List Flexibility in Graph Coloring

• List flexibility number is a graph invariant that quantifies the minimum list sizes required to satisfy a prescribed fraction of vertex color requests, thus refining standard choosability.
• It is defined as the smallest k for which a graph is (k,1/ρ(G))-flexible, linking list coloring with adversarial requests and structural parameters like the Hall ratio and fractional list packing.
• This invariant has practical applications in analyzing various graph classes, including planar, bipartite, and complete multipartite graphs, under both normalized and weighted formulations.

List flexibility number is a graph invariant from flexible list coloring that measures how large vertex lists must be to guarantee that a prescribed fraction of vertex requests can be honored in every proper list-coloring instance. In one widely used normalization, the parameter is the smallest $k$ such that the graph is $(k,1/\rho(G))$-flexible, where $\rho(G)$ is the Hall ratio; in a related weighted formulation, the analogous parameter is the smallest $k$ for which the graph is weighted $1/k$-flexibly $k$-choosable, and this coincides with the fractional list packing number $\chi_\ell^\bullet(G)$. Across these formulations, the invariant refines ordinary choosability by incorporating adversarial requests, and it is closely tied to Hall ratio, list packing, degeneracy, graph products, and structural graph classes such as planar and complete multipartite graphs (Kaul et al., 2022, Bowdoin et al., 28 Sep 2025, Cambie et al., 2024).

1. Definitions and competing conventions

Let $G$ be a finite simple graph and $L$ a list assignment. A request of $L$ is a function $(k,1/\rho(G))$0 with nonempty domain $(k,1/\rho(G))$1 such that $(k,1/\rho(G))$2 for each $(k,1/\rho(G))$3. The triple $(k,1/\rho(G))$4 is $(k,1/\rho(G))$5-satisfiable if there exists a proper $(k,1/\rho(G))$6-coloring $(k,1/\rho(G))$7 such that

$(k,1/\rho(G))$8

A graph $(k,1/\rho(G))$9 is $\rho(G)$0-flexible if $\rho(G)$1 is $\rho(G)$2-satisfiable whenever $\rho(G)$3 is a $\rho(G)$4-assignment for $\rho(G)$5 and $\rho(G)$6 is a request of $\rho(G)$7 (Kaul et al., 2022, Bennett et al., 2024).

The term “list flexibility number” is used in more than one, closely related, sense.

Convention	Parameter	Representative source
Hall-ratio normalized	$\rho(G)$8	(Kaul et al., 2022, Bowdoin et al., 28 Sep 2025, Bennett et al., 2024)
Weighted/fractional-packing	smallest $\rho(G)$9 such that $k$0 is weighted $k$1-flexibly $k$2-choosable; equals $k$3	(Cambie et al., 2024)
Positive-$k$4 threshold	$k$5	(Choi et al., 2020)

Weighted flexibility is formulated probabilistically: $k$6 is weighted $k$7-flexible for $k$8 if there exists a probability distribution on $k$9-colorings such that for every vertex $1/k$0 and every color $1/k$1,

$1/k$2

This suggests that the terminology is not completely standardized: some papers normalize by the Hall-ratio barrier $1/k$3, whereas others normalize by the extremal balanced value $1/k$4 that arises from fractional packings (Cambie et al., 2023, Cambie et al., 2024).

2. Hall ratio, universal thresholds, and basic inequalities

The Hall ratio of a graph $1/k$5 is

$1/k$6

and equivalently

$1/k$7

This parameter gives the exact universal obstruction to flexibility: if $1/k$8, then $1/k$9 is not $k$0-flexible for any $k$1; if $k$2, then $k$3 is $k$4-flexible, so the optimal fraction $k$5 is always achievable once lists have size $k$6 (Kaul et al., 2022, Bennett et al., 2024).

The Hall-ratio-normalized list flexibility number is therefore

$k$7

It satisfies

$k$8

The lower bound by $k$9 is immediate, because any positive-flexibility guarantee implies $\chi_\ell^\bullet(G)$0-choosability, while the upper bound comes from the $\chi_\ell^\bullet(G)$1-flexibility theorem (Kaul et al., 2022, Bowdoin et al., 28 Sep 2025, Bennett et al., 2024).

