---
title: Fibonacci Cordial Labeling
url: https://www.emergentmind.com/topics/fibonacci-cordial-labeling
type: topic
---

# Fibonacci Cordial Labeling

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 [2509.01823]. 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 \(p\) [2601.10561].

## 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 \(G\) is an injective map
\[
f:V(G)\to \{F_0,F_1,\dots,F_n\},
\]
with induced edge labels
\[
f^*:E(G)\to\{0,1\},\qquad f^*(uv)=(f(u)+f(v)) \bmod 2,
\]
such that
\[
|e_f(0)-e_f(1)|\le 1.
\]
The notation
\[
\tilde{\varepsilon}:=|\varepsilon_1-\varepsilon_0|
\]
is also used for the same balance quantity [2509.01823].

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 \(f:V(G)\to\{0,1\}\) with induced edge labels \(f^*(uv)=|f(u)-f(v)|\), both vertex labels and edge labels being balanced up to \(1\). 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 [2509.01823].

A key structural fact used throughout the parity-based constructions is that \(F_i\) is even exactly when \(i\equiv 0 \pmod 3\), and otherwise \(F_i\) is odd. Consequently, an edge receives label \(0\) when its endpoint labels have the same parity and label \(1\) when they have opposite parity. In practical terms, many existence proofs reduce to controlled placement of Fibonacci indices divisible by \(3\) so that same-parity and opposite-parity adjacencies are balanced [2509.01823].

## 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 [2509.01823].

| Graph family | Result |
|---|---|
| Generalized Petersen graph \((n,1)\) | Fibonacci cordial for all \(n\) |
| Helm graph \(H_n\) | Fibonacci cordial for all \(n\ge 3\) |
| Closed helm graphs | Fibonacci cordial for all \(n\ge 3\) |
| \(C_m\boxplus P_n\) | Fibonacci cordial |
| \(F_m\boxplus P_n\) | Fibonacci cordial |

For generalized Petersen graphs \((n,1)\), the construction is split into the cases \(n=3p\), \(n=3p+1\), and \(n=3p+2\), with further dependence on \(p\pmod 4\). In the detailed \(n=3p\) case, the proof states that the construction assigns even Fibonacci numbers to exactly \(p_1+p_2\) vertices, with edge counts
\[
\varepsilon_0=3n-3p_1-p_2,\qquad \varepsilon_1=3p_1+p_2,
\]
hence
\[
\tilde{\varepsilon}=|\varepsilon_0-\varepsilon_1|=|6p_1+2p_2-3n|\le 1.
\]
The same parity-counting strategy is then said to apply to the other two residue classes. Figure 1 provides a concrete labeling of \(\mathrm{GP}(6,1)\), where the outer cycle receives \(F_5,F_{10},F_8,F_6,F_7,F_3\) and the inner cycle receives \(F_1,F_0,F_{12},F_9,F_4,F_2\) [2509.01823].

For helm graphs \(H_n\), the vertex set is
\[
V(H_n)=\{v\}\cup\{v_1,\dots,v_n\}\cup\{u_1,\dots,u_n\},
\]
where \(v\) is the apex, \(v_i\) are cycle vertices, and \(u_i\) are pendant vertices adjacent to \(v_i\). The constructions depend on \(n\pmod 3\) and auxiliary parameters \(p_1,p_2,p_3\). The paper singles out \(H_4\), \(H_8\), and \(H_{16}\) as special cases; for \(H_4\), Figure 2 gives the explicit labeling with apex \(F_7\), cycle vertices \(F_5,F_4,F_3,F_0\), and pendant vertices \(F_6,F_2,F_1,F_9\) [2509.01823].

