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 $G \leq \mathrm{Aut}(\Gamma)$0—i.e., $G \leq \mathrm{Aut}(\Gamma)$1 permutes the cycles of $G \leq \mathrm{Aut}(\Gamma)$2.

Since $G \leq \mathrm{Aut}(\Gamma)$3 is cubic, the edge complement of $G \leq \mathrm{Aut}(\Gamma)$4 in $G \leq \mathrm{Aut}(\Gamma)$5 forms a perfect matching $G \leq \mathrm{Aut}(\Gamma)$6, whose edges are called links, whereas the edges of $G \leq \mathrm{Aut}(\Gamma)$7 are non-links. Denote the quotient graph by $G \leq \mathrm{Aut}(\Gamma)$8, whose vertex set is $G \leq \mathrm{Aut}(\Gamma)$9 itself, and $2$0 is adjacent to $2$1 in $2$2 whenever a link joins some vertex of $2$3 to one in $2$4. If $2$5 and $2$6, then $2$7 (or $2$8) has cycle-quotient type.

For bialternating cycle-quotient type, cycles are labeled so that $2$9 in $\mathcal{C} = \{C_0, \ldots, C_{m-1}\}$0 (indices modulo $\mathcal{C} = \{C_0, \ldots, C_{m-1}\}$1). Label the vertices of $\mathcal{C} = \{C_0, \ldots, C_{m-1}\}$2 as $\mathcal{C} = \{C_0, \ldots, C_{m-1}\}$3, where $\mathcal{C} = \{C_0, \ldots, C_{m-1}\}$4 runs once around $\mathcal{C} = \{C_0, \ldots, C_{m-1}\}$5 (modulo $\mathcal{C} = \{C_0, \ldots, C_{m-1}\}$6), and $\mathcal{C} = \{C_0, \ldots, C_{m-1}\}$7 denotes the unique outside neighbor (along a link) of $\mathcal{C} = \{C_0, \ldots, C_{m-1}\}$8. The bialternating property requires that, as one moves along $\mathcal{C} = \{C_0, \ldots, C_{m-1}\}$9 in increasing $m$0 from $m$1, the outside neighbor $m$2 lies alternately in $m$3 for two consecutive values of $m$4 and then in $m$5 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) $m$6: $m$7 is the $m$8-vertex prism or Möbius ladder, with $m$9s linked in alternating fashion.
• (b) $\Gamma$0:
  • (i) $\Gamma$1 with $\Gamma$2 or $\Gamma$3 (two sporadic 8-cycle families).
  • (ii) $\Gamma$4 odd, $\Gamma$5 with $\Gamma$6.
  • (iii) $\Gamma$7 even $\Gamma$8, $\Gamma$9, $\mathcal{C}$0.
• (c) $\mathcal{C}$1:
  • $\mathcal{C}$2, $\mathcal{C}$3, $\mathcal{C}$4, 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 $\mathcal{C}$5 is constructed as follows: let $\mathcal{C}$6, $\mathcal{C}$7, $\mathcal{C}$8 with $\mathcal{C}$9 or $G$0, $G$1, $G$2 such that $G$3, $G$4, $G$5, and $G$6 meet one of the three congruence conditions set out for $G$7 cases.

Vertices are $G$8 with $G$9 mod $g \in G$0, $g \in G$1 mod $g \in G$2. Edges are:

• Non-links: $g \in G$3, i.e., edges within cycles.
• Links connecting $g \in G$4 to $g \in G$5: $g \in G$6, for $g \in G$7, $g \in G$8.
• Wrap-around links ($g \in G$9): if $g(C_i)$0 even, $g(C_i)$1; if $g(C_i)$2 odd, $g(C_i)$3.

The resulting graph is cubic, and $g(C_i)$4 acts regularly, preserving $g(C_i)$5 and the bialternating structure. When $g(C_i)$6, closed $g(C_i)$7-cycles exist (girth at most $g(C_i)$8); if $g(C_i)$9, shortest cycles are $G \leq \mathrm{Aut}(\Gamma)$00-cycles.

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

The regular group $G \leq \mathrm{Aut}(\Gamma)$01 is generated by three involutions with relations:

• $G \leq \mathrm{Aut}(\Gamma)$02.
• Define $G \leq \mathrm{Aut}(\Gamma)$03, $G \leq \mathrm{Aut}(\Gamma)$04, $G \leq \mathrm{Aut}(\Gamma)$05, each sending $G \leq \mathrm{Aut}(\Gamma)$06 to its three neighbors.
• Relations: $G \leq \mathrm{Aut}(\Gamma)$07, $G \leq \mathrm{Aut}(\Gamma)$08, $G \leq \mathrm{Aut}(\Gamma)$09.

The group presentation $G \leq \mathrm{Aut}(\Gamma)$10 yields $G \leq \mathrm{Aut}(\Gamma)$11, making every such graph a Cayley graph for a group generated by three involutions.

5. Explicit Example

For $G \leq \mathrm{Aut}(\Gamma)$12, all required congruences are satisfied, with $G \leq \mathrm{Aut}(\Gamma)$13, $G \leq \mathrm{Aut}(\Gamma)$14, $G \leq \mathrm{Aut}(\Gamma)$15, $G \leq \mathrm{Aut}(\Gamma)$16, and $G \leq \mathrm{Aut}(\Gamma)$17 odd, $G \leq \mathrm{Aut}(\Gamma)$18, $G \leq \mathrm{Aut}(\Gamma)$19. Vertices $G \leq \mathrm{Aut}(\Gamma)$20 connect as:

• $G \leq \mathrm{Aut}(\Gamma)$21 within cycles.
• $G \leq \mathrm{Aut}(\Gamma)$22 for links between cycles.
• $G \leq \mathrm{Aut}(\Gamma)$23 as wrap-around.

This produces a graph $G \leq \mathrm{Aut}(\Gamma)$24 on $G \leq \mathrm{Aut}(\Gamma)$25 vertices, with $G \leq \mathrm{Aut}(\Gamma)$26 a $G \leq \mathrm{Aut}(\Gamma)$27-cycle and $G \leq \mathrm{Aut}(\Gamma)$28 regular. The $G \leq \mathrm{Aut}(\Gamma)$29-factor $G \leq \mathrm{Aut}(\Gamma)$30 is $G \leq \mathrm{Aut}(\Gamma)$31-invariant, and the girth is $G \leq \mathrm{Aut}(\Gamma)$32.

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 $G \leq \mathrm{Aut}(\Gamma)$33-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 $G \leq \mathrm{Aut}(\Gamma)$34-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.