Proper Conflict-Free Coloring

• Proper conflict-free coloring is a vertex-coloring method that requires adjacent vertices to differ while ensuring each non-isolated vertex has a uniquely appearing color in its open neighborhood.
• It generalizes traditional coloring by incorporating h-conflict-free extensions and unique neighborhood conditions, with exact evaluations shown on trees, cycles, and subdivision graphs.
• The topic yields asymptotically tight bounds, robust list-coloring results, and NP-completeness proofs, underscoring its impact on modern graph theory research.

Proper conflict-free coloring is a vertex-coloring problem on graphs that combines ordinary properness with a neighborhood-uniqueness constraint: adjacent vertices must receive different colors, and every non-isolated vertex must have a color that appears exactly once in its open neighborhood. The resulting parameter, usually written $\chi_{\rm pcf}(G)$, has developed into a distinct topic between ordinary proper coloring, odd coloring, and distance-2 coloring. Its theory now includes exact evaluations on basic graph classes, asymptotically sharp bounds in terms of the maximum degree $\Delta$, sparse-graph threshold theorems, list and degree-choosability variants, and hardness results for fixed-color decision problems (Caro et al., 2022, Liu et al., 2024).

1. Definitions, parameters, and scope

For a graph $G$ and a vertex $v$, the standard neighborhood in this literature is the open neighborhood $N(v)$. A proper conflict-free coloring is a proper coloring $\varphi$ such that for every non-isolated vertex $v$, there exists a color $c$ with

$\bigl|\{u\in N(v):\varphi(u)=c\}\bigr|=1.$

Equivalently, every non-isolated vertex has a neighbor whose color is unique in its neighborhood. The minimum number of colors is the proper conflict-free chromatic number $\chi_{\rm pcf}(G)$ (Caro et al., 2022).

A central generalization is the $\Delta$0-conflict-free proper parameter. In one formulation, a proper coloring is $\Delta$1-conflict-free if every vertex $\Delta$2 has at least $\Delta$3 colors appearing exactly once in $\Delta$4, and the minimum number of colors is denoted $\Delta$5 (Chuet et al., 7 May 2025). Earlier work on graphs without isolated vertices uses the same notation with the requirement that every vertex have at least $\Delta$6 such colors, and explicitly notes that $\Delta$7 (Kamyczura et al., 2022). This family interpolates between ordinary proper conflict-free coloring and distance-2 phenomena.

The open-neighborhood condition is essential. In the proper closed-neighborhood variant, the conflict-free requirement collapses to ordinary proper coloring, because in any proper coloring the color of $\Delta$8 itself appears exactly once in $\Delta$9; accordingly, the proper closed-neighborhood parameter equals $G$0 (Fabrici et al., 2022). By contrast, the open-neighborhood conflict-free literature without properness is a separate line of work: adjacent vertices may share colors there, so its results do not transfer automatically to proper conflict-free coloring (Bhyravarapu et al., 2019).

2. Canonical examples and extremal behavior

Several basic graph classes already exhibit the main phenomena of the theory (Caro et al., 2022).

Graph class	Proper conflict-free chromatic number
$G$1	$G$2
Nontrivial tree $G$3	$G$4
$G$5	$G$6 if $G$7; $G$8 if $G$9 and $v$0; $v$1 if $v$2
$v$3, $v$4	$v$5
$v$6, $v$7	$v$8

These examples already separate proper conflict-free coloring sharply from ordinary chromatic behavior. The complete subdivision $v$9 is bipartite, hence has ordinary chromatic number $N(v)$0, yet $N(v)$1; consequently, the ratio $N(v)$2 can be arbitrarily large (Caro et al., 2022). The parameter is also not monotone under taking subgraphs: $N(v)$3, but

$N(v)$4

(Caro et al., 2022).

The 5-cycle plays a recurrent exceptional role. Since $N(v)$5, conjectures of Brooks type are typically stated only for connected graphs with $N(v)$6 (Caro et al., 2022). Sharpness of the conjectural $N(v)$7 upper bound is witnessed by subdivision constructions: if $N(v)$8, then

$N(v)$9

while $\varphi$0, so even connected bipartite graphs can require $\varphi$1 colors (Chuet et al., 7 May 2025).

The $\varphi$2-conflict-free extension adds a second extremal regime. For fixed $\varphi$3, the leading term is linear in $\varphi$4, but when $\varphi$5 is close to $\varphi$6 the problem approaches square coloring. In particular, for infinitely many $\varphi$7, there are graphs with

$\varphi$8

so the parameter can be quadratic in $\varphi$9 when $v$0 (Kamyczura et al., 2022).

