ch_{ad}(G)\) or \(\mathrm{ch}_a(G)\) in the literature. The notion sits between ordinary list coloring, choosability with separation, and more general conflict-coloring frameworks, and it has become a useful intermediary in planar graph coloring, sparse graph theory, and asymptotic coloring of high-girth multigraphs [2010.16190, 2107.04253, 2509.13913]. ## 1. Formal definition and basic mechanism Given a graph \(G=(V,E)\), a list assignment \(L\) assigns to each vertex \(v\) a set \(L(v)\) of admissible colors. An \(L\)-coloring is a coloring \(c\) with \(c(v)\in L(v)\) for all \(v\). If \(F:E(G)\to\mathbb N\) is a possibly improper edge coloring, then \(c\) is adapted to \(F\) when no edge \(uv\) has both endpoints and the edge itself colored the same; equivalently, \(c(u)=c(v)=F(uv)\) is forbidden on every edge. A graph is adaptably \(k\)-choosable if for every list assignment \(L\) with \(|L(v)|\ge k\) for all \(v\), and every edge coloring \(F\), there exists an \(L\)-coloring adapted to \(F\) [2010.16190]. This local rule is weaker than proper coloring on a fixed instance, because equality on adjacent vertices is allowed unless it matches the edge label. At the same time, the quantification over all edge colorings \(F\) makes the invariant robust. In the multigraph formulation, the same definition is given using an edge labelling \(\tau:E\to\mathbb N\), and \(\mathrm{ch}_a(G)\) is the minimum \(k\) such that every size-\(k\) list assignment and every \(\tau\) admit a coloring \(\sigma\) with no edge \(e=uv\) for which \(\tau(e)=\sigma(u)=\sigma(v)\) [2107.04253]. A standard comparison is with choosability with separation. If \(G\) is adaptably \(k\)-choosable, then \(G\) is \((k,1)\)-choosable. The proof idea is explicit: from a \((k,1)\)-list assignment, each edge is colored by its unique shared color, if such a color exists; any coloring adapted to that edge coloring is then proper as a list coloring [2010.16190]. This observation makes adaptable choosability a sufficient condition for separation choosability with overlap parameter \(1\). | Notion | Local constraint | Relation stated in the literature | |---|---|---| | Adaptable choosability \(ch_{ad}(G)\) | forbid \(c(u)=c(v)=F(uv)\) | implies \((k,1)\)-choosability | | Separation choosability \(ch_{sep}(G)\) | adjacent lists share at most one color | special case of adaptable \(k\)-choosability | | Union separation \((k,t)\) | \(|L(u)\cup L(v)|\ge t\) on edges | complementary local diversity condition | ## 2. Separation choosability and union-local variants Choosability with separation prescribes an upper bound on \(|L(u)\cap L(v)|\) for adjacent vertices. A \((k,d)\)-list assignment requires \(|L(v)|\ge k\) for all \(v\) and \(|L(u)\cap L(v)|\le d\) on every edge. For planar graphs, it is known that they are \((4,1)\)-choosable, while whether planar graphs are \((3,1)\)-choosable remains open; one strengthening allows an independent set \(I\) to receive lists of size \(3\) and all other vertices lists of size \(4\), still under the condition \(|L(u)\cap L(v)|\le 1\) for every edge [1306.5283]. The same paper shows a sharp obstruction to pushing list sizes lower: for every \(k\), there exists a planar graph \(G\) and a \((*,1)\)-list assignment \(L\) with \(|L(v)|\ge 2\) for every vertex and \(|L(u)\cup L(v)|\ge k\) for every edge \(uv\), such that \(G\) is not \(L\)-colorable [1306.5283]. This demonstrates that very large edge-local diversity in the union of adjacent lists does not compensate for lists of size \(2\) in planar graphs. Choosability with union separation makes that edge-local diversity the primary hypothesis. For \(t\ge k\), a \((k,t)\)-list assignment requires \(|L(v)|\ge k\) for all vertices and \(|L(u)\cup L(v)|\ge t\) for every edge \(uv\). A graph is \((k,t)\)-choosable if every such assignment is colorable. When \(t=k\), this is ordinary \(k\)-choosability; increasing \(t\) forces more diversity between adjacent lists [1512.07847]. The resulting theory is nontrivial even on bipartite graphs: for all \(t\ge k\ge 2\), there