For closed helm graphs, the paper states that \(|V|=2n+1\) and \(|E|=4n\), so any Fibonacci cordial labeling must satisfy \(\varepsilon_0=\varepsilon_1\). In the \(n=3p\) case, it explicitly computes
\[
\varepsilon_0=4n-3p_1-4p_2-2p_3,\qquad \varepsilon_1=3p_1+4p_2+2p_3,
\]
and, using \(p_1+p_2+p_3=2p\), obtains
\[
\tilde{\varepsilon}=\left|6p_1+8p_2+4p_3-4n\right|=\left|2p_1+4p_2-4p\right|=0.
\]
The remaining congruence classes are treated similarly. Figure 3 gives a full Fibonacci cordial labeling of the closed helm on \(14\) outer cycle vertices, with apex \(F_{28}\) [2509.01823].

The joint sum results concern \(C_m\boxplus P_n\) and \(F_m\boxplus P_n\), where \(G_1\boxplus G_2\) is obtained by connecting a vertex of \(G_1\) with a vertex of \(G_2\). The theorem for \(F_m\boxplus P_n\) is organized into six cases by \(m\pmod 6\), with \(f(u)=F_0\) at the fan apex and case-dependent formulas for the remaining vertices. Figure 5 gives a concrete labeling of \(F_{10}\boxplus P_{12}\) [2509.01823].

## 3. Circulant graphs, exact classifications, and congruence obstructions

A substantial part of the parity-based theory concerns circulant graphs
\[
\Gamma(n,S),
\]
with vertex set \(\mathbb Z_n\) and edge set
\[
E(\Gamma)=\{(u,v):v-u\in S\},
\]
where \(S=S^{-1}\) and \(0\notin S\). The paper studies small symmetric connection sets such as \(\pm\{1,2\}\), \(\pm\{1,2,3\}\), and \(\pm\{1,2,3,4\}\) [2509.01823].

A central structural lemma states that for any two injective Fibonacci labelings \(f\) and \(g\) on \(\Gamma(n,\pm\{1,2,\ldots,j\})\),
\[
\tilde{\varepsilon}_f-\tilde{\varepsilon}_g\equiv 0 \pmod 4.
\]
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 \(4\). This yields modular obstructions to cordiality [2509.01823].

The paper proves the following nonexistence results:

| Family | Nonexistence condition |
|---|---|
| \(\Gamma(n,\pm\{1,2,\dots,4m+1\})\) | not Fibonacci cordial whenever \(n\equiv 2\pmod 4\) and \(m\le \frac{n-2}{4}\) |
| \(\Gamma(n,\pm\{1,2,\dots,j\})\) | not Fibonacci cordial when \(j\equiv 2\pmod 4\) and \(n\) is odd |
| \(\Gamma(n,\pm\{1,2,\dots,j\})\) | not Fibonacci cordial when \(j\equiv 3\pmod 4\) and \(n\equiv 2\pmod 4\) |

Beyond obstruction results, the paper gives exact or near-exact classifications. It proves that
\[
\Gamma(n,\pm\{1,2\}) \text{ is Fibonacci cordial if and only if } n\not\equiv 1\pmod 2,
\]
so this family is Fibonacci cordial exactly for even \(n\). Figure 6 provides the concrete cyclic labeling
\[
F_0,\ F_3,\ F_1,\ F_6,\ F_2,\ F_4,\ F_5,\ F_7
\]
for \(\Gamma(8,\pm\{1,2\})\) [2509.01823].

For \(\Gamma(n,\pm\{1,2,3\})\), the result is
\[
\Gamma(n,\pm\{1,2,3\}) \text{ is Fibonacci cordial if and only if } n=6 \text{ or } n\equiv 0,1,3\pmod 4.
\]
The paper states that \(\Gamma(6,\pm\{1,2,3\})\cong K_6\), \(\Gamma(7,\pm\{1,2,3\})\) is also complete, computational verification confirms Fibonacci cordiality for \(7\le n\le 28\) whenever appropriate, and asymptotic-style constructive patterns are given for \(n\ge 29\) [2509.01823].

For \(\Gamma(n,\pm\{1,2,3,4\})\), the theorem states:
\[
\Gamma(n,\pm\{1,2,3,4\}) \text{ is Fibonacci cordial for all } n>8.
\]
The construction is explicit for \(n=10\), using \(f(v_i)=F_i\) for \(0\le i\le 10\), and a unified labeling strategy is said to work for all \(n\ge 10\) [2509.01823].

