5-Regular Circulant Graphs

Updated 9 December 2025

5-regular circulant graphs are vertex-transitive Cayley graphs defined over cyclic groups with each vertex connected to five others via a symmetric jump set.
They are characterized by semi-transitive orientability, which ensures word-representability and relates to specific colorability and algebraic conditions.
These graphs play a pivotal role in extremal graph theory, particularly in the degree–diameter problem, with algorithmic methods identifying near-optimal constructions.

A 5-regular circulant graph is a vertex-transitive Cayley graph of the cyclic group $\mathbb{Z}_n$ in which each vertex has degree five, constructed by a symmetric generating (jump) set of size five. In standard parametrization, with $n$ even, these graphs take the form $C_n(a_1,a_2,a_3)$ with $0<a_1<a_2<a_3\le n/2$ and $a_3=n/2$ , so each vertex is connected to the vertices at distances $a_1$ , $a_2$ , and $n/2$ modulo $n$ . 5-regular circulant graphs are central in the study of algebraically defined networks, extremal graph theory (notably the degree–diameter problem), and word-representability through semi-transitive orientability.

1. Fundamental Definitions and Algebraic Structure

Let $n\ge6$ be even, and consider the vertex set $V = \{0,1,\ldots,n-1\}$ . The edge set is given by $E = \{\{i,j\} : |i-j|\bmod n\in\{a_1,a_2,a_3\}\}.$ A connection set $S = \{\pm a_1, \pm a_2, n/2\}$ (with sizes adjusted to ensure undirectedness and regularity) yields a Cayley graph over $\mathbb{Z}_n$ . If $a_3=n/2$ , the graph is simple and connected provided that $\gcd(a_1, n), \gcd(a_2, n)$ , and parity conditions are satisfied. These graphs are especially amenable to combinatorial, algebraic, and spectral analysis, including network symmetry and automorphism characterization (Feria-Puron et al., 2015).

2. Semi-Transitive Orientability and Word-Representability

A graph is word-representable if there is a word over its vertex alphabet such that adjacency coincides with the alternation property in the word. Semi-transitive orientation, equivalent to word-representability, requires that the orientation is acyclic and that for every directed path $v_0\to v_1\to\ldots\to v_m$ ( $m\ge2$ ), either there is no arc $v_0\to v_m$ , or if there is, then all arcs $v_i\to v_j$ are present for $1\le i<j\le m$ .

For 5-regular circulants $C_n(a_1,a_2,n/2)$ :

Sufficient condition (Theorem 4.1): If $a_1\ge (n+1)/4$ , the total order orientation $i\to j$ for $i<j$ is semi-transitive (Srinivasan et al., 5 Jun 2024).
Consecutive-steps-plus-antipode family (Theorem 4.3): All $C_n(t, t+1, n/2)$ for $1\le t\le n/2$ are semi-transitive under the same orientation.
Negative condition: Triples of the form $\{a, a+1, 2a\}$ , with $2< (n+1)/5 \le a\le (n-1)/4$ , induce $W_5$ subgraphs and are not semi-transitive.
Semi-transitive orientability corresponds precisely to word-representability, so these results yield infinite families that are (or are not) word-representable.

A full classification of all 5-regular circulant graphs regarding semi-transitive orientation remains open, although two infinite positive families and one infinite negative family are established (Srinivasan et al., 5 Jun 2024).

3. Representation Number and Word Constructions

The representation number $R(G)$ of a graph $G$ is the smallest $k$ such that $G$ is $k$ -word-representable (there is a representing word in which each letter occurs exactly $k$ times).

Key results:

For $k=2$ , cycles $C_n$ satisfy $R=2$ .
For $k=3$ , connected circulants $C_{2n}(a, n)$ satisfy $R\le3$ .
For $k=4$ , all circulant graphs are word-representable, with $R\le4$ in major families.
For 5-regular circulants $C_{2n}(a,b,n)$ , explicit morphisms using arithmetic on the cyclic group yield $R\le5$ in significant families, specifically for $x\in (\frac n2, \frac{2n}{3}]$ in $C_{2n}(x,1, n)$ . For special cases, $R=3$ or $R=1$ can occur (e.g., $C_{2n}(2,1,n)$ and $n=3$ gives $K_6$ ) (Roy et al., 5 Dec 2025).
The question of whether all 5-regular word-representable circulant graphs are 5-word-representable is still open (Roy et al., 5 Dec 2025, Srinivasan et al., 5 Jun 2024).