3. General bounds and asymptotic theory

A foundational general result is the bound

$v$1

for connected graphs, together with the conjecture that every connected graph with maximum degree $v$2 satisfies

$v$3

(Caro et al., 2022). This conjecture has become the central benchmark for the subject.

The first major asymptotic improvement beyond the $v$4 bound came from a list-coloring framework for proper conflict-free coloring of a pair $v$5. Specializing to ordinary graphs gives, for sufficiently large $v$6,

$v$7

and for all graphs with $v$8,

$v$9

(Cranston et al., 2022).

The asymptotically optimal leading constant was then established by Liu and Reed, who proved that for sufficiently large $c$0,

$c$1

hence

$c$2

for every graph of maximum degree $c$3 (Liu et al., 2024). This asymptotically confirms the Caro–Petruševski–Škrekovski conjecture in first order, although it does not settle the exact additive form $c$4.

For the $c$5-parameter, a later theorem shows that for every fixed $c$6,

$c$7

with explicit form

$c$8

(Chuet et al., 7 May 2025). The same work proves a matching first-order lower bound: for infinitely many values of $c$9, there exists a graph with

$\bigl|\{u\in N(v):\varphi(u)=c\}\bigr|=1.$0

and accordingly conjectures that

$\bigl|\{u\in N(v):\varphi(u)=c\}\bigr|=1.$1

for sufficiently large $\bigl|\{u\in N(v):\varphi(u)=c\}\bigr|=1.$2, which would be tight (Chuet et al., 7 May 2025).

The large-$\bigl|\{u\in N(v):\varphi(u)=c\}\bigr|=1.$3 regime behaves differently. If $\bigl|\{u\in N(v):\varphi(u)=c\}\bigr|=1.$4, then $\bigl|\{u\in N(v):\varphi(u)=c\}\bigr|=1.$5-proper-conflict-free coloring becomes essentially a distance-2 coloring requirement, and for $\bigl|\{u\in N(v):\varphi(u)=c\}\bigr|=1.$6-regular graphs with $\bigl|\{u\in N(v):\varphi(u)=c\}\bigr|=1.$7 or $\bigl|\{u\in N(v):\varphi(u)=c\}\bigr|=1.$8 it essentially coincides with square coloring; this explains why $\bigl|\{u\in N(v):\varphi(u)=c\}\bigr|=1.$9 can be quadratic when $\chi_{\rm pcf}(G)$0 is close to $\chi_{\rm pcf}(G)$1 (Kamyczura et al., 2022).

4. Sparse graphs, minimum degree, and planar structure

One major line of work studies proper conflict-free coloring through sparsity parameters, especially maximum average degree. For $\chi_{\rm pcf}(G)$2, if

$\chi_{\rm pcf}(G)$3

then $\chi_{\rm pcf}(G)$4 has a proper conflict-free $\chi_{\rm pcf}(G)$5-coloring unless it contains the $\chi_{\rm pcf}(G)$6-subdivision of $\chi_{\rm pcf}(G)$7; for $\chi_{\rm pcf}(G)$8, if

$\chi_{\rm pcf}(G)$9

and $\Delta$00 has no induced $\Delta$01-cycle, then $\Delta$02 is proper conflict-free $\Delta$03-colorable (Cho et al., 2022). The $\Delta$04 statement was later sharpened to a full extremal characterization: among graphs with

$\Delta$05

the non-PCF-$\Delta$06-colorable graphs are exactly those containing an induced $\Delta$07 whose subdivision vertices remain 2-vertices in the ambient graph (Wang et al., 2022).

A complementary line shows that large minimum degree radically improves the asymptotics. If $\Delta$08 and $\Delta$09, then

$\Delta$10

In particular, for sufficiently large $\Delta$11-regular graphs and $\Delta$12,

$\Delta$13

so for $\Delta$14 one gets

$\Delta$15

under regularity (Kamyczura et al., 2022). A later refinement proves that when

$\Delta$16

one may further reduce the excess to

$\Delta$17

with explicit bound

$\Delta$18

in the regime $\Delta$19 (Chuet et al., 7 May 2025).

Planar and outerplanar graphs form a separate geometric branch of the subject. For the proper open-neighborhood version, outerplanar graphs satisfy

$\Delta$20

while planar graphs satisfy

$\Delta$21

(Fabrici et al., 2022). A later planar-girth result proves that if $\Delta$22 is planar with girth at least $\Delta$23, then

$\Delta$24

(Anderson et al., 2024). Another sparse-planar consequence is that every planar graph with girth at least $\Delta$25 is proper conflict-free $\Delta$26-colorable (Cho et al., 2022).