## 4. \((a,b)\)-Fibonacci-Legendre cordial labeling

The 2026 paper introduces a number-theoretic refinement of Fibonacci cordial labeling. It works with the generalized Fibonacci sequence
\[
F_0=a,\quad F_1=b,\quad F_n=F_{n-1}+F_{n-2}\ \text{for } n\ge 2,
\]
with \((0,1)\) yielding the classical Fibonacci sequence and \((2,1)\) yielding the classical Lucas sequence. A central ingredient is the \((a,b)\)-Pisano period \(\pi_m(a,b)\), the least positive integer \(k\) such that
\[
F_{n+k}\equiv F_n \pmod m\quad \text{for all }n
\]
[2601.10561].

For a simple connected graph \(G=(V,E)\) of order \(n=|V(G)|\), the labeling framework again starts from a bijection
\[
f:V(G)\to \{0,1,\dots,|V(G)|-1\}.
\]
Fix an odd prime \(p\). The induced edge labeling \(f_p^*:E(G)\to\{0,1\}\) is defined by
\[
f_p^*(uv)=\frac{1+\left(\frac{F_{f(u)}+F_{f(v)}}{p}\right)}{2}
\quad\text{when }F_{f(u)}+F_{f(v)}\not\equiv 0\pmod p,
\]
and
\[
f_p^*(uv)=0\quad\text{when }F_{f(u)}+F_{f(v)}\equiv 0\pmod p.
\]
Here \(\left(\frac{x}{p}\right)\) is the Legendre symbol:
\[
\left(\frac{x}{p}\right)=
\begin{cases}
1,& x \text{ is a quadratic residue mod }p,\\
-1,& x \text{ is a quadratic nonresidue mod }p,\\
0,& x\equiv 0\pmod p.
\end{cases}
\]
Thus edge label \(1\) means that \(F_{f(u)}+F_{f(v)}\) is a nonzero quadratic residue mod \(p\), while edge label \(0\) means that the sum is either a quadratic nonresidue or zero modulo \(p\) [2601.10561].

The cordiality condition remains
\[
|e_{f_p^*}(0)-e_{f_p^*}(1)|\le 1.
\]
A graph admitting such a labeling is called an \((a,b)\)-FLC graph modulo \(p\). Relative to the broader literature, the paper identifies three novelties: the Legendre-symbol criterion modulo an odd prime \(p\) rather than parity, general initial values \((a,b)\) rather than only the classical Fibonacci or Lucas sequences, and a connection to the newly defined \(k\)-Pisano-Legendre primes [2601.10561].

The arithmetic control parameters are defined via the Pisano index set
\[
S=\{0,1,\ldots,\pi_p(a,b)-1\},
\]
and
\[
\Lambda_j^p(a,b)=\left\{\xi\in S:\left(\tfrac{F_\xi}{p}\right)=j\right\},
\qquad j\in\{-1,0,1\}.
\]
Then \(p\) is a \(k\)-Pisano-Legendre prime relative to \((a,b)\) if
\[
k=|\Lambda_1^p(a,b)|-|\Lambda_{-1}^p(a,b)|-|\Lambda_0^p(a,b)|.
\]
The set of attainable values is
\[
\chi(a,b)=\{k:\exists\text{ odd prime }p\text{ that is }k\text{-PL relative to }(a,b)\}.
\]
The case \(k=0\),
\[
|\Lambda_1^p(a,b)|=|\Lambda_{-1}^p(a,b)|+|\Lambda_0^p(a,b)|,
\]
is especially important because it drives the exact balancing in many graph constructions [2601.10561].

