Bialternating Cycle Quotient Type

• Bialternating Cycle Quotient Type is defined by a G-invariant 2-factor in cubic graphs with cycles linked alternately to adjacent cycles.
• It encompasses a five-parameter family of Cayley graphs, classified by explicit congruence conditions and cycle quotient isomorphism to cycles.
• The construction employs group-theoretic methods using three involutions, advancing insights into automorphism structures and graph decompositions.

A bialternating cycle quotient type is a structural property of certain cubic, vertex-transitive graphs characterized by a specific configuration of cycles and links corresponding to a quotient graph that is itself a cycle. This property arises in the study of factor-invariant cubic graphs, expanding the theory of cubic vertex-transitive graphs and providing novel insights into their automorphism and decomposition structures. The class has recently been classified as forming a previously unknown infinite family of Cayley graphs described by five parameters and constructed explicitly in (Šparl, 16 Jan 2026).

1. Formal Definition and Cycle Quotient Structure

Let $\Gamma$ be a connected cubic graph, and $G \leq \mathrm{Aut}(\Gamma)$ a vertex-transitive subgroup. A $2$-factor $\mathcal{C} = \{C_0, \ldots, C_{m-1}\}$ is a set of $m$ pairwise disjoint cycles covering all vertices of $\Gamma$. The set $\mathcal{C}$ is $G$-invariant if for every $g \in G$, $g(C_i)$ equals some $C_j$—i.e., $G$ permutes the cycles of $\mathcal{C}$.

Since $\Gamma$ is cubic, the edge complement of $\mathcal{C}$ in $E(\Gamma)$ forms a perfect matching $I$, whose edges are called links, whereas the edges of $\mathcal{C}$ are non-links. Denote the quotient graph by $\Gamma_{\mathcal{C}}$, whose vertex set is $\mathcal{C}$ itself, and $C_i$ is adjacent to $C_j$ in $\Gamma_{\mathcal{C}}$ whenever a link joins some vertex of $C_i$ to one in $C_j$. If $|\mathcal{C}| = m \geq 3$ and $\Gamma_{\mathcal{C}} \cong C_m$, then $\Gamma$ (or $(\Gamma, G, \mathcal{C})$) has cycle-quotient type.

For bialternating cycle-quotient type, cycles are labeled so that $C_i \sim C_{i \pm 1}$ in $\Gamma_{\mathcal{C}}$ (indices modulo $m$). Label the vertices of $\Gamma$ as $$V(\Gamma) = \{v_{i, j}: i \in \mathbb{Z}_m, j \in \mathbb{Z}_n\}$$, where $j \mapsto v_{i, j}$ runs once around $C_i$ (modulo $n$), and $v_{i, j}^*$ denotes the unique outside neighbor (along a link) of $v_{i, j}$. The bialternating property requires that, as one moves along $C_i$ in increasing $j$ from $j \equiv 0$, the outside neighbor $v_{i, j}^*$ lies alternately in $C_{i+1}$ for two consecutive values of $j$ and then in $C_{i-1}$ for the next two, and so forth.

2. Classification Theorem and Boundary Cases

A comprehensive classification (Šparl, 16 Jan 2026) establishes all possible cubic, vertex-transitive graphs of bialternating cycle-quotient type:

• (a) $n=4$: $\Gamma$ is the $4m$-vertex prism or Möbius ladder, with $C_4$s linked in alternating fashion.
• (b) $n=8$:
  • (i) $m \equiv 0 \pmod{3}$ with $\Gamma \cong X^1(m)$ or $X^2(m)$ (two sporadic 8-cycle families).
  • (ii) $m$ odd, $\Gamma \cong X(m, 8, 5, 4, \ell)$ with $\ell \in \{3, 7\}$.
  • (iii) $m$ even $\geq4$, $\Gamma \cong X(m, 8, a, 4, \ell)$, $a \in \{1,5\}, \ell \in \{2,6\}$.
• (c) $n \geq 12$:
  • $n = 4n_0$, $n_0 \geq 3$, $\Gamma \cong X(m, n, a, b, \ell)$, where the parameters satisfy one of three specific congruence conditions involving parity and divisibility.

Conversely, any graph constructed with such parameters possesses the required cubic, vertex-transitive properties and the bialternating cycle-quotient type.