5. List coloring and degree-choosability

The list-theoretic analogue asks for proper conflict-free colorings from prescribed lists. A graph $\Delta$27 is proper conflict-free $\Delta$28-choosable if every list assignment $\Delta$29 with

$\Delta$30

admits a proper conflict-free $\Delta$31-coloring. This degree-based formulation has become a parallel program to the $\Delta$32 conjecture.

A general degeneracy theorem states that every $\Delta$33-degenerate graph is proper conflict-free $\Delta$34-choosable (Kashima et al., 16 Sep 2025). The same paper proves a sharp improvement for trees: $\Delta$35 and shows that $\Delta$36 fails even for stars (Kashima et al., 16 Sep 2025).

For sparse graphs measured by maximum average degree, two sharp list-type thresholds are known: $\Delta$37 and

$\Delta$38

(Kashima et al., 22 Jan 2026). As corollaries, every planar graph of girth at least $\Delta$39 is proper conflict-free $\Delta$40-choosable, and every planar graph of girth at least $\Delta$41 is proper conflict-free $\Delta$42-choosable (Kashima et al., 22 Jan 2026).

Outerplanar graphs admit a stronger exact statement: every connected outerplanar graph other than $\Delta$43 is proper conflict-free $\Delta$44-choosable, while there are infinitely many connected outerplanar graphs that are not proper conflict-free $\Delta$45-choosable (Kashima et al., 8 Sep 2025). More recently, three further confirmations of the degree-$\Delta$46 conjecture were obtained: connected $\Delta$47-minor-free graphs with maximum degree at most $\Delta$48, connected outer-1-planar graphs with maximum degree at most $\Delta$49, and planar graphs with girth at least $\Delta$50 are all proper conflict-free $\Delta$51-choosable; in addition, every outer-1-planar graph is proper conflict-free $\Delta$52-choosable, and both planar graphs of girth at least $\Delta$53 and outer-1-planar graphs are proper conflict-free $\Delta$54-choosable (Wang et al., 28 Dec 2025).

These results support a broader conjecture that every connected graph other than $\Delta$55 is proper conflict-free $\Delta$56-choosable (Wang et al., 28 Dec 2025).

6. Complexity, related variants, and open directions

The fixed-$\Delta$57 decision problem is computationally hard. For every integer $\Delta$58, deciding whether a graph admits a proper conflict-free $\Delta$59-coloring is NP-complete, even when the input graph is bipartite; moreover, $\Delta$60 is NP-complete on planar graphs (Ahn et al., 2022). The easy boundary cases are also explicit: for $\Delta$61, the problem is polynomial-time solvable, and

$\Delta$62

(Ahn et al., 2022).

Proper conflict-free coloring is closely related to odd coloring. Every proper conflict-free $\Delta$63-coloring is an odd $\Delta$64-coloring, so

$\Delta$65

and several sparse-graph arguments exploit this relaxation (Wang et al., 2022). The relation becomes especially tight in local degree-2 and degree-4 structures, where “odd” and “unique” neighborhood witnesses coincide (Anderson et al., 2024).

A persistent source of confusion is the distinction from nonproper open-neighborhood conflict-free coloring. There is now a theorem that every planar graph admits a full open-neighborhood conflict-free coloring with $\Delta$66 colors, and this bound is tight, but the proof uses a matching-contraction lift that assigns the same color to the endpoints of matching edges, which is fundamentally incompatible with properness (Hliněný et al., 24 Jun 2026). Consequently, the nonproper planar $\Delta$67-color theorem does not settle the proper variant.

Several central problems remain open. The exact Caro–Petruševski–Škrekovski conjecture

$\Delta$68

for connected graphs with $\Delta$69 is still unresolved in general (Caro et al., 2022). In planar graph theory, the proper open-neighborhood parameter is still only pinned down by

$\Delta$70

and the conjectured value is $\Delta$71 (Fabrici et al., 2022). In list coloring, the conjecture that every connected graph other than $\Delta$72 is proper conflict-free $\Delta$73-choosable remains open (Wang et al., 28 Dec 2025). For the $\Delta$74-parameter, the transition between the linear regime and the near-square-coloring regime is also unresolved: existing results show $\Delta$75 when $\Delta$76 and $\Delta$77, but examples with $\Delta$78 force quadratic behavior, leaving the threshold problem open (Kamyczura et al., 2022).

Proper conflict-free coloring therefore occupies a sharply defined position in modern graph coloring: it is now asymptotically linear in $\Delta$79, structurally well understood on several sparse classes, list-theoretically rich, and computationally hard even on highly restricted inputs, while still retaining a small set of conjectures whose resolution would substantially clarify the boundary between proper coloring and neighborhood-uniqueness constraints.