Fibonacci Cordial Labeling

• Fibonacci cordial labeling is a graph-labeling method that assigns distinct Fibonacci numbers to vertices and uses parity to balance edge labels.
• It employs an injective mapping of Fibonacci numbers, with edge labels determined by the parity (or Legendre symbol modulo an odd prime) of the sum of vertex labels.
• Recent studies provide explicit constructions on structured graphs, detailing both existence proofs and modular obstructions in the labelings.

Fibonacci cordial labeling is a graph-labeling notion in which Fibonacci numbers determine a binary edge labeling subject to a cordial balance condition. In the parity-based form, a graph $G$ admits a Fibonacci cordial labeling if there is an injective function $f:V(G)\to\{F_0,F_1,\dots,F_n\}$ such that the induced edge labeling $f^*:E(G)\to\{0,1\}$, given by $f^*(uv)=(f(u)+f(v))\bmod 2$, satisfies $|e_f(0)-e_f(1)|\le 1$; a graph with such a labeling is called a Fibonacci cordial graph (Mitra et al., 1 Sep 2025). A later refinement replaces parity by a Legendre-symbol criterion modulo an odd prime $p$ and replaces the classical Fibonacci sequence by the $(a,b)$-Fibonacci sequence, producing the notion of $(a,b)$-Fibonacci-Legendre cordial labeling modulo $p$ (Andoyo, 15 Jan 2026).

1. Classical definition and parity mechanism

The parity-based theory begins with the Fibonacci sequence

$F_n=F_{n-1}+F_{n-2};\ \ F_0=0,\ F_1=F_2=1.$

A Fibonacci cordial labeling of a graph $f:V(G)\to\{F_0,F_1,\dots,F_n\}$0 is an injective map

$f:V(G)\to\{F_0,F_1,\dots,F_n\}$1

with induced edge labels

$f:V(G)\to\{F_0,F_1,\dots,F_n\}$2

such that

$f:V(G)\to\{F_0,F_1,\dots,F_n\}$3

The notation

$f:V(G)\to\{F_0,F_1,\dots,F_n\}$4

is also used for the same balance quantity (Mitra et al., 1 Sep 2025).

This notion is situated as a sequence-based variant of cordial labeling. The paper recalls Cahit’s notion of cordial labeling as a binary vertex labeling $f:V(G)\to\{F_0,F_1,\dots,F_n\}$5 with induced edge labels $f:V(G)\to\{F_0,F_1,\dots,F_n\}$6, both vertex labels and edge labels being balanced up to $f:V(G)\to\{F_0,F_1,\dots,F_n\}$7. Fibonacci cordial labeling differs in two ways: vertices are labeled by distinct Fibonacci numbers, and edge labels depend only on the parity of the sum of endpoint labels (Mitra et al., 1 Sep 2025).

A key structural fact used throughout the parity-based constructions is that $f:V(G)\to\{F_0,F_1,\dots,F_n\}$8 is even exactly when $f:V(G)\to\{F_0,F_1,\dots,F_n\}$9, and otherwise $f^*:E(G)\to\{0,1\}$0 is odd. Consequently, an edge receives label $f^*:E(G)\to\{0,1\}$1 when its endpoint labels have the same parity and label $f^*:E(G)\to\{0,1\}$2 when they have opposite parity. In practical terms, many existence proofs reduce to controlled placement of Fibonacci indices divisible by $f^*:E(G)\to\{0,1\}$3 so that same-parity and opposite-parity adjacencies are balanced (Mitra et al., 1 Sep 2025).

2. Constructive results for structured graph families

The 2025 study develops explicit Fibonacci cordial labelings for several structured families and presents them as constructive existence theorems (Mitra et al., 1 Sep 2025).

Graph family	Result
Generalized Petersen graph $f^*:E(G)\to\{0,1\}$4	Fibonacci cordial for all $f^*:E(G)\to\{0,1\}$5
Helm graph $f^*:E(G)\to\{0,1\}$6	Fibonacci cordial for all $f^*:E(G)\to\{0,1\}$7
Closed helm graphs	Fibonacci cordial for all $f^*:E(G)\to\{0,1\}$8
$f^*:E(G)\to\{0,1\}$9	Fibonacci cordial
$f^*(uv)=(f(u)+f(v))\bmod 2$0	Fibonacci cordial