## 5. Graph constructions controlled by \(k\)-Pisano-Legendre primes

The \((a,b)\)-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 [2601.10561].

| Graph family | Result and hypotheses |
|---|---|
| \(P_{q\pi_p(a,b)+1}\) | \((a,b)\)-FLC modulo \(p\) if \(0\in\chi(a,b)\) and \(p\) is \(0\)-PL relative to \((a,b)\) |
| \(S_{q\pi_p(0,b)+1}\) | \((0,b)\)-FLC modulo \(p\) if \(0\in\chi(0,b)\) and \(p\) is \(0\)-PL relative to \((0,b)\) |
| \(W_{q\pi_p(0,b)+1}\) | \((0,b)\)-FLC modulo \(p\) under the same \(0\)-PL hypothesis |
| \(C_{q\pi_p(a,b)}\otimes G\), \(C_{q\pi_p(a,b)}\square G\), \(C_{q\pi_p(a,b)}\boxtimes G\) | \((a,b)\)-FLC modulo \(p\) if \(0\in\chi(a,b)\), \(p\) is \(0\)-PL, and \(p\equiv \pm1\pmod 8\) |
| \(C_{q\pi_p(a,b)}\times G\) | \((a,b)\)-FLC modulo \(p\) if \(0\in\chi(a,b)\) and \(p\) is \(0\)-PL; no condition \(p\equiv\pm1\pmod 8\) is needed |
| \(G\circ P_{\pi_p(0,b)-1}\), special \(G+H\), special \(G\circ H\) | \((0,b)\)-FLC modulo \(p\) under \(k\)-PL hypotheses and explicit size conditions involving \(k\) and \(\varepsilon\in\{-1,0,1\}\) |

The proof strategy is highly uniform. Vertex labels are arranged in blocks of length \(\pi_p(a,b)\), exploiting
\[
F_{m+n\pi_p(a,b)}\equiv F_m \pmod p.
\]
Edge sums are then reduced to a small number of forms:
\[
F_i+F_{i+1}=F_{i+2},\qquad F_0+F_j=F_j \text{ when } F_0=0,\qquad F_j+F_j=2F_j.
\]
This turns the graph-labeling problem into counting residue types over one Pisano period. When edge sums reduce to \(2F_j\), the paper uses
\[
\left(\frac{2}{p}\right)=
\begin{cases}
-1,& p\equiv \pm 3\pmod 8,\\
1,& p\equiv \pm 1\pmod 8,
\end{cases}
\]
so the congruence condition \(p\equiv\pm1\pmod 8\) is required precisely when a \(2F_j\)-term appears [2601.10561].

The path theorem illustrates the mechanism most transparently. For \(P_{q\pi_p(a,b)+1}\), one labels \(v_i\) by \(i\), so each path edge has sum \(F_i+F_{i+1}=F_{i+2}\). Over each block of length \(\pi_p(a,b)\), the edge sums run through
\[
F_2,F_3,\ldots,F_{\pi_p-1},F_0,F_1.
\]
Hence
\[
e_{f_p^*}(0)=q\bigl(|\Lambda_{-1}^p(a,b)|+|\Lambda_0^p(a,b)|\bigr),\qquad
e_{f_p^*}(1)=q|\Lambda_1^p(a,b)|.
\]
If \(p\) is \(0\)-PL relative to \((a,b)\), then the two counts are equal [2601.10561].

The special corona and join theorems show how general \(k\)-PL primes are used. In Theorem 3.5, for instance, if \(G\) has order \(n\) and size
\[
n(2k-1)+\varepsilon,\qquad \varepsilon\in\{-1,0,1\},
\]
and if
\[
\left(\frac{F_1}{p}\right)=\left(\frac{F_2}{p}\right)=1,
\]
then
\[
G\circ P_{\pi_p(0,b)-1}
\]
is \((0,b)\)-FLC modulo \(p\), where \(p\) is a \(k\)-PL prime relative to \((0,b)\). The size parameter is tuned so that the final imbalance becomes exactly \(\varepsilon\) [2601.10561].