4. Word-Representability Criteria and Colorability

Several sufficient criteria for word-representability and semi-transitive orientability draw on colorings and algebraic reductions:

Parity criterion: For $n$ odd, if $a$ and $b$ have the same parity in $C_{2n}(a, b, n)$ , the graph is word-representable; often, this involves Cartesian product decompositions with $P_2$ and 4-regular circulants (Roy et al., 5 Dec 2025).
Reduction to $C_{2n}(x,1,n)$ : If $\gcd(b, 2n)=1$ , any such circulant is isomorphic to $C_{2n}(x,1,n)$ for unique $x$ .
3-colorability criterion: If $3\nmid x$ , $3\nmid (n-x)$ , and $3\nmid n$ , then $C_{2n}(x, 1, n)$ is 3-colorable and hence word-representable.
Extended colorability: Detailed partitioning using cyclic generators provides further sufficient criteria, covering additional classes that are word-representable.
Cartesian product factorization: Under certain divisibility and parity conditions, $C_{2n}(a, b, n)$ decomposes as a product of a 3-regular and a 2-regular circulant, implying upper bounds on $R(G)$ (Roy et al., 5 Dec 2025).

No non-word-representable 5-regular circulant is currently known, despite such examples for higher regularities (Roy et al., 5 Dec 2025).

5. Extremal Order, Diameter, and Structural Properties

The degree–diameter problem seeks the largest possible order $n$ of a 5-regular circulant graph of a given diameter $d$ .

For degree 5 and diameter $d$ :

The exact upper bound is $N_{5,d}^{\text{circ}} \le 4d^2+2$ , a result from the so-called Delannoy-type bound.
The best-known and empirically optimal construction is via the "double-loop" family:

$S_d = \{\pm1, \pm d, n/2\},\quad n=4d^2$

for $d\ge2$ . For each $d\le 10$ , such graphs attain $\approx 99\%$ of the bound, and evidence suggests $N_{5,d}^{\text{circ}}=4d^2$ for all $d\ge2$ (Feria-Puron et al., 2015).

No construction exceeding this order is known, and a general proof that the quadratic bound is always sharp is an open problem. Structural uniqueness of the double-loop extremals is also an open question (Feria-Puron et al., 2015).

Table: Extremal 5-regular Circulant Graphs by Diameter

Diameter $d$	Order $n$	Connection Set $S$
1	6	$\{\pm1, \pm2, 3\}$
2	16	$\{\pm1, \pm2, 8\}$
3	36	$\{\pm1, \pm3, 18\}$
4	64	$\{\pm1, \pm4, 32\}$
5	100	$\{\pm1, \pm5, 50\}$

6. Algorithmic Generation and Search Techniques

The state-of-the-art approach for identifying extremal or large 5-regular circulant graphs is a depth-first backtracking search over all symmetric generating sets $S$ of the cyclic group. Key algorithmic features include:

Enforcing connectivity by requiring $1\in S$ .
Tree search over possible generator sets up to the required size, with pruning based on path-count and diameter constraints.
Explicit stack management for effective depth traversal.
Diameter verification via breadth-first search upon reaching the required generator size.
The method efficiently rediscovers all previously known optimums and pushes the known records for higher degrees, with statistical analysis confirming the closeness to the theoretical maximum (Feria-Puron et al., 2015).

This approach generalizes to other degrees and can be adapted to directed or mixed (arc-and-edge) circulant graphs, but degree 5 remains distinctive in the tight alignment of theory, computation, and construction.

7. Open Problems and Research Directions

Central open questions and topics include:

Full classification of 5-regular circulant graphs regarding semi-transitive orientability and word-representability.
Uniform representation number: whether all word-representable 5-regular circulants are 5-word-representable.
Proof or disproof that $N_{5,d}^{\text{circ}} = 4d^2$ for all $d\ge2$ .
Structural characterization and uniqueness for extremal examples in the double-loop family.
Extension of search heuristics, particularly for large order and additional algebraic constraints.
Potential for new phenomena in directed or mixed circulant settings (Srinivasan et al., 5 Jun 2024, Roy et al., 5 Dec 2025, Feria-Puron et al., 2015).