5-Regular Circulant Graphs

• 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., 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., 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., 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., 2024, Roy et al., 5 Dec 2025, Feria-Puron et al., 2015).

These avenues remain at the frontier of combinatorics, algebraic graph theory, and algorithmic network design.