Odd Graceful Chromatic Number

• Odd Graceful Chromatic Number is the minimum k for which a bipartite graph admits a proper vertex labeling yielding injective and exclusively odd edge differences.
• It leverages parity separation by assigning odd labels to one partition and even labels to the other, ensuring every edge difference is odd and unique.
• General constructions use square graphs and Brooks’ theorem to derive bounds, with exact values achieved for complete and near-complete bipartite families.

The odd graceful chromatic number of a graph $G$, denoted $\chi_{og}(G)$, is the minimum integer $k$ for which $G$ admits a vertex labeling $\lambda:V(G)\to\{0,1,\dots,k\}$ that is simultaneously a proper coloring, induces an injective edge labeling $\lambda'(uv)=|\lambda(u)-\lambda(v)|$, and assigns an odd label to every edge. If no such labeling exists, one sets $\chi_{og}(G)=\infty$. In the formulation developed for this parameter, finiteness is equivalent to bipartiteness: non-bipartite graphs admit no odd graceful coloring, while every bipartite graph does admit one (Afifurrahman et al., 25 Aug 2025).

1. Formal definition and parity structure

The parameter is rooted in the labeling tradition of graceful graph theory. A graph labeling in this sense is a map

$\lambda:V(G)\to \mathbb{Z},$

with induced edge labeling

$\lambda'(uv)=|\lambda(u)-\lambda(v)|$

for each edge $uv\in E(G)$. A labeling is graceful if the vertex labels use exactly $\chi_{og}(G)$0 and the induced edge labels are exactly $\chi_{og}(G)$1. It is odd graceful if the vertex labels use $\chi_{og}(G)$2 and the induced edge labels are exactly

$\chi_{og}(G)$3

The coloring variant studied for $\chi_{og}(G)$4 relaxes the classical graceful-labeling range but preserves the edge-difference viewpoint (Afifurrahman et al., 25 Aug 2025).

For a positive integer $\chi_{og}(G)$5, a $\chi_{og}(G)$6-graceful coloring of $\chi_{og}(G)$7 is a labeling

$\chi_{og}(G)$8

such that $\chi_{og}(G)$9 is a proper coloring and the induced edge labels are injective. A $k$0-odd graceful coloring is a $k$1-graceful coloring in which every edge label is odd. The odd graceful chromatic number is therefore

$k$2

with the convention $k$3 when no such $k$4 exists (Afifurrahman et al., 25 Aug 2025).

A decisive structural fact is that odd edge differences force a parity split across any bipartition. If $k$5 is bipartite with bipartition $k$6 and $k$7 is an odd graceful coloring, then exactly one of the following holds: either all vertices of $k$8 receive odd labels and all vertices of $k$9 receive even labels, or all vertices of $G$0 receive even labels and all vertices of $G$1 receive odd labels. This parity separation is not an auxiliary observation; it is the organizing principle behind the main constructions and bounds (Afifurrahman et al., 25 Aug 2025).

2. Relation to graceful coloring and to odd coloring

The odd graceful chromatic number sits at the intersection of two distinct traditions: graceful colorings defined by edge differences, and odd colorings defined by neighborhood parity. These should not be conflated.

In the graceful-coloring literature, a graceful $G$2-coloring is a proper vertex coloring

$G$3

that induces a proper edge coloring

$G$4

with values in $G$5; the corresponding minimum $G$6 is the graceful chromatic number $G$7 (D. et al., 2024, Laavanya et al., 2022). The odd graceful chromatic number is more restrictive: every odd graceful coloring is graceful, but not vice versa (Afifurrahman et al., 25 Aug 2025). The restriction is global and arithmetic: edge differences must be injective and odd.

