Odd Coloring in Graph Theory

• Odd coloring is defined as a proper vertex coloring where every non-isolated vertex has at least one color appearing an odd number of times in its neighborhood.
• The gap between odd chromatic number and classical chromatic number can be arbitrarily large, illustrated by examples such as subdivided complete graphs and cycles.
• Research explores structural bounds, sparsity conditions, and NP-completeness, using techniques from product structure, degeneracy, and parameterized complexity.

Searching arXiv for recent and foundational papers on odd coloring to ground the article. Odd coloring is a refinement of proper vertex coloring in which local parity in open neighborhoods is constrained. A proper coloring $\varphi$ of a graph $G$ is odd if, for every non-isolated vertex $v$, there exists a color $c$ such that $\bigl|\varphi^{-1}(c)\cap N(v)\bigr|$ is odd; the minimum number of colors in such a coloring is the odd chromatic number $\chi_o(G)$. The notion was introduced by Petruševski and Škrekovski and has since developed into a distinct branch of graph coloring theory, with structural, extremal, topological, and algorithmic aspects that diverge sharply from those of ordinary chromatic number (Petruševski et al., 2021).

1. Definition and formal framework

Throughout this literature, graphs are simple, finite, and undirected. For a vertex $v$, the open neighborhood is

$N(v)=\{u\in V(G):uv\in E(G)\}.$

A proper $k$-coloring is a map $\varphi:V(G)\to\{1,\dots,k\}$ such that adjacent vertices receive distinct colors. An odd $G$0-coloring is a proper $G$1-coloring with the additional requirement that every non-isolated vertex $G$2 has some color appearing an odd number of times in $G$3; equivalently,

$G$4

The odd chromatic number is

$G$5

A useful formalization is

$G$6

so that $G$7 is odd exactly when $G$8 for every non-isolated vertex (Kitano, 2024).

The relation to ordinary coloring is immediate: every odd coloring is proper, hence $G$9. The difference is not merely quantitative. Odd coloring is governed by multiplicities of colors in neighborhoods rather than only adjacency constraints, and this makes parity at even-degree vertices especially significant. If a vertex has odd degree, then in any proper coloring some neighbor color must appear an odd number of times, since the total size of the neighborhood is odd; much of the combinatorial difficulty is therefore concentrated at even-degree vertices (Cho et al., 2022).

2. Fundamental behavior and separation from ordinary coloring

Several basic examples show that odd coloring behaves differently from classical chromatic theory. The gap $v$0 can be arbitrarily large: if $v$1 is obtained from $v$2 by subdividing each edge once, then $v$3 is bipartite and hence $v$4, but $v$5 (Petruševski et al., 2021). In the notation $v$6, one has

$v$7

and the same example also witnesses a large gap between odd coloring and proper conflict-free coloring only in one direction, since every proper conflict-free coloring is odd but not conversely (Ahn et al., 2022).

Odd chromatic number is also not monotone under taking subgraphs. A standard example is

$v$8

even though $v$9 is a subgraph of $c$0 (Petruševski et al., 2021). This suggests that odd coloring is controlled less by containment than by delicate local parity interactions.

Explicit values are known for several classical families. For cycles,

$c$1

In particular, $c$2, which underlies the conjecture that 5 is the optimal universal bound for planar graphs (Caro et al., 2022). For trees $c$3,

$c$4

and for hypercubes,

$c$5

These formulas show that parity of vertex degrees can determine odd colorability in ways with no analogue for $c$6 (Caro et al., 2022).

3. Structural bounds for sparse, planar, and thickness-constrained graphs

A major line of work relates odd coloring to sparsity. Cranston studied graphs of bounded maximum average degree $c$7, where

$c$8

For $c$9, Cho, Choi, Kwon, and Park resolved Cranston’s conjecture in a stronger form: if

$\bigl|\varphi^{-1}(c)\cap N(v)\bigr|$0

then $\bigl|\varphi^{-1}(c)\cap N(v)\bigr|$1 is odd $\bigl|\varphi^{-1}(c)\cap N(v)\bigr|$2-colorable unless it contains the subdivided complete graph $\bigl|\varphi^{-1}(c)\cap N(v)\bigr|$3 as a subgraph (Cho et al., 2022). For $\bigl|\varphi^{-1}(c)\cap N(v)\bigr|$4, the conjecture fails because $\bigl|\varphi^{-1}(c)\cap N(v)\bigr|$5 and more generally graphs whose every block is a 5-cycle are not odd 4-colorable; nevertheless, if

$\bigl|\varphi^{-1}(c)\cap N(v)\bigr|$6

and $\bigl|\varphi^{-1}(c)\cap N(v)\bigr|$7 has no induced 5-cycle, then $\bigl|\varphi^{-1}(c)\cap N(v)\bigr|$8 (Cho et al., 2022). Wang and Yang sharpened this by proving that if $\bigl|\varphi^{-1}(c)\cap N(v)\bigr|$9, then $\chi_o(G)$0 if and only if $\chi_o(G)$1 belongs to the class of graphs whose component blocks are all 5-cycles (Wang et al., 2022).