exists a bipartite graph that is not \((k,t)\)-choosable, and for all \(t\ge 2\) there exists a bipartite planar graph that is not \((2,t)\)-choosable. There also exists a planar graph that is not \((3,5)\)-choosable. On the positive side, all planar graphs are \((3,11)\)-choosable and \((4,9)\)-choosable, and if a planar graph contains no chorded \(4\)-cycles, then it is \((4,7)\)-choosable [1512.07847]. The paper presenting union separation explicitly places it in the broad landscape of adaptable choosability, as a complementary way of controlling edge-local interactions between lists. ## 3. General upper bounds from arboricity and maximum average degree Two structural mechanisms recur in adaptable choosability: orientations and sparsity. If the edge-arboricity \(a(G)\) of a graph is at most \(k\), then \(G\) is adaptably \((k+1)\)-choosable. The proof proceeds via an orientation with \(d^+(v)\le k\): one chooses the color of \(v\) to avoid the colors appearing on outgoing edges in the prescribed edge coloring [2010.16190]. This gives immediate corollaries. Every planar graph is adaptably \(4\)-choosable because \(a(G)\le 3\), every triangle-free planar graph is adaptably \(3\)-choosable because \(a(G)\le 2\), and every \(K_5\)-minor-free graph is adaptably \(4\)-choosable [2010.16190]. A second bound is stated in terms of maximum average degree: \[ ch_{ad}(G)\le \left\lfloor \frac{Mad(G)}{2}\right\rfloor+1. \] Consequently, every graph \(G\) is \(\big(\left\lfloor \frac{Mad(G)}{2}\right\rfloor+1,1\big)\)-choosable [2010.16190]. In practice, many planar and near-planar results are obtained by proving \(Mad(G)<4\), after which adaptable \(3\)-choosability follows. For union separation, sparsity yields an analogous threshold theorem. For \(k\ge 2\) and \(t\ge 2k-1\), if \[ Mad(G) < 2k\left(1-\frac{k}{t+1}\right), \] then \(G\) is \((k,t)\)-choosable [1512.07847]. This gives a parallel sparsity principle: low average density guarantees colorability when adjacent lists have sufficiently large unions. ## 4. Planar graphs and the \((3,1)\)-choosability program A central motivation for adaptable choosability is the conjecture that every planar graph is \((3,1)\)-choosable. Since adaptable \(3\)-choosability implies \((3,1)\)-choosability, sufficient conditions for the former yield progress on the latter [2010.16190]. Several such conditions are known. If \(G\) is a planar graph with no intersecting triangles and every triangle is adjacent to at most one \(4\)-cycle, then \(Mad(G)<4\), hence \(G\) is adaptably \(3\)-choosable and therefore \((3,1)\)-choosable. The same conclusion holds when \(G\) has no intersecting triangles and no intersecting \(4\)-cycles. It also holds when no triangle is adjacent to any triangle or \(4\)-cycle and every \(5\)-cycle is adjacent to at most three triangles. Another result states that if any two triangles in \(G\) have distance at least \(2\) and no triangle is adjacent to a \(4\)-cycle, then \(G\) is adaptably \(3\)-choosable [2010.16190]. These proofs are based on discharging arguments designed to establish \(Mad(G)<4\), followed by the general maximum-average-degree bound. The method is materially shorter than many classical reducibility arguments in planar list coloring, and this is one of the conceptual contributions of the adaptable framework [2010.16190]. The landscape is not monotone with respect to forbidding short cycles in the simplest possible way. Triangle-free planar graphs are adaptably \(3\)-choosable, but planar graphs without \(4\)-cycles may fail to be adaptably \(3\)-choosable in general [2010.16190]. This guards against the misconception that forbidding a single short cycle length is automatically enough. At the same time, some nontrivial planar subclasses do satisfy the desired conclusion: every Halin graph is \((3,1)\)-choosable [2010.16190]. ## 5. High-degree sparse multigraphs and conflict-coloring methods A different regime emerges for multigraphs of large maximum degree \(\Delta\) and no cycles of length \(3\) or \(4\). In that setting, \[ \mathrm{ch}_a(G)\le (2\sqrt{2}+o(1))\sqrt{\frac{\Delta}{\ln\Delta}}. \] Thus, adaptable choosability grows on the order of \(\sqrt{\Delta/\ln\Delta}\) rather than linearly in \(\Delta\) [2107.04253]. The same work states that this is within a factor of \(4\) of optimal, and that the asymptotically best possible leading constant under these assumptions lies between \(\sqrt{2}\) and \(2\sqrt{2}\) [2107.04253]. The proof is obtained from a more general conflict-choosability theorem. In conflict coloring, an edge labelling \(\tau:E\to\mathbb N\times\mathbb N\) assigns an ordered pair of forbidden colors to each edge, and a coloring \(\sigma\) is proper if no edge \(uv\) satisfies \((\sigma(u),\sigma(v))=\tau(uv)\). The conflict degree \(D(\tau)\) measures how many edges between a fixed pair of vertices can impose the same local forbidden color. Adaptable coloring is the symmetric conflict-coloring case with conflict degree \(1\), and consequently \(\mathrm{ch}_a(G)\le \mathrm{ch}_{\nleftrightarrow}(G)\) [2107.04253]. The general theorem states that for any \(\epsilon>0\), there exists \(\Delta_0\) such that if \(G\) has maximum degree \(\Delta\ge\Delta_0\), no cycles of length \(3\) or \(4\), and \(D(\tau)\le \Delta^{\frac{1}{4}\epsilon^2/(\epsilon+5)^2}\), then every list assignment with \[ |L(v)|\ge (2\sqrt{2}+\epsilon)\sqrt{\Delta/\ln\Delta} \] admits a proper conflict coloring. A corollary gives the cleaner numerical bound that list size \(50\sqrt{\Delta/\ln\Delta}\) suffices when \(D(\tau)\le \Delta^{1/5}\) [2107.04253]. Technically, the proof uses a truncation lemma to remove overly problematic colors, an iterative semi-random coloring procedure, and concentration inequalities including Talagrand, Chernoff, and the Lovász Local Lemma. The evolving state is tracked by parameters \(L_i\) and \(T_i\), representing list sizes and badness measures, and a Reed-type finishing lemma completes the coloring when the remaining conflict load is small relative to the lists [2107.04253]. ## 6. Comparative invariants, exact small cases, and planar classifications Recent work places adaptable choosability in a sharper comparative framework together with separation choosability \(ch_{sep}(G)\) and the single conflict chromatic number \(\chi_{\nleftrightarrow}(G)\). The basic inequalities are \[ \chi_{\nleftrightarrow}(G)\ge ch_{ad}(G)\ge ch_{sep}(G), \qquad ch(G)\ge ch_{ad}(G)\ge ch_{sep}(G). \] This suggests that adaptable choosability is an intermediate invariant: stronger than separation choosability, but weaker than single-conflict and ordinary list coloring [2509.13913]. At the low end, the parameter \(ch_{ad}(G)=2\) is completely characterized for connected graphs of minimum degree at least \(2\): such a graph is adaptably \(2\)-choosable if and only if it consists of two or three internally disjoint paths connecting two distinct vertices. The same characterization holds for single conflict \(2\)-colorability. By contrast, the class with \(ch_{sep}(G)=2\) is broader; for example, \(K_4\) is separation \(2\)-choosable but not adaptably \(2\)-choosable [2509.13913]. There are also explicit families where the three parameters coincide. If \(G\) is connected, \(\Delta(G)\le 4\), and \(G\) contains at least two disjoint cycles of length at least \(4\), then \[ ch_{sep}(G)=ch_{ad}(G)=\chi_{\nleftrightarrow}(G)=ch(G)=3. \] Conversely, there are explicit families with \(\chi_{\nleftrightarrow}(G)=ch_{ad}(G)>ch_{sep}(G)\), and the paper poses the open problem of whether \(ch_{ad}(G)-ch_{sep}(G)\) can be arbitrarily large [2509.13913]. For planar graphs, the attainable triples \((ch_{sep}(G),ch_{ad}(G),\chi_{\nleftrightarrow}(G))\) are highly constrained. Excluding the unresolved cases \((3,3,4)\) and \((4,4,4)\), a simple planar graph has such a triple if and only if it is one of \[ (1,1,1),\ (2,2,2),\ (2,3,3),\ (3,3,3),\ (3,4,4). \] There is a planar multigraph with triple \((3,3,4)\), but whether such a simple planar graph exists remains open. Another planar impossibility result states that no planar graph satisfies \[ ch_{sep}(G)