Cyclically 4-Edge Connected Cubic Planar Graphs

• Cyclically 4-edge connected cubic planar graphs are defined as planar graphs with all vertices of degree 3 and no cyclic edge cuts smaller than 4, ensuring robust connectivity.
• They employ constructive methods such as two-edge and one-edge insertions that maintain cubic regularity and proper 3-edge-coloring, thereby enhancing cycle and matching properties.
• Their applications range from network design to chemical modeling, with ongoing research addressing challenges in Hamiltonicity, embeddings, and decycling configurations.

Cyclically 4-edge connected cubic planar graphs are a distinguished class of graphs in topological, combinatorial, and algebraic graph theory. These are planar graphs with every vertex of degree three, admitting no cyclic edge cut of size three or less; that is, for any removal of up to three edges, the resulting subgraphs cannot both contain cycles. This connectivity threshold imparts notably rigid structure to the cycle space, chromatic invariants, perfect matching properties, decycling configurations, and topological embeddings of these graphs. The following sections survey major facts, constructions, and theoretical outcomes from recent research.

1. Structural Definition and Construction Principles

A cubic graph is one where all vertex degrees are three, and a cyclically 4-edge connected graph is one for which any nontrivial edge cut separating the graph into two cyclic components must be of size at least 4. In the planar case, such graphs are triangle-free with at least 8 vertices, excluding the Q8 (ladder) and V8 (Möbius ladder) graphs. The connectivity concept refines traditional edge-connectivity by requiring robustness specifically with respect to cycle-separating cuts.

Construction methods for cubic planar graphs allowing for controlled edge coloring are given in (Kurapov, 2010). Two inductive insertions suffice:

• Two-Edge Insertion: On two distinct edges, insert new vertices and join them by a new edge, yielding one extra edge and two vertices. This preserves cubic regularity and enables tracking of cycles essential for coloring invariants.
• One-Edge Insertion: Insert two new vertices on one edge and connect them, likewise maintaining cubic and planar properties.

In both methods, careful bookkeeping ensures that the newly added edge receives a color maintaining proper 3-edge-coloring, as guaranteed by an existence theorem for colored discs: in any bridgeless planar cubic graph and any pair of linked edges on an elementary cycle, there exists a colored disc traversing both edges. The Klein four-group structure governs the additive coloring, with colors R (red), B (blue), G (green), and W (white) satisfying $W = R + B + G$.

2. Cycle Space, Shortness Coefficient, and Hamiltonicity

Cyclically 4-edge connected cubic planar graphs have a robust cycle structure. The shortness coefficient $c_4$ quantifies the normalized circumference (maximum cycle length) via $c_4 = \liminf \frac{\mathrm{circ}(G)}{|G|}$ for families of such graphs. From (Markström, 2013), there exists an infinite family with $c_4 \leq 12/13$, realized by assembling snark blocks into 4-regular 4-edge connected multigraphs, with each block limiting cycle passage. The method produces snarks whose oddness—minimum number of odd circuits in any 2-factor—increases linearly with the graph order.

Hamiltonicity questions—whether every such graph contains a Hamiltonian cycle—form a central inquiry. While not universally Hamiltonian, these graphs guarantee long cycles: for $n$-vertex essentially 4-connected planar graphs (a broader class), the circumference meets the lower bound $\lceil (2n+6)/3 \rceil$ (Wigal et al., 2020). Leapfrog extensions and facial 2-factor contractions provide partial solutions to Barnette's Conjecture for cubic bipartite cases (Gh. et al., 2018): if $G_0$ is cyclically 4-edge connected and bipartite, the leapfrog extension is Hamiltonian.

In contrast, (Pivotto et al., 2019) demonstrates a spectrum for the number of Hamilton cycles in this class: infinite families exist with as few as four Hamilton cycles (via ladder extensions), and others with exponentially many (e.g., rings of ladders, nanotube fullerenes).

3. Coloring Properties and Choosability

The interplay between cycle structure and coloring is particularly strong. For cyclically 4-edge connected cubic planar graphs, proper 3-edge-coloring is always possible, barring exceptional snarks (Kurapov, 2010, Goddyn et al., 2012). The polynomial method (edge monomial, star labelings, and Combinatorial Nullstellensatz application) underpins choosability claims: every edge may be assigned a list of size 3, yielding 3-edge-choosability for all cyclically 4-edge connected planar cubic graphs.