3. Algebraic Construction and Parameterization

The X-graph $X(m, n, a, b, \ell)$ is constructed as follows: let $m \geq 3$, $n=4n_0 \geq 8$, $\ell \in \mathbb{Z}_n$ with $\ell \equiv 2$ or $3 \pmod{4}$, $a=4a_0+1$, $b=4b_0$ such that $\gcd(n_0, b_0)=1$, $bb_0 \equiv 4 \pmod{n}$, $ba_0 + a \equiv 1 \pmod{n}$, and $(\ell, b, a)$ meet one of the three congruence conditions set out for $n\geq12$ cases.

Vertices are $v_{i,j}$ with $i$ mod $m$, $j$ mod $n$. Edges are:

• Non-links: $(v_{i,j}, v_{i,j \pm 1})$, i.e., edges within cycles.
• Links connecting $C_i$ to $C_{i+1}$: $$(v_{i,2i + 4j_0 + \delta}, v_{i+1,2i + b j_0 + \delta a})$$, for $j_0 = 0,\ldots,n_0-1$, $\delta \in \{0,1\}$.
• Wrap-around links ($C_{m-1}, C_0$): if $\ell$ even, $$(v_{m-1,2(m-1)+4j_0+\delta}, v_{0, \ell + b j_0 + \delta a})$$; if $\ell$ odd, $$(v_{m-1,2(m-1)+4j_0+\delta},v_{0,\ell - b j_0 - \delta a})$$.

The resulting graph is cubic, and $G = \langle \rho, \alpha, \gamma \rangle$ acts regularly, preserving $\mathcal{C}$ and the bialternating structure. When $a \neq 1$, closed $10$-cycles exist (girth at most $10$); if $a=1$, shortest cycles are $4$-cycles.

4. Group-Theoretic Characterization: Cayley Graphs with Three Involutions

The regular group $G = \langle \rho, \alpha, \gamma \rangle$ is generated by three involutions with relations:

• $\rho^2 = 1,\;\alpha^2 = 1,\;\gamma^2 = 1$.
• Define $x = \alpha \gamma$, $y = \gamma \rho$, $z = \rho \alpha$, each sending $v_{0,0}$ to its three neighbors.
• Relations: $(xy)^{n/2} = 1$, $(yz)^{m} = 1$, $(zx)^{\ell + 2 } = 1$.

The group presentation $$G \cong \langle x,y,z \mid x^2 = y^2 = z^2 = 1;\; (xy)^{n/2} = (yz)^{m} = (zx)^{\ell + 2} = 1 \rangle$$ yields $$X(m, n, a, b, \ell) \cong \mathrm{Cay}(G, \{x, y, z\})$$, making every such graph a Cayley graph for a group generated by three involutions.

5. Explicit Example

For $m=5, n=12, a=1, b=8, \ell=7$, all required congruences are satisfied, with $gcd(3,2)=1$, $8\cdot2\equiv4\pmod{12}$, $a=1$, $\ell\equiv3\pmod{4}$, and $m,\ell$ odd, $b\equiv-4$, $a=2\ell-1 = 13\equiv1$. Vertices $v_{i,j}$ connect as:

• $(v_{i,j}, v_{i,j\pm1})$ within cycles.
• $(v_{i,2i+4j_0+\delta}, v_{i+1,2i+8j_0+\delta})$ for links between cycles.
• $(v_{4,8+4j_0+\delta}, v_{0,7+8j_0+\delta})$ as wrap-around.

This produces a graph $X(5,12,1,8,7)$ on $60$ vertices, with $\Gamma_{\mathcal{C}}$ a $5$-cycle and $G = \langle \rho, \alpha, \gamma\rangle$ regular. The $2$-factor $\mathcal{C}$ is $G$-invariant, and the girth is $4$.

6. Context and Significance in Graph Theory

The study of bialternating cycle quotient types advances the understanding of factor-invariant cubic graphs, extending prior work focused on single or double-cycle $2$-factors and alternating quotient types. The infinite five-parameter family unifies diverse configurations—including prisms, Möbius ladders, sporadic 8-cycle families, and richer high-girth structures—under the Cayley graph umbrella with groups generated by involutions. This provides new instances and template constructions of cubic, vertex-transitive graphs with prescribed cycle-decomposition and quotient graph properties, facilitating investigations on structure, automorphisms, and applications in algebraic graph theory.

7. Further Implications

The explicit construction and group-theoretic characterization suggest a wealth of possible extensions to higher degree, additional cycle decompositions, and analysis of automorphism group actions. A plausible implication is that analogous quotient-type classifications might exist for quartic and higher regular graphs, by decomposing edge sets into invariant $k$-factors. The interplay between bialternation in cycle connection pattern and the five parameter family enables the systematic generation of new examples for testing conjectures involving vertex-transitivity, graph factorization, and Cayley graph constructions.