For generalized Petersen graphs $f^*(uv)=(f(u)+f(v))\bmod 2$1, the construction is split into the cases $f^*(uv)=(f(u)+f(v))\bmod 2$2, $f^*(uv)=(f(u)+f(v))\bmod 2$3, and $f^*(uv)=(f(u)+f(v))\bmod 2$4, with further dependence on $f^*(uv)=(f(u)+f(v))\bmod 2$5. In the detailed $f^*(uv)=(f(u)+f(v))\bmod 2$6 case, the proof states that the construction assigns even Fibonacci numbers to exactly $f^*(uv)=(f(u)+f(v))\bmod 2$7 vertices, with edge counts

$f^*(uv)=(f(u)+f(v))\bmod 2$8

hence

$f^*(uv)=(f(u)+f(v))\bmod 2$9

The same parity-counting strategy is then said to apply to the other two residue classes. Figure 1 provides a concrete labeling of $|e_f(0)-e_f(1)|\le 1$0, where the outer cycle receives $|e_f(0)-e_f(1)|\le 1$1 and the inner cycle receives $|e_f(0)-e_f(1)|\le 1$2 (Mitra et al., 1 Sep 2025).

For helm graphs $|e_f(0)-e_f(1)|\le 1$3, the vertex set is

$|e_f(0)-e_f(1)|\le 1$4

where $|e_f(0)-e_f(1)|\le 1$5 is the apex, $|e_f(0)-e_f(1)|\le 1$6 are cycle vertices, and $|e_f(0)-e_f(1)|\le 1$7 are pendant vertices adjacent to $|e_f(0)-e_f(1)|\le 1$8. The constructions depend on $|e_f(0)-e_f(1)|\le 1$9 and auxiliary parameters $p$0. The paper singles out $p$1, $p$2, and $p$3 as special cases; for $p$4, Figure 2 gives the explicit labeling with apex $p$5, cycle vertices $p$6, and pendant vertices $p$7 (Mitra et al., 1 Sep 2025).

For closed helm graphs, the paper states that $p$8 and $p$9, so any Fibonacci cordial labeling must satisfy $(a,b)$0. In the $(a,b)$1 case, it explicitly computes

$(a,b)$2

and, using $(a,b)$3, obtains

$(a,b)$4

The remaining congruence classes are treated similarly. Figure 3 gives a full Fibonacci cordial labeling of the closed helm on $(a,b)$5 outer cycle vertices, with apex $(a,b)$6 (Mitra et al., 1 Sep 2025).

The joint sum results concern $(a,b)$7 and $(a,b)$8, where $(a,b)$9 is obtained by connecting a vertex of $(a,b)$0 with a vertex of $(a,b)$1. The theorem for $(a,b)$2 is organized into six cases by $(a,b)$3, with $(a,b)$4 at the fan apex and case-dependent formulas for the remaining vertices. Figure 4 gives a concrete labeling of $(a,b)$5 (Mitra et al., 1 Sep 2025).

3. Circulant graphs, exact classifications, and congruence obstructions

A substantial part of the parity-based theory concerns circulant graphs

$(a,b)$6

with vertex set $(a,b)$7 and edge set

$(a,b)$8

where $(a,b)$9 and $p$0. The paper studies small symmetric connection sets such as $p$1, $p$2, and $p$3 (Mitra et al., 1 Sep 2025).

A central structural lemma states that for any two injective Fibonacci labelings $p$4 and $p$5 on $p$6,

$p$7

The proof idea is that changing the parity of one vertex flips the parity relation on all incident edges, altering the imbalance by a multiple of $p$8. This yields modular obstructions to cordiality (Mitra et al., 1 Sep 2025).

The paper proves the following nonexistence results:

Family	Nonexistence condition
$p$9	not Fibonacci cordial whenever $F_n=F_{n-1}+F_{n-2};\ \ F_0=0,\ F_1=F_2=1.$0 and $F_n=F_{n-1}+F_{n-2};\ \ F_0=0,\ F_1=F_2=1.$1
$F_n=F_{n-1}+F_{n-2};\ \ F_0=0,\ F_1=F_2=1.$2	not Fibonacci cordial when $F_n=F_{n-1}+F_{n-2};\ \ F_0=0,\ F_1=F_2=1.$3 and $F_n=F_{n-1}+F_{n-2};\ \ F_0=0,\ F_1=F_2=1.$4 is odd
$F_n=F_{n-1}+F_{n-2};\ \ F_0=0,\ F_1=F_2=1.$5	not Fibonacci cordial when $F_n=F_{n-1}+F_{n-2};\ \ F_0=0,\ F_1=F_2=1.$6 and $F_n=F_{n-1}+F_{n-2};\ \ F_0=0,\ F_1=F_2=1.$7