For planar graphs, the first universal bound was $\chi_o(G)$2 (Petruševski et al., 2021), later improved to $\chi_o(G)$3 (Caro et al., 2022). The central conjecture remains that every planar graph is odd 5-colorable (Petruševski et al., 2021). Girth assumptions yield stronger results. Because planar graphs of girth at least $\chi_o(G)$4 satisfy

$\chi_o(G)$5

Cho et al. obtained, among other consequences, that planar graphs with girth at least 7 are odd 5-colorable, planar graphs with girth at least 6 are odd 6-colorable, and planar graphs with girth at least 11 are odd 4-colorable (Cho et al., 2022). Using the forb-flex method, later work improved the last threshold to girth at least 10: $\chi_o(G)$6 (Anderson et al., 2024). Wang and Yang also proved that a planar graph without $\chi_o(G)$7-cycles adjacent to $\chi_o(G)$8-cycles is odd 6-colorable (Wang et al., 2022).

Thickness provides a different structural axis. If $\chi_o(G)$9 denotes the thickness of $v$0, then Kitano proved that if every minor of $v$1 has thickness at most $v$2, then

$v$3

More sharply, if

$v$4

then

$v$5

This yields, for example, odd 6-colorability for planar graphs with girth at least 6 and minimum degree at least 1, and odd 12-colorability for biplanar graphs with girth at least 6 and minimum degree at least 3 (Kitano, 2024). At the same time, thickness alone does not bound odd chromatic number: biplanar graphs can have unbounded $v$6, witnessed by subdivided complete graphs $v$7 with $v$8 and $v$9 (Kitano, 2024).

4. Exact and near-exact results for specific graph classes

The literature now contains a broad range of exact values, sharp bounds, and class-specific characterizations.

Graph class	Result	Source
Outerplanar graphs	Odd 5-colorable; odd 4-colorable iff some block is not $N(v)=\{u\in V(G):uv\in E(G)\}.$0	(Kashima et al., 2024)
1-planar graphs	Odd 23-colorable, later improved to odd 13-colorable	(Cranston et al., 2022, Liu et al., 2022)
Toroidal graphs	Odd 9-colorable	(Metrebian, 2022)
$N(v)=\{u\in V(G):uv\in E(G)\}.$1-trees	Odd $N(v)=\{u\in V(G):uv\in E(G)\}.$2-colorable; 2-trees odd 4-colorable; 3-trees odd 5-colorable	(Kashima et al., 29 Apr 2025)
Graphs with product structure	If $N(v)=\{u\in V(G):uv\in E(G)\}.$3 with $N(v)=\{u\in V(G):uv\in E(G)\}.$4 a $N(v)=\{u\in V(G):uv\in E(G)\}.$5-tree, then odd coloring number at most $N(v)=\{u\in V(G):uv\in E(G)\}.$6	(Dujmović et al., 2022)

Outerplanar graphs exhibit one of the cleanest exact classifications. Every outerplanar graph is odd 5-colorable, and a connected outerplanar graph is odd 4-colorable if and only if it has a block that is not a copy of $N(v)=\{u\in V(G):uv\in E(G)\}.$7. Equivalently, graphs all of whose blocks are 5-cycles are the precise obstructions to odd 4-colorability in the outerplanar class (Kashima et al., 2024). This gives a structural explanation for the special role of $N(v)=\{u\in V(G):uv\in E(G)\}.$8 already visible in planar conjectures.

For 1-planar graphs, Cranston, Lafferty, and Song proved $N(v)=\{u\in V(G):uv\in E(G)\}.$9 (Cranston et al., 2022), and this was improved to $k$0 (Liu et al., 2022). The lower-bound example $k$1, obtained by subdividing each edge of $k$2 once, is 1-planar and satisfies $k$3 (Liu et al., 2022). On the torus, every graph embeddable in the torus admits an odd 9-coloring (Metrebian, 2022).

For bounded-treewidth structures, the picture is tightening. It follows from a minor-closed degeneracy bound that every $k$4-tree is odd $k$5-colorable, and this was improved to

$k$6

colors for all $k$7-trees (Kashima et al., 29 Apr 2025). The same work proves the tight small cases: every 2-tree is odd 4-colorable and every 3-tree is odd 5-colorable, and conjectures that every $k$8-tree is odd $k$9-colorable (Kashima et al., 29 Apr 2025).