## 6. Examples, arithmetic distribution, and limitations

The 2026 paper includes a concrete labeling example on \(C_3\) with \((a,b)=(7,4)\) and \(p=5\). With
\[
f(u_1)=0,\qquad f(u_2)=1,\qquad f(u_3)=2,
\]
the relevant residues are
\[
F_0\equiv 2,\qquad F_1\equiv 4,\qquad F_2\equiv 1 \pmod 5.
\]
The edge sums satisfy
\[
F_0+F_1\equiv 1,\qquad F_0+F_2\equiv 3,\qquad F_1+F_2\equiv 0\pmod 5,
\]
so
\[
f_5^*(u_1u_2)=1,\qquad f_5^*(u_1u_3)=0,\qquad f_5^*(u_2u_3)=0.
\]
Thus
\[
e_{f_p^*}(0)=2,\qquad e_{f_p^*}(1)=1,
\]
and \(C_3\) is \((7,4)\)-FLC modulo \(5\). The same paper also computes that \(3\) is a \((-2)\)-PL prime relative to \((0,1)\), because \(\pi_3(0,1)=8\) and
\[
|\Lambda_1|=3,\qquad |\Lambda_{-1}|=3,\qquad |\Lambda_0|=2,
\]
hence
\[
k=3-3-2=-2
\]
[2601.10561].

The arithmetic side is developed further through the minimal-prime function
\[
\zeta_{(a,b)}(k)=\min\{p:\text{$p$ is a $k$-PL prime relative to $(a,b)$}\}.
\]
For the classical Fibonacci case \((0,1)\), the paper reports sample values
\[
\zeta_{(0,1)}(-4)=5,\quad \zeta_{(0,1)}(-2)=7,\quad \zeta_{(0,1)}(0)=11,\quad \zeta_{(0,1)}(12)=17,\quad \zeta_{(0,1)}(28)=73,\quad \zeta_{(0,1)}(42)=14969.
\]
For \(|k|<600\) and \(3<\zeta_{(0,1)}(k)<25000\), it reports the empirical inequality
\[
\zeta_{(0,1)}(k)>0.055k^2.
\]
Analogous empirical lower curves are noted for \((2,1)\), \((-2,7)\), and \((-2,-4)\), leading to the conjecture that for any \((a,b)\ne(0,0)\), there exists \(0<c<1\) such that
\[
\zeta_{(a,b)}(k)>ck^2 \quad\text{for all }k\in\chi(a,b)
\]
[2601.10561].

Two further conjectural directions are explicit. First,
\[
|\chi(a,b)|=\aleph_0 \quad \text{for all }(a,b)\ne(0,0).
\]
Second, if \(0\in\chi(a,b)\), then
\[
|\vartheta_{(a,b)}(0)|=\aleph_0,
\]
where
\[
\vartheta_{(a,b)}(k)=\{p:\text{$p$ is a $k$-PL prime relative to $(a,b)$}\}.
\]
For \((0,1)\), the first ten elements of \(\vartheta_{(0,1)}(0)\) listed in the paper are
\[
11,29,31,71,131,191,229,251,271,281
\]
[2601.10561].

The parity-based paper also ends with an open direction: for large \(n\), almost every circulant graph \(\Gamma(n,S)\) with a “small” connection set \(S\) admits a Fibonacci cordial labeling [2509.01823]. This suggests a contrast between the two strands of the subject. The parity-based theory emphasizes combinatorial parity placement on structured families, whereas the \((a,b)\)-Fibonacci-Legendre theory is a number-theoretic refinement governed by Pisano periods, quadratic residues, and \(k\)-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 \(a=0\), \(p\equiv\pm1\pmod 8\), \(0\)-PL or \(k\)-PL conditions, or auxiliary assumptions such as
\[
\left(\frac{F_1}{p}\right)=\left(\frac{F_2}{p}\right)=1.
\]
It also presents empirical conjectures rather than density theorems or infinitude theorems [2601.10561]. The 2025 paper includes proven nonexistence statements for circulants, but some constructions appear typographically incomplete or inconsistent: the theorem for \(C_m\boxplus P_n\) has an incomplete proof as printed, the formula for \(F_m\boxplus P_n\) includes a likely typo \(p_2=p+1x\), and some proofs say “the reader can verify” rather than supplying full edge counts [2509.01823].

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 \(3\) 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.

Source: https://www.emergentmind.com/topics/fibonacci-cordial-labeling