Several monotonicity properties are immediate from the definitions: if $\chi_\ell^\bullet(G)$2 is $\chi_\ell^\bullet(G)$3-flexible, then it is $\chi_\ell^\bullet(G)$4-flexible for every $\chi_\ell^\bullet(G)$5 and every $\chi_\ell^\bullet(G)$6; and every subgraph of a $\chi_\ell^\bullet(G)$7-flexible graph is again $\chi_\ell^\bullet(G)$8-flexible (Bennett et al., 2024).

3. Exact values and model examples

A number of graphs admit exact Hall-ratio-normalized list flexibility numbers. Trees with at least one edge satisfy $\chi_\ell^\bullet(G)$9. Complete graphs satisfy $G$0. Odd cycles satisfy $G$1. The ladder $G$2 is $G$3-flexible, hence $G$4 (Kaul et al., 2022).

For complete bipartite graphs, the exact $G$5-flexibility function is known for the cases $G$6. Since $G$7, the universal ceiling is $G$8. One has $G$9, and indeed $L$0 for all $L$1. For $L$2, one has $L$3, while $L$4 for all $L$5; moreover $L$6 for every $L$7. The small cases $L$8 and $L$9 are especially instructive: they are $L$0-choosable, but with $L$1 they admit no positive uniform $L$2, so the optimal Hall-ratio fraction is first attained at $L$3 (Bennett et al., 2024).

Balanced complete bipartite graphs also exhibit strict separation between list flexibility number and coloring number: for $L$4, $L$5, whereas $L$6 (Bennett et al., 2024). This shows that the coloring number upper bound can be far from tight even inside highly structured families.

4. Packing, weighted flexibility, and fractional list packing

Flexible list coloring is tightly connected to packing-type parameters. If $L$7 denotes the list packing number, then every graph $L$8 is $L$9-flexible. More generally, if there are $(k,1/\rho(G))$00 proper $(k,1/\rho(G))$01-colorings such that each vertex-color pair appears exactly $(k,1/\rho(G))$02 times, then any request is $(k,1/\rho(G))$03-satisfiable (Kaul et al., 2022).

The weighted formulation sharpens this connection. A fractional packing of a $(k,1/\rho(G))$04-fold list-cover yields a probability distribution on proper colorings in which every color at every vertex is used with probability exactly $(k,1/\rho(G))$05, and therefore implies weighted $(k,1/\rho(G))$06-flexibility with $(k,1/\rho(G))$07. In this framework, the smallest $(k,1/\rho(G))$08 for which such a perfectly balanced distribution always exists is the fractional list packing number $(k,1/\rho(G))$09; the paper on layered graphs explicitly identifies this quantity with the relevant “list flexibility number” in the weighted $(k,1/\rho(G))$10 sense (Cambie et al., 2023, Cambie et al., 2024).

This alternative normalization places the parameter in the chain

$(k,1/\rho(G))$11

and yields strong structural bounds. If $(k,1/\rho(G))$12, then $(k,1/\rho(G))$13, hence $(k,1/\rho(G))$14 is weighted $(k,1/\rho(G))$15-flexibly $(k,1/\rho(G))$16-choosable. If $(k,1/\rho(G))$17, then $(k,1/\rho(G))$18, so $(k,1/\rho(G))$19 is weighted $(k,1/\rho(G))$20-flexibly $(k,1/\rho(G))$21-choosable. For Cartesian products,

$(k,1/\rho(G))$22

and similarly for the correspondence version $(k,1/\rho(G))$23 (Cambie et al., 2024).

5. Structural results for graph classes

Degeneracy gives uniform but generally nonoptimal guarantees. Every $(k,1/\rho(G))$24-degenerate graph is $(k,1/\rho(G))$25-flexible, and the same bound holds in the weighted setting, even for correspondence coloring. For bipartite $(k,1/\rho(G))$26-degenerate graphs, every single-vertex request is 1-satisfiable from every $(k,1/\rho(G))$27-assignment, extending an earlier prime-case theorem to all $(k,1/\rho(G))$28 in the bipartite setting (Kaul et al., 2022, Cambie et al., 2024).

Complete multipartite graphs admit a stronger Hall-ratio-normalized theorem. If

$(k,1/\rho(G))$29

then

$(k,1/\rho(G))$30

and $(k,1/\rho(G))$31. Thus every complete $(k,1/\rho(G))$32-partite graph achieves the optimal fraction $(k,1/\rho(G))$33 with lists of size at most its coloring number (Bennett et al., 2024).