By contrast, the odd chromatic number $G$8 is a vertex-coloring parameter with a neighborhood parity condition. A proper coloring $G$9 is odd if every non-isolated vertex $\lambda:V(G)\to\{0,1,\dots,k\}$0 has some color $\lambda:V(G)\to\{0,1,\dots,k\}$1 appearing an odd number of times in $\lambda:V(G)\to\{0,1,\dots,k\}$2; the minimum number of colors in such a coloring is $\lambda:V(G)\to\{0,1,\dots,k\}$3 (Caro et al., 2022). This parameter is unrelated to graceful labelings in that paper, and the phrase “odd graceful chromatic number” does not appear there (Caro et al., 2022). The distinction is substantial: $\lambda:V(G)\to\{0,1,\dots,k\}$4 is defined for arbitrary graphs and can behave non-monotonically under subgraphs, whereas $\lambda:V(G)\to\{0,1,\dots,k\}$5 is finite exactly on bipartite graphs and is used with a monotonicity lemma of the form $\lambda:V(G)\to\{0,1,\dots,k\}$6 for subgraphs $\lambda:V(G)\to\{0,1,\dots,k\}$7 (Caro et al., 2022, Afifurrahman et al., 25 Aug 2025).

A further neighboring notion is the strong odd chromatic number, where every color present in a neighborhood must appear there an odd number of times; this is again a neighborhood-parity parameter rather than a graceful-labeling parameter (Manattu et al., 3 Feb 2026). The shared adjective “odd” therefore covers several inequivalent constructions: parity of edge differences in $\lambda:V(G)\to\{0,1,\dots,k\}$8, parity of neighborhood color counts in $\lambda:V(G)\to\{0,1,\dots,k\}$9, and universal neighborhood-parity constraints in the strong odd chromatic number.

3. General upper bounds for bipartite graphs

Let $\lambda'(uv)=|\lambda(u)-\lambda(v)|$0 be bipartite with bipartition $\lambda'(uv)=|\lambda(u)-\lambda(v)|$1. The principal general bound is expressed in terms of square graphs on the two sides of the bipartition. The square $\lambda'(uv)=|\lambda(u)-\lambda(v)|$2 has the same vertex set as $\lambda'(uv)=|\lambda(u)-\lambda(v)|$3, and $\lambda'(uv)=|\lambda(u)-\lambda(v)|$4 whenever the distance between $\lambda'(uv)=|\lambda(u)-\lambda(v)|$5 and $\lambda'(uv)=|\lambda(u)-\lambda(v)|$6 in $\lambda'(uv)=|\lambda(u)-\lambda(v)|$7 is at most $\lambda'(uv)=|\lambda(u)-\lambda(v)|$8. The induced subgraphs $\lambda'(uv)=|\lambda(u)-\lambda(v)|$9 and $\chi_{og}(G)=\infty$0 are then used as auxiliary coloring objects (Afifurrahman et al., 25 Aug 2025).

The central theorem states that

$\chi_{og}(G)=\infty$1

The construction colors $\chi_{og}(G)=\infty$2 and $\chi_{og}(G)=\infty$3, then assigns odd labels to one side and even labels to the other: $\chi_{og}(G)=\infty$4 Because adjacent vertices lie in opposite parts, every edge difference is odd; because the square-graph colorings separate vertices at distance $\chi_{og}(G)=\infty$5, the induced edge differences are injective (Afifurrahman et al., 25 Aug 2025).

Two immediate consequences follow. Since $\chi_{og}(G)=\infty$6 for any graph $\chi_{og}(G)=\infty$7,

$\chi_{og}(G)=\infty$8

for every bipartite graph $\chi_{og}(G)=\infty$9. This is improved for the non-complete case: $\lambda:V(G)\to \mathbb{Z},$0 whenever $\lambda:V(G)\to \mathbb{Z},$1 is bipartite but not complete bipartite. The proof uses the fact that any non-complete bipartite graph embeds into a graph of the form $\lambda:V(G)\to \mathbb{Z},$2, together with exact values for such near-complete bipartite graphs (Afifurrahman et al., 25 Aug 2025).