Product-structure methods provide another route to bounded odd chromatic number. If $\varphi:V(G)\to\{1,\dots,k\}$0 is a subgraph of a strong product $\varphi:V(G)\to\{1,\dots,k\}$1, where $\varphi:V(G)\to\{1,\dots,k\}$2 is a $\varphi:V(G)\to\{1,\dots,k\}$3-tree and $\varphi:V(G)\to\{1,\dots,k\}$4 is a path, then $\varphi:V(G)\to\{1,\dots,k\}$5 has a proper odd coloring using at most $\varphi:V(G)\to\{1,\dots,k\}$6 colors (Dujmović et al., 2022). Since $\varphi:V(G)\to\{1,\dots,k\}$7-planar graphs admit such product structure with $\varphi:V(G)\to\{1,\dots,k\}$8, this implies bounded odd coloring number for all $\varphi:V(G)\to\{1,\dots,k\}$9-planar graphs (Dujmović et al., 2022). A plausible implication is that structural decompositions used in modern sparse graph theory can often be converted into odd-coloring bounds.

5. Algorithmic and parameterized complexity

From the decision perspective, Odd $G$00-Coloring asks whether $G$01. Complexity separates sharply between $G$02 and $G$03. For $G$04, a graph is odd 2-colorable if and only if it is bipartite and every vertex has odd degree or degree 0, giving a polynomial-time characterization (Ahn et al., 2022). For every fixed $G$05, however, Odd $G$06-Coloring is NP-complete, even on bipartite graphs; NP-completeness already holds for $G$07 on subcubic bipartite planar graphs (Ahn et al., 2022). Computing $G$08 is NP-hard (Caro et al., 2022).

Parameterized complexity refines this picture. Odd Coloring is fixed-parameter tractable on bounded-treewidth graphs via CMSO logic, and one explicit bound gives

$G$09

(Bhyravarapu et al., 7 Mar 2025). More recently, the problem was shown to admit a polynomial kernel when parameterized by distance to clique, but not to admit a polynomial kernel when parameterized by vertex cover number unless

$G$10

(Bhyravarapu et al., 7 Mar 2025). It is fixed-parameter tractable when parameterized by distance to cluster, distance to co-cluster, or neighborhood diversity, and $G$11-hard when parameterized by clique-width (Bhyravarapu et al., 7 Mar 2025).

These results coexist with a more specialized algorithmic observation: for fixed $G$12 and bounded clique-width $G$13, Odd $G$14-Coloring can be solved in $G$15 time using monadic second-order logic tools (Ahn et al., 2022). This does not contradict the $G$16-hardness by clique-width, since the latter concerns the parameterized problem with $G$17 as part of the input rather than fixed in advance. On restricted graph classes, the parameterized study also shows polynomial-time solvability on cographs and split graphs, while NP-completeness persists on certain subclasses of bipartite graphs (Bhyravarapu et al., 7 Mar 2025).

6. Related notions and current directions

Odd coloring is closely related to several neighborhood-sensitive coloring notions. A proper conflict-free coloring requires that every non-isolated vertex have a color appearing exactly once in its neighborhood; every such coloring is automatically odd, so

$G$18

but the converse fails in general (Ahn et al., 2022). Much recent sparse-graph work has therefore developed in parallel for odd and proper conflict-free colorings, often with shared extremal constructions (Wang et al., 2022).

A stronger parity-based variant is strong odd coloring. Here a proper coloring is required to satisfy that, for every non-isolated vertex $G$19, every color appearing in $G$20 appears an odd number of times there. Denoting the corresponding parameter by $G$21, one has

$G$22

so strong odd coloring is both a strengthening of odd coloring and a relaxation of square coloring (Kwon et al., 2024). For sparse graphs,

$G$23

and if $G$24 with $G$25, then

$G$26

(Kwon et al., 2024). This suggests that odd coloring sits naturally inside a larger hierarchy of parity-constrained neighborhood colorings.

Another related notion is odd-sum coloring, where a proper coloring $G$27 must satisfy

$G$28

for every vertex $G$29. Its odd-sum chromatic number $G$30 is always at most $G$31, but on planar and surface-embedded graphs its extremal behavior differs substantially from odd coloring (Cranston, 2022). The comparison is conceptually useful: both odd coloring and odd-sum coloring impose parity in neighborhoods, but the former constrains multiplicities of colors in open neighborhoods, whereas the latter constrains parity of sums on closed neighborhoods.

Current open directions are sharply defined. For planar graphs, the main conjecture remains $G$32 (Petruševski et al., 2021). For outerplanar graphs, the odd 4-colorability classification is complete (Kashima et al., 2024). For $G$33-trees, the conjecture that every $G$34-tree is odd $G$35-colorable remains open (Kashima et al., 29 Apr 2025). For thickness, it is open whether the upper bounds $G$36 and $G$37 under girth and degree hypotheses are strict or sharp (Kitano, 2024). On the algorithmic side, the complexity of odd coloring on planar graphs for $G$38 is explicitly posed as open (Ahn et al., 2022).

Taken together, these developments show that odd coloring is not merely a variant of proper coloring but a separate parameter system with its own extremal obstructions, sparsity thresholds, topological phenomena, and complexity landscape. The recurring role of subdivided complete graphs, 5-cycles, discharging, and product structure suggests that parity in neighborhoods is structurally rigid in exactly the way ordinary chromatic number is not.