Beyond obstruction results, the paper gives exact or near-exact classifications. It proves that

$F_n=F_{n-1}+F_{n-2};\ \ F_0=0,\ F_1=F_2=1.$8

so this family is Fibonacci cordial exactly for even $F_n=F_{n-1}+F_{n-2};\ \ F_0=0,\ F_1=F_2=1.$9. Figure 5 provides the concrete cyclic labeling

$f:V(G)\to\{F_0,F_1,\dots,F_n\}$00

for $f:V(G)\to\{F_0,F_1,\dots,F_n\}$01 (Mitra et al., 1 Sep 2025).

For $f:V(G)\to\{F_0,F_1,\dots,F_n\}$02, the result is

$f:V(G)\to\{F_0,F_1,\dots,F_n\}$03

The paper states that $f:V(G)\to\{F_0,F_1,\dots,F_n\}$04, $f:V(G)\to\{F_0,F_1,\dots,F_n\}$05 is also complete, computational verification confirms Fibonacci cordiality for $f:V(G)\to\{F_0,F_1,\dots,F_n\}$06 whenever appropriate, and asymptotic-style constructive patterns are given for $f:V(G)\to\{F_0,F_1,\dots,F_n\}$07 (Mitra et al., 1 Sep 2025).

For $f:V(G)\to\{F_0,F_1,\dots,F_n\}$08, the theorem states: $f:V(G)\to\{F_0,F_1,\dots,F_n\}$09 The construction is explicit for $f:V(G)\to\{F_0,F_1,\dots,F_n\}$10, using $f:V(G)\to\{F_0,F_1,\dots,F_n\}$11 for $f:V(G)\to\{F_0,F_1,\dots,F_n\}$12, and a unified labeling strategy is said to work for all $f:V(G)\to\{F_0,F_1,\dots,F_n\}$13 (Mitra et al., 1 Sep 2025).

4. $f:V(G)\to\{F_0,F_1,\dots,F_n\}$14-Fibonacci-Legendre cordial labeling

The 2026 paper introduces a number-theoretic refinement of Fibonacci cordial labeling. It works with the generalized Fibonacci sequence

$f:V(G)\to\{F_0,F_1,\dots,F_n\}$15

with $f:V(G)\to\{F_0,F_1,\dots,F_n\}$16 yielding the classical Fibonacci sequence and $f:V(G)\to\{F_0,F_1,\dots,F_n\}$17 yielding the classical Lucas sequence. A central ingredient is the $f:V(G)\to\{F_0,F_1,\dots,F_n\}$18-Pisano period $f:V(G)\to\{F_0,F_1,\dots,F_n\}$19, the least positive integer $f:V(G)\to\{F_0,F_1,\dots,F_n\}$20 such that

$f:V(G)\to\{F_0,F_1,\dots,F_n\}$21

(Andoyo, 15 Jan 2026).

For a simple connected graph $f:V(G)\to\{F_0,F_1,\dots,F_n\}$22 of order $f:V(G)\to\{F_0,F_1,\dots,F_n\}$23, the labeling framework again starts from a bijection

$f:V(G)\to\{F_0,F_1,\dots,F_n\}$24

Fix an odd prime $f:V(G)\to\{F_0,F_1,\dots,F_n\}$25. The induced edge labeling $f:V(G)\to\{F_0,F_1,\dots,F_n\}$26 is defined by

$f:V(G)\to\{F_0,F_1,\dots,F_n\}$27

and

$f:V(G)\to\{F_0,F_1,\dots,F_n\}$28

Here $f:V(G)\to\{F_0,F_1,\dots,F_n\}$29 is the Legendre symbol: $f:V(G)\to\{F_0,F_1,\dots,F_n\}$30 Thus edge label $f:V(G)\to\{F_0,F_1,\dots,F_n\}$31 means that $f:V(G)\to\{F_0,F_1,\dots,F_n\}$32 is a nonzero quadratic residue mod $f:V(G)\to\{F_0,F_1,\dots,F_n\}$33, while edge label $f:V(G)\to\{F_0,F_1,\dots,F_n\}$34 means that the sum is either a quadratic nonresidue or zero modulo $f:V(G)\to\{F_0,F_1,\dots,F_n\}$35 (Andoyo, 15 Jan 2026).