The paper also gives degree-based estimates through Brooks’ theorem. If $\lambda:V(G)\to \mathbb{Z},$3 is bipartite with maximum degree $\lambda:V(G)\to \mathbb{Z},$4, then

$\lambda:V(G)\to \mathbb{Z},$5

A corollary states that if $\lambda:V(G)\to \mathbb{Z},$6 has diameter at least $\lambda:V(G)\to \mathbb{Z},$7, $\lambda:V(G)\to \mathbb{Z},$8, and $\lambda:V(G)\to \mathbb{Z},$9, then automatically $\lambda'(uv)=|\lambda(u)-\lambda(v)|$0 and $\lambda'(uv)=|\lambda(u)-\lambda(v)|$1 are neither odd cycles nor complete graphs, so

$\lambda'(uv)=|\lambda(u)-\lambda(v)|$2

In particular, a cubic bipartite graph with diameter at least $\lambda'(uv)=|\lambda(u)-\lambda(v)|$3 satisfies

$\lambda'(uv)=|\lambda(u)-\lambda(v)|$4

The same framework yields explicit bounds for Möbius ladders: for odd $\lambda'(uv)=|\lambda(u)-\lambda(v)|$5,

$\lambda'(uv)=|\lambda(u)-\lambda(v)|$6

These are obtained by identifying $\lambda'(uv)=|\lambda(u)-\lambda(v)|$7 and then computing the required chromatic numbers (Afifurrahman et al., 25 Aug 2025).

4. Exact values for complete and near-complete bipartite families

The strongest exact results currently available in the source material concern complete bipartite and near-complete bipartite graphs. They show that $\lambda'(uv)=|\lambda(u)-\lambda(v)|$8 is controlled not merely by the sizes of the two parts, but also by arithmetic features of those sizes and by the pattern of deleted edges (Afifurrahman et al., 25 Aug 2025).

Family	Condition	$\lambda'(uv)=|\lambda(u)-\lambda(v)|$9
$uv\in E(G)$0	$uv\in E(G)$1	$uv\in E(G)$2
$uv\in E(G)$3	$uv\in E(G)$4, $uv\in E(G)$5 or $uv\in E(G)$6	$uv\in E(G)$7
$uv\in E(G)$8	$uv\in E(G)$9, otherwise	$\chi_{og}(G)$00
$\chi_{og}(G)$01	listed special arithmetic cases	$\chi_{og}(G)$02
$\chi_{og}(G)$03	otherwise	$\chi_{og}(G)$04

For complete bipartite graphs, the full formula is: $\chi_{og}(G)$05 for integers $\chi_{og}(G)$06, together with the star value $\chi_{og}(G)$07 (Afifurrahman et al., 25 Aug 2025).

The extremal labelings in the $\chi_{og}(G)$08 cases are also characterized. If $\chi_{og}(G)$09, then, up to swapping the bipartition,

$\chi_{og}(G)$10

If $\chi_{og}(G)$11, then

$\chi_{og}(G)$12

again up to permuting the two sides (Afifurrahman et al., 25 Aug 2025).

For near-complete bipartite graphs obtained by deleting $\chi_{og}(G)$13 edges incident to one vertex in the $\chi_{og}(G)$14-part,

$\chi_{og}(G)$15

the value drops to

$\chi_{og}(G)$16

in three specific situations:

1. $\chi_{og}(G)$17 and $\chi_{og}(G)$18;
2. $\chi_{og}(G)$19 and $\chi_{og}(G)$20;
3. $\chi_{og}(G)$21, $\chi_{og}(G)$22, and $\chi_{og}(G)$23 is odd.

Otherwise,

$\chi_{og}(G)$24

These families are not isolated curiosities: they are used to deduce the general bound $\chi_{og}(G)$25 for non-complete bipartite graphs (Afifurrahman et al., 25 Aug 2025).

5. Methods of construction and proof

The recurring constructive device is parity separation across the bipartition. One side receives odd labels and the other even labels. This automatically forces every edge difference to be odd, so the essential difficulty is not parity itself but injectivity of the induced edge labeling (Afifurrahman et al., 25 Aug 2025).