Generalizations to surfaces of higher genus confirm these properties with topological nuance (Inoue et al., 26 May 2024, Inoue et al., 11 May 2025, Weiß et al., 18 Sep 2025). Every cyclically 4-edge connected cubic graph strongly embeddable on the projective plane or torus (except for "Petersen-like" obstructions) is 3-edge-colorable and admits a nowhere-zero 4-flow, generalized from planar cases.

Coloring-flow duality statements have also been established: for cubic graphs embedded in the projective plane, being 3-edge-colorable is equivalent to the dual graph being 5-vertex-colorable (Inoue et al., 26 May 2024).

4. Perfect Matchings, 2-Factors, and Chord Structures

The complementarity between perfect matchings and 2-factors, as governed by Petersen's Theorem, is central. In cyclically 4-edge connected planar cubic graphs (excluding trivial cases), every edge lies in a perfect matching whose removal disconnects the graph, i.e., creates a disconnected 2-factor (Diwan, 2021). Characterizations specify which edges force Hamiltonian 2-factors (only in certain $H_n$ graphs) and link this structure to dual properties in triangulations.

Robustness under edge deletion is formalized in an extension of Plesnik's theorem (Lukoťka et al., 2017): after deleting up to four edges (for cubic graphs), a cyclically 4-edge connected graph still admits a perfect matching unless isolated vertices or large independent sets emerge—a rare phenomenon given their high connectivity.

[Berge's conjecture] on covering all edges with five perfect matchings is confirmed for cyclically 4-edge connected cubic graphs with coloring defect 3, and with further restrictions, only four suffice unless the graph is Petersen (Karabáš et al., 2022).

5. Decycling Numbers, Embeddings, and Topological Properties

Decycling sets—vertex sets whose removal yields an acyclic graph—are characterized in cyclically 4-edge connected cubic planar graphs via topological methods (Nedela et al., 2023). The minimum decycling number equals $\lceil (n+2)/4 \rceil$, attainable via coherent decycling partitions (tree plus near-independent set). Topological embedding theory links this to maximum genus and cellular embeddings: equality holds if and only if the graph is upper-embeddable in an orientable surface with one or two faces.

Strong embeddings (where each face is a cycle) into surfaces of positive genus are always possible for cyclically 4-edge connected cubic planar graphs except when the dual is an Apollonian network (Weiß et al., 18 Sep 2025). The embedding is achieved by modifying the rotation system and using an even subgraph of the dual to twist selected edges, with the genus computable via Euler characteristic changes. Non-orientability arises only for Apollonian duals.

6. Generation Algorithms and Classification Results

The constructive characterization of cyclically 4-edge connected cubic graphs leverages the "bridging" operation: starting from ladders (or the Q8 graph for non-planar cases), bridging pairs of edges with cycle spread at least (1,2) generates all such graphs except for certain exceptions (Kingan et al., 2021). Algorithms implemented using McKay’s nauty system efficiently check isomorphism and generate all graphs up to given order, confirming theoretical classification.

Tables summarizing key construction and classification criteria can be found below:

Base Graph	Operation	Resulting Class
Ladder Q2k	Bridging edges (spread ≥ (1,2))	All planar cyclically 4-edge connected cubic graphs ≥10 vertices except ladders
Q8	Bridging edges (spread ≥ (1,2))	Non-planar cyclically 4-edge connected cubic graphs ≥10 vertices (except for ladders, Petersen, Möbius ladders)

The systematic use of cycle spread refines previous methods and makes graph generation computationally feasible.

7. Applications and Open Problems

Cyclically 4-edge connected cubic planar graphs shape resource allocation, frequency assignment, and chemical graph modeling (fullerenes, nanotubes), owing to their predictable cycle, matching, and coloring properties (Pivotto et al., 2019). The spectrum of Hamilton cycle counts, decycling configurations, and chromatic invariants contributes to extremal graph theory and graph minor classification.

Open questions include:

• Whether the shortness coefficient can be driven to zero (i.e., arbitrarily short maximum cycles relative to order) (Markström, 2013).
• The full extension of the "two edges in one cycle" conjecture for 2-factors in broader classes (Diwan, 2021).
• The enumeration of strong embeddings by automorphism orbit of subgraphs of the dual (Weiß et al., 18 Sep 2025).

Further research is directed at refining algebraic and combinatorial criteria for these graphs and advancing practical algorithms for their construction, classification, and coloring.

---

Cyclically 4-edge connected cubic planar graphs thus represent a keystone in structural graph theory, linking local constraints (edge-connectivity, cycle spread) to global coloring, matching, cycle, and embedding properties, with deep implications for both pure mathematics and applied network design.