The cordiality condition remains

$f:V(G)\to\{F_0,F_1,\dots,F_n\}$36

A graph admitting such a labeling is called an $f:V(G)\to\{F_0,F_1,\dots,F_n\}$37-FLC graph modulo $f:V(G)\to\{F_0,F_1,\dots,F_n\}$38. Relative to the broader literature, the paper identifies three novelties: the Legendre-symbol criterion modulo an odd prime $f:V(G)\to\{F_0,F_1,\dots,F_n\}$39 rather than parity, general initial values $f:V(G)\to\{F_0,F_1,\dots,F_n\}$40 rather than only the classical Fibonacci or Lucas sequences, and a connection to the newly defined $f:V(G)\to\{F_0,F_1,\dots,F_n\}$41-Pisano-Legendre primes (Andoyo, 15 Jan 2026).

The arithmetic control parameters are defined via the Pisano index set

$f:V(G)\to\{F_0,F_1,\dots,F_n\}$42

and

$f:V(G)\to\{F_0,F_1,\dots,F_n\}$43

Then $f:V(G)\to\{F_0,F_1,\dots,F_n\}$44 is a $f:V(G)\to\{F_0,F_1,\dots,F_n\}$45-Pisano-Legendre prime relative to $f:V(G)\to\{F_0,F_1,\dots,F_n\}$46 if

$f:V(G)\to\{F_0,F_1,\dots,F_n\}$47

The set of attainable values is

$f:V(G)\to\{F_0,F_1,\dots,F_n\}$48

The case $f:V(G)\to\{F_0,F_1,\dots,F_n\}$49,

$f:V(G)\to\{F_0,F_1,\dots,F_n\}$50

is especially important because it drives the exact balancing in many graph constructions (Andoyo, 15 Jan 2026).

5. Graph constructions controlled by $f:V(G)\to\{F_0,F_1,\dots,F_n\}$51-Pisano-Legendre primes

The $f:V(G)\to\{F_0,F_1,\dots,F_n\}$52-Fibonacci-Legendre theory proves cordiality for paths, stars, wheels, and several structured graph products, but always under explicit number-theoretic conditions rather than as unrestricted closure statements (Andoyo, 15 Jan 2026).

Graph family	Result and hypotheses
$f:V(G)\to\{F_0,F_1,\dots,F_n\}$53	$f:V(G)\to\{F_0,F_1,\dots,F_n\}$54-FLC modulo $f:V(G)\to\{F_0,F_1,\dots,F_n\}$55 if $f:V(G)\to\{F_0,F_1,\dots,F_n\}$56 and $f:V(G)\to\{F_0,F_1,\dots,F_n\}$57 is $f:V(G)\to\{F_0,F_1,\dots,F_n\}$58-PL relative to $f:V(G)\to\{F_0,F_1,\dots,F_n\}$59
$f:V(G)\to\{F_0,F_1,\dots,F_n\}$60	$f:V(G)\to\{F_0,F_1,\dots,F_n\}$61-FLC modulo $f:V(G)\to\{F_0,F_1,\dots,F_n\}$62 if $f:V(G)\to\{F_0,F_1,\dots,F_n\}$63 and $f:V(G)\to\{F_0,F_1,\dots,F_n\}$64 is $f:V(G)\to\{F_0,F_1,\dots,F_n\}$65-PL relative to $f:V(G)\to\{F_0,F_1,\dots,F_n\}$66
$f:V(G)\to\{F_0,F_1,\dots,F_n\}$67	$f:V(G)\to\{F_0,F_1,\dots,F_n\}$68-FLC modulo $f:V(G)\to\{F_0,F_1,\dots,F_n\}$69 under the same $f:V(G)\to\{F_0,F_1,\dots,F_n\}$70-PL hypothesis
$f:V(G)\to\{F_0,F_1,\dots,F_n\}$71, $f:V(G)\to\{F_0,F_1,\dots,F_n\}$72, $f:V(G)\to\{F_0,F_1,\dots,F_n\}$73	$f:V(G)\to\{F_0,F_1,\dots,F_n\}$74-FLC modulo $f:V(G)\to\{F_0,F_1,\dots,F_n\}$75 if $f:V(G)\to\{F_0,F_1,\dots,F_n\}$76, $f:V(G)\to\{F_0,F_1,\dots,F_n\}$77 is $f:V(G)\to\{F_0,F_1,\dots,F_n\}$78-PL, and $f:V(G)\to\{F_0,F_1,\dots,F_n\}$79
$f:V(G)\to\{F_0,F_1,\dots,F_n\}$80	$f:V(G)\to\{F_0,F_1,\dots,F_n\}$81-FLC modulo $f:V(G)\to\{F_0,F_1,\dots,F_n\}$82 if $f:V(G)\to\{F_0,F_1,\dots,F_n\}$83 and $f:V(G)\to\{F_0,F_1,\dots,F_n\}$84 is $f:V(G)\to\{F_0,F_1,\dots,F_n\}$85-PL; no condition $f:V(G)\to\{F_0,F_1,\dots,F_n\}$86 is needed
$f:V(G)\to\{F_0,F_1,\dots,F_n\}$87, special $f:V(G)\to\{F_0,F_1,\dots,F_n\}$88, special $f:V(G)\to\{F_0,F_1,\dots,F_n\}$89	$f:V(G)\to\{F_0,F_1,\dots,F_n\}$90-FLC modulo $f:V(G)\to\{F_0,F_1,\dots,F_n\}$91 under $f:V(G)\to\{F_0,F_1,\dots,F_n\}$92-PL hypotheses and explicit size conditions involving $f:V(G)\to\{F_0,F_1,\dots,F_n\}$93 and $f:V(G)\to\{F_0,F_1,\dots,F_n\}$94

