Strong Odd Coloring in Graph Theory

• Strong odd coloring is a vertex-coloring method where, for every vertex, each color in its neighborhood appears an odd number of times or not at all.
• It lies between odd coloring and square coloring, with significant impact on sparse graphs, planar, outerplanar, and minor-closed classes through explicit chromatic bounds.
• Research on strong odd coloring employs techniques such as discharging, reducible configurations, and layered proofs to achieve tight bounds and extend the theoretical framework.

Strong odd coloring is a vertex-coloring notion in which properness is combined with a neighborhood parity constraint. For a simple graph $G$, a strong odd coloring is a proper coloring such that for every vertex $v$ and every color $c$, either no neighbor of $v$ has color $c$, or the number of neighbors of $v$ colored $c$ is odd. The associated extremal parameter, called the strong odd chromatic number, is the minimum number of colors in such a coloring. Introduced as a strengthened version of odd coloring and a relaxation of square coloring, the notion has developed into a distinct line of research spanning sparse graphs, minor-closed and bounded-expansion classes, planar and outerplanar graphs, and several exact computations for special graph families (Kwon et al., 2024).

1. Definition, notation, and basic inequalities

Let $G$ be a finite simple graph and let $N_G(v)$ denote the open neighborhood of a vertex $v$. A proper $v$0-coloring is a strong odd $v$1-coloring if for every non-isolated vertex $v$2, whenever a color appears in $v$3, it appears an odd number of times in $v$4. The minimum such $v$5 is denoted $v$6 in the original sparse-graph paper. The same parameter is also denoted $v$7 in later work. The defining comparison with adjacent parameters is

$v$8

where $v$9 is the odd chromatic number and $c$0 is the chromatic number of the square of $c$1; every square coloring is automatically a strong odd coloring (Kwon et al., 2024).

Later treatments place strong odd coloring within a longer chain of inequalities,

$c$2

Here $c$3 is the ordinary chromatic number and $c$4 the maximum degree. One further refinement recorded in the literature is that in claw-free graphs, the strong odd chromatic number coincides with the chromatic number of the square graph (Caro et al., 2024).

2. Position within neighborhood-parity coloring theory

Strong odd coloring arose from the earlier notion of odd coloring. In an odd coloring, one asks only that for each non-isolated vertex, at least one color appear an odd number of times in its neighborhood; Petruševski and Škrekovski introduced this parameter and proved that every simple planar graph admits an odd $c$5-coloring, while conjecturing that $c$6 colors always suffice for planar graphs (Petruševski et al., 2021). Strong odd coloring tightens this requirement from “there exists a color” to “every color used in the neighborhood occurs oddly,” so it is strictly more restrictive in formulation (Kwon et al., 2024).

This stronger parity condition interacts naturally with other local coloring notions. Conflict-free coloring is stronger than odd coloring because it requires a color appearing exactly once in each nonempty neighborhood, and a 2022 note showed

$c$7

implying bounded odd and conflict-free chromatic numbers on classes of bounded expansion and yielding $c$8 bounds for $c$9-planar graphs through bounds on the $v$0-strong coloring number (Hickingbotham, 2022). Strong odd coloring fits between odd coloring and square coloring rather than between odd and conflict-free coloring, but this background is important because it locates strong odd coloring within a broader family of neighborhood-certified colorings.

A notable separation phenomenon is known for strong odd versus odd coloring. In general graphs, the strong odd chromatic number does not admit a function of the odd chromatic number, and conversely: for each $v$1, there exists a bipartite graph $v$2 with $v$3 while the odd chromatic number is at most $v$4. This excludes any simple transfer principle from odd-coloring bounds to strong odd-coloring bounds (Goetze et al., 5 May 2025).

3. Sparse graphs and maximum-average-degree bounds

The first systematic quantitative results for strong odd coloring were obtained for sparse graphs in terms of maximum average degree. If $v$5, then

$v$6

and this bound is tight: there exists a planar subcubic graph with $v$7 and $v$8. If $v$9 and $c$0, then

$c$1

A further theorem states that if $c$2 is a $c$3-free subcubic graph with $c$4, then $c$5. For planar graphs, these translate into girth conditions: if $c$6, then $c$7, and if $c$8, then $c$9 (Kwon et al., 2024).

The sparse-graph proofs combine several standard and nonstandard ingredients. The paper uses reducible configurations in minimal counterexamples, discharging with initial charge equal to degree, color-extension lemmas, and a technical odd representative system lemma. In the final step it also invokes Brooks’ theorem and square-coloring arguments. The overall pattern is characteristic of modern sparse-graph coloring: one first constrains local structure by forbidding reducible patterns, then converts the global sparsity hypothesis into a contradiction via discharging, and finally handles the residual core by explicit coloring arguments (Kwon et al., 2024).

These results establish the first sharp threshold behavior for strong odd coloring under explicit $v$0 bounds. They also make precise the role of strong odd coloring as a relaxation of square coloring: the additive bounds $v$1 and $v$2 are far below the generic quadratic upper bound inherited from $v$3 on the sparse side of the theory (Kwon et al., 2024).

4. Planar, outerplanar, minor-closed, and bounded-expansion classes

A central early question asked whether there exists a constant $v$4 such that $v$5 for all planar graphs. One answer showed that for every planar graph,

$v$6

and for every outerplanar graph,

$v$7

The same work supplied explicit lower bounds: two planar graphs with $v$8, and an outerplanar graph $v$9 with $c$0 (Caro et al., 2024).