For planar graphs and their subclasses, the literature mostly proves positive $(k,1/\rho(G))$34-flexibility at specific list sizes rather than exact Hall-ratio-normalized values. Every planar graph without $(k,1/\rho(G))$35-cycles is weighted $(k,1/\rho(G))$36-flexible for $(k,1/\rho(G))$37-lists for some absolute $(k,1/\rho(G))$38 (Masařík, 2019). Planar graphs without $(k,1/\rho(G))$39 are weighted $(k,1/\rho(G))$40-flexible for $(k,1/\rho(G))$41-lists, and planar graphs without $(k,1/\rho(G))$42-cycles and with $(k,1/\rho(G))$43-cycle distance at least $(k,1/\rho(G))$44, as well as planar graphs without $(k,1/\rho(G))$45, are weighted $(k,1/\rho(G))$46-flexible for $(k,1/\rho(G))$47-lists (Choi et al., 2020). Weak flexibility at list size $(k,1/\rho(G))$48 is also known for planar graphs forbidding $(k,1/\rho(G))$49, and full weighted flexibility is proved for the class forbidding $(k,1/\rho(G))$50 (Lidický et al., 2020).

Fractional-packing methods give explicit optimal weighted constants for several planar families. Triangle-free planar graphs are weighted $(k,1/\rho(G))$51-flexibly $(k,1/\rho(G))$52-choosable, planar graphs of girth $(k,1/\rho(G))$53 are weighted $(k,1/\rho(G))$54-flexibly $(k,1/\rho(G))$55-choosable, and planar graphs of girth at least $(k,1/\rho(G))$56 are weighted $(k,1/\rho(G))$57-flexibly $(k,1/\rho(G))$58-choosable; the last value is optimal (Cambie et al., 2023).

6. Large joins, separations, and open problems

A recent structural theorem characterizes when large joins with complete graphs preserve flexible chromatic-choosability. For a graph $(k,1/\rho(G))$59, the following are equivalent: there exists $(k,1/\rho(G))$60 such that

$(k,1/\rho(G))$61

and every subset $(k,1/\rho(G))$62 of size $(k,1/\rho(G))$63 contains $(k,1/\rho(G))$64 together with a proper $(k,1/\rho(G))$65-coloring of $(k,1/\rho(G))$66 that colors $(k,1/\rho(G))$67 and $(k,1/\rho(G))$68 the same. In particular, for all odd cycles,

$(k,1/\rho(G))$69

The same paper also shows that an Ohba-like statement fails in general when ordinary list chromatic number is replaced by list flexibility number (Bowdoin et al., 28 Sep 2025).

The main open problems remain structural. A central question asks whether there exists a graph $(k,1/\rho(G))$70 with

$(k,1/\rho(G))$71

This is ruled out for complete multipartite graphs, but no general answer is known (Bennett et al., 2024). Another question asks whether there exists a $(k,1/\rho(G))$72-degenerate graph with list flexibility number greater than $(k,1/\rho(G))$73 (Kaul et al., 2022). There is also no universal constant $(k,1/\rho(G))$74 such that $(k,1/\rho(G))$75 for all graphs, so any upper bound in terms of $(k,1/\rho(G))$76 would have to be genuinely nonadditive (Kaul et al., 2022). On the planar side, it remains open whether all planar graphs are weighted $(k,1/\rho(G))$77-flexible for $(k,1/\rho(G))$78-lists, and whether planar graphs without $(k,1/\rho(G))$79-cycles are weighted $(k,1/\rho(G))$80-flexible for $(k,1/\rho(G))$81-lists (Masařík, 2019). In the fractional-packing direction, pathwidth bounds are sharp, but analogous treewidth bounds remain unresolved (Cambie et al., 2024).

Taken together, these results show that list flexibility number sits at an intersection of extremal coloring, probabilistic coloring, and structural graph theory. The Hall-ratio-normalized invariant $(k,1/\rho(G))$82, the weighted balanced-distribution parameter $(k,1/\rho(G))$83, and the older positive-$(k,1/\rho(G))$84 threshold $(k,1/\rho(G))$85 encode different aspects of the same phenomenon: how much list size must increase beyond ordinary choosability before adversarial requests can be satisfied at a guaranteed rate.