The square-graph bound resolves injectivity by encoding distance-$\chi_{og}(G)$26 conflicts as ordinary graph-coloring constraints. Coloring $\chi_{og}(G)$27 and $\chi_{og}(G)$28 ensures that vertices in the same part that can participate in competing edge differences are kept apart numerically. The labels are then placed in two arithmetic progressions, one odd and one even, with a controlled offset between the two sides (Afifurrahman et al., 25 Aug 2025).

For complete and near-complete bipartite graphs, the proofs use additive combinatorics. If $\chi_{og}(G)$29 and $\chi_{og}(G)$30 are label sets on the two sides, sumset estimates of the form

$\chi_{og}(G)$31

are combined with the equality criterion that $\chi_{og}(G)$32 and $\chi_{og}(G)$33 are arithmetic progressions with the same difference. These arguments, together with parity classes modulo $\chi_{og}(G)$34, force rigid forms for extremal labelings and yield the exact values above (Afifurrahman et al., 25 Aug 2025).

A key obstruction to collisions comes from a graceful-coloring lemma: if $\chi_{og}(G)$35 is a graceful coloring and $\chi_{og}(G)$36 form a path of length two, then

$\chi_{og}(G)$37

This prevents two edges in a length-$\chi_{og}(G)$38 path from receiving the same absolute difference. The same kind of arithmetic exclusion is central to odd graceful constructions (Afifurrahman et al., 25 Aug 2025).

The degree-based bounds combine a combinatorial estimate

$\chi_{og}(G)$39

with Brooks’ theorem. This produces quadratic bounds in $\chi_{og}(G)$40 after substitution into the main square-graph inequality (Afifurrahman et al., 25 Aug 2025).

Methodologically, this differs from some graceful-chromatic-number results for complete graphs, where graceful colorings are characterized by 3-AP-free vertex-color sets rather than by bipartite parity separation (D. et al., 2024). The odd graceful chromatic number is therefore not just a parity-decorated version of $\chi_{og}(G)$41; its natural habitat is the bipartite setting, and its proofs rely on bipartition-specific arithmetic.

6. Scope, limitations, and current landscape

The parameter is finite if and only if the graph is bipartite, so the general theory is necessarily a theory of bipartite graphs (Afifurrahman et al., 25 Aug 2025). Within that domain, the available results divide into two types: exact formulas for highly structured families, and general upper bounds in terms of $\chi_{og}(G)$42, $\chi_{og}(G)$43, and chromatic numbers of square-induced auxiliary graphs.

Prior work summarized in the source material had already established lower bounds such as

$\chi_{og}(G)$44

for bipartite graphs, and had computed exact values for paths, cycles, some caterpillars, generalized stars, ladders, prism graphs, and related families. The newer upper-bound theory complements those computations by proving existence of odd graceful colorings for all bipartite graphs and by quantifying the label range needed in broad structural terms (Afifurrahman et al., 25 Aug 2025).

The present state of the subject remains incomplete for arbitrary bipartite graphs. The source explicitly notes that exact values of $\chi_{og}(G)$45 are not known in general, and that lower bounds beyond $\chi_{og}(G)$46 are known only in specific cases. It also identifies several open directions: the tightness of the bounds $\chi_{og}(G)$47 and $\chi_{og}(G)$48, characterization of all bipartite graphs achieving these bounds, and the possibility of sharper estimates in terms of other invariants such as girth or matching number (Afifurrahman et al., 25 Aug 2025).

A final terminological caution is essential. The phrase odd graceful chromatic number refers to the labeling parameter $\chi_{og}(G)$49 just defined. It is distinct from the odd chromatic number $\chi_{og}(G)$50, which is a neighborhood-parity coloring parameter (Caro et al., 2022), and from the strong odd chromatic number, which imposes odd multiplicity for every color present in every neighborhood (Manattu et al., 3 Feb 2026). The current literature uses all three notions, but they answer different combinatorial questions and require different techniques.