The proof strategy is highly uniform. Vertex labels are arranged in blocks of length $f:V(G)\to\{F_0,F_1,\dots,F_n\}$95, exploiting

$f:V(G)\to\{F_0,F_1,\dots,F_n\}$96

Edge sums are then reduced to a small number of forms: $f:V(G)\to\{F_0,F_1,\dots,F_n\}$97 This turns the graph-labeling problem into counting residue types over one Pisano period. When edge sums reduce to $f:V(G)\to\{F_0,F_1,\dots,F_n\}$98, the paper uses

$f:V(G)\to\{F_0,F_1,\dots,F_n\}$99

so the congruence condition $f^*:E(G)\to\{0,1\}$00 is required precisely when a $f^*:E(G)\to\{0,1\}$01-term appears (Andoyo, 15 Jan 2026).

The path theorem illustrates the mechanism most transparently. For $f^*:E(G)\to\{0,1\}$02, one labels $f^*:E(G)\to\{0,1\}$03 by $f^*:E(G)\to\{0,1\}$04, so each path edge has sum $f^*:E(G)\to\{0,1\}$05. Over each block of length $f^*:E(G)\to\{0,1\}$06, the edge sums run through

$f^*:E(G)\to\{0,1\}$07

Hence

$f^*:E(G)\to\{0,1\}$08

If $f^*:E(G)\to\{0,1\}$09 is $f^*:E(G)\to\{0,1\}$10-PL relative to $f^*:E(G)\to\{0,1\}$11, then the two counts are equal (Andoyo, 15 Jan 2026).

The special corona and join theorems show how general $f^*:E(G)\to\{0,1\}$12-PL primes are used. In Theorem 3.5, for instance, if $f^*:E(G)\to\{0,1\}$13 has order $f^*:E(G)\to\{0,1\}$14 and size

$f^*:E(G)\to\{0,1\}$15

and if

$f^*:E(G)\to\{0,1\}$16

then

$f^*:E(G)\to\{0,1\}$17

is $f^*:E(G)\to\{0,1\}$18-FLC modulo $f^*:E(G)\to\{0,1\}$19, where $f^*:E(G)\to\{0,1\}$20 is a $f^*:E(G)\to\{0,1\}$21-PL prime relative to $f^*:E(G)\to\{0,1\}$22. The size parameter is tuned so that the final imbalance becomes exactly $f^*:E(G)\to\{0,1\}$23 (Andoyo, 15 Jan 2026).