This was sharpened substantially in the minor-closed setting. For every proper minor-closed graph class $c$1, there exists a constant $c$2 such that $c$3 for all $c$4. In particular, if $c$5, then

$c$6

improving the earlier outerplanar upper bound $c$7. For planar graphs, the known range became

$c$8

The proof proceeds through bounded treewidth, bounded row-treewidth, and clique-sum decompositions of proper minor-closed classes (Goetze et al., 5 May 2025).

The boundedness phenomenon extends further. For every $c$9 and every graph class of bounded expansion $G$0, there exists $G$1 such that every graph in $G$2 admits a proper coloring with at most $G$3 colors satisfying the zero-or-odd condition in every ball of radius $G$4. For $G$5, this gives bounded strong odd chromatic number on every graph class of bounded expansion and answers a question raised by Goetze, Klute, Knauer, Parada, Peña, and Ueckerdt (Pilipczuk, 21 May 2025).

At the same time, attempts to pin down a small planar constant have met explicit obstructions. A later paper constructed an infinite family of planar graphs serving as counterexamples to a recent conjecture that every planar graph is strongly odd $G$6-colorable. The constructions give planar graphs with strong odd chromatic number greater than $G$7, and examples up to $G$8 are listed in that work (Manattu et al., 3 Feb 2026).

5. Exact values for special graph classes and graph products

For several basic graph families, the strong odd chromatic number is known exactly. Every tree $G$9 satisfies $N_G(v)$0, and $N_G(v)$1 if and only if $N_G(v)$2 is an odd tree, meaning every vertex has odd degree. If $N_G(v)$3 is a connected unicyclic graph other than $N_G(v)$4, then $N_G(v)$5, while $N_G(v)$6. For cycles,

$N_G(v)$7

The same paper gives linear-time algorithms for optimal strong odd colorings of trees and connected unicyclic graphs (Caro et al., 2024).

For standard graph products, multiplicative upper bounds are available. If $N_G(v)$8 denotes the Cartesian, direct, or strong product, then

$N_G(v)$9

For the lexicographic product,

$v$0

These estimates are sharp in several cases: $v$1, and for $v$2 the value depends on the parities of $v$3 and $v$4. Yet nonmultiplicative behavior also occurs: $v$5 admits a strong odd $v$6-coloring, far below $v$7 (Caro et al., 2024).

Graph class or construction	Strong odd chromatic number	Source
Trees	$v$8; equals $v$9 iff every degree is odd	(Caro et al., 2024)
Connected unicyclic graphs	$v$00, except $v$01 which needs $v$02	(Caro et al., 2024)
Outerplanar graphs	$v$03	(Goetze et al., 5 May 2025)
Planar graphs	$v$04	(Goetze et al., 5 May 2025)

These exact and near-exact results show that strong odd coloring is neither merely a sparse-graph perturbation of ordinary coloring nor simply a weak form of square coloring. Trees require at most $v$05 colors, cycles exhibit a modular pattern, outerplanar graphs are confined to a constant range, and product graphs can behave either multiplicatively or unexpectedly economically (Caro et al., 2024).

6. Proof methods, extensions, and open directions

Three methodological streams dominate the subject. The sparse-graph results use reducible configurations, discharging, coloring-extension arguments, odd representative systems, and in some cases Brooks’ theorem or square-coloring reductions (Kwon et al., 2024). The minor-closed boundedness theorem uses layering arguments for bounded treewidth graphs, product structure for bounded row-treewidth graphs, and clique-sum decompositions of proper minor-closed classes (Goetze et al., 5 May 2025). The bounded-expansion theorem translates the problem to set systems of graph balls and then bounds the strong odd chromatic number in terms of semi-ladder index, $v$06VC dimension, and hereditary subchromatic number, with Ding–Seymour–Winkler duality as one of the key ingredients (Pilipczuk, 21 May 2025).

The theory also now extends beyond open neighborhoods. For every fixed radius $v$07, bounded-expansion classes admit proper colorings in which, inside every ball of radius $v$08, every color appears either zero times or an odd number of times. The $v$09 case specializes to strong odd coloring, while the general statement places the subject within the combinatorics of set systems and sparse-graph logic (Pilipczuk, 21 May 2025).

Several problems remain open in the literature. For outerplanar graphs, the exact maximum is not known: current bounds leave $v$10 equal to either $v$11 or $v$12 (Goetze et al., 5 May 2025). For planar graphs, the known interval $v$13 leaves a large gap, and later counterexamples show that $v$14 colors do not suffice in general (Goetze et al., 5 May 2025). Questions also remain about the best constants for planar graphs, the possibility of sharper asymptotic bounds in terms of $v$15, and the extent to which strong odd coloring can be characterized structurally across hereditary graph classes.

A broader conceptual conclusion already follows from the existing results. Strong odd coloring is now known to be bounded on every proper minor-closed class and on every class of bounded expansion, but its behavior remains genuinely independent of ordinary odd coloring on general graphs. That combination of locality, parity, and sparsity is what distinguishes the parameter from older coloring notions and explains the diversity of techniques that have emerged in its study (Goetze et al., 5 May 2025).