6. Examples, arithmetic distribution, and limitations

The 2026 paper includes a concrete labeling example on $f^*:E(G)\to\{0,1\}$24 with $f^*:E(G)\to\{0,1\}$25 and $f^*:E(G)\to\{0,1\}$26. With

$f^*:E(G)\to\{0,1\}$27

the relevant residues are

$f^*:E(G)\to\{0,1\}$28

The edge sums satisfy

$f^*:E(G)\to\{0,1\}$29

so

$f^*:E(G)\to\{0,1\}$30

Thus

$f^*:E(G)\to\{0,1\}$31

and $f^*:E(G)\to\{0,1\}$32 is $f^*:E(G)\to\{0,1\}$33-FLC modulo $f^*:E(G)\to\{0,1\}$34. The same paper also computes that $f^*:E(G)\to\{0,1\}$35 is a $f^*:E(G)\to\{0,1\}$36-PL prime relative to $f^*:E(G)\to\{0,1\}$37, because $f^*:E(G)\to\{0,1\}$38 and

$f^*:E(G)\to\{0,1\}$39

hence

$f^*:E(G)\to\{0,1\}$40

(Andoyo, 15 Jan 2026).

The arithmetic side is developed further through the minimal-prime function

$f^*:E(G)\to\{0,1\}$41

For the classical Fibonacci case $f^*:E(G)\to\{0,1\}$42, the paper reports sample values

$f^*:E(G)\to\{0,1\}$43

For $f^*:E(G)\to\{0,1\}$44 and $f^*:E(G)\to\{0,1\}$45, it reports the empirical inequality

$f^*:E(G)\to\{0,1\}$46

Analogous empirical lower curves are noted for $f^*:E(G)\to\{0,1\}$47, $f^*:E(G)\to\{0,1\}$48, and $f^*:E(G)\to\{0,1\}$49, leading to the conjecture that for any $f^*:E(G)\to\{0,1\}$50, there exists $f^*:E(G)\to\{0,1\}$51 such that

$f^*:E(G)\to\{0,1\}$52

(Andoyo, 15 Jan 2026).

Two further conjectural directions are explicit. First,

$f^*:E(G)\to\{0,1\}$53

Second, if $f^*:E(G)\to\{0,1\}$54, then

$f^*:E(G)\to\{0,1\}$55

where

$f^*:E(G)\to\{0,1\}$56

For $f^*:E(G)\to\{0,1\}$57, the first ten elements of $f^*:E(G)\to\{0,1\}$58 listed in the paper are

$f^*:E(G)\to\{0,1\}$59

(Andoyo, 15 Jan 2026).

The parity-based paper also ends with an open direction: for large $f^*:E(G)\to\{0,1\}$60, almost every circulant graph $f^*:E(G)\to\{0,1\}$61 with a “small” connection set $f^*:E(G)\to\{0,1\}$62 admits a Fibonacci cordial labeling (Mitra et al., 1 Sep 2025). This suggests a contrast between the two strands of the subject. The parity-based theory emphasizes combinatorial parity placement on structured families, whereas the $f^*:E(G)\to\{0,1\}$63-Fibonacci-Legendre theory is a number-theoretic refinement governed by Pisano periods, quadratic residues, and $f^*:E(G)\to\{0,1\}$64-Pisano-Legendre primes.

Several cautions are explicit in the source material. The 2026 paper proves results only for certain graph families and structured products, not all graphs; several theorems require $f^*:E(G)\to\{0,1\}$65, $f^*:E(G)\to\{0,1\}$66, $f^*:E(G)\to\{0,1\}$67-PL or $f^*:E(G)\to\{0,1\}$68-PL conditions, or auxiliary assumptions such as

$f^*:E(G)\to\{0,1\}$69

It also presents empirical conjectures rather than density theorems or infinitude theorems (Andoyo, 15 Jan 2026). The 2025 paper includes proven nonexistence statements for circulants, but some constructions appear typographically incomplete or inconsistent: the theorem for $f^*:E(G)\to\{0,1\}$70 has an incomplete proof as printed, the formula for $f^*:E(G)\to\{0,1\}$71 includes a likely typo $f^*:E(G)\to\{0,1\}$72, and some proofs say “the reader can verify” rather than supplying full edge counts (Mitra et al., 1 Sep 2025).

Within those limits, Fibonacci cordial labeling currently comprises two closely related but technically distinct frameworks. One is parity-based and injective, using the residue class of Fibonacci indices modulo $f^*:E(G)\to\{0,1\}$73 to force balance on structured graphs. The other is modular and arithmetic, replacing parity by the Legendre symbol and tying cordiality to the residue distribution of generalized Fibonacci sequences over a Pisano period.