Small Path Double Cover Conjecture

• The Small Path Double Cover Conjecture is a graph theory problem proposing that every suitably restricted graph can have its edges double-covered by a ‘small’ (often linear) number of paths.
• It generalizes the cycle double cover and path covering problems by leveraging techniques like Gallai’s path decomposition, Hamiltonian constructions, and combinatorial coloring.
• Recent research connects the conjecture with flow-tension duality and oriented cover variations, offering practical bounds and decomposition strategies for complex graphs.

The Small Path Double Cover Conjecture (SPDCC) is a central open problem in the theory of graph coverings that generalizes path and cycle double cover conjectures. This conjecture asks whether every (suitably restricted) graph admits a cover of its edge set by a “small” number of paths such that every edge is covered at least twice. The notion of “small” typically refers to a bound linear or sublinear in the number of vertices or edges, analogous to established results for cycle covers. The literature draws deep connections between path covers, cycle double covers (CDC), flow-tension dualities, combinatorial decompositions, and matching theory, using constructions and results around Gallai’s path decomposition, Hamiltonian path tables in complete graphs, and intricate coloring and decomposition methods.

1. Definitions and Foundations

For a finite undirected graph $G = (V, E)$:

• A path double cover (PDC) is a collection $\mathcal{P}$ of (possibly overlapping) paths such that every edge $e \in E$ appears in at least two distinct paths in $\mathcal{P}$.
• A small path double cover (SPDCC) strengthens this by insisting $|\mathcal{P}| \le f(n)$ for a “small” function $f$, often $O(n)$, where $n = |V|$.
• The conjecture is often stated for connected graphs, sometimes excluding trivial exceptions (e.g., small complete graphs such as $K_3$, $K_4$, $K_5$, $K_6$ depending on orientation and structure).

Contrast this with the cycle double cover (CDC): a collection of cycles such that every edge is in exactly two cycles. The “small” variant, SCDC, seeks bounds like at most $n - 1$ cycles, as in Seyffarth’s theorem for planar 4-connected graphs (Jooken et al., 12 Jun 2025). Various orientation constraints lead to the oriented perfect path double cover (OPPDC) and small oriented cycle double cover (SOCDC) notions (Gh. et al., 2012, Gh. et al., 2012).

2. Connection to Path and Cycle Decomposition Results

The techniques that settle Gallai’s path decomposition conjecture (Mehendale, 2011) underpin much of the research on SPDCC:

• Gallai’s Conjecture: Every connected $n$-vertex graph decomposes into at most $\lfloor (n+1)/2 \rfloor$ paths. This is realized via spanning tree decompositions and the systematic extension to cover all edges.

  $$|\mathcal{P}| \le \left\lfloor\frac{n+1}{2}\right\rfloor.$$

• Edge-based bounds: By associating pseudo-trees with the edge set, the count of paths can be driven to $O(e)$ (e.g., $2(e+2)$ for a graph with $e$ edges).
• For complete graphs $K_{2n}$, one constructs $n$ edge-disjoint Hamiltonian paths, covering every edge once (Mehendale, 2011). By combining two such decompositions (permuting endpoints or using distinct trees), one may ensure every edge is double-covered with only $2n$ paths.

Implication: These decompositions offer direct strategies to engineer “small” collections of paths with controlled coverage properties, and the extension principles (spanning tree skeleton, Hamiltonian path tables, pseudo-tree constructions) are robust for adaptation to SPDCC requirements.

3. Interaction with Double Cover Conjectures and Minimal Counterexample Structure

Recent work on OPPDC and SOCDC deepens the foundation for SPDCC (Gh. et al., 2012, Gh. et al., 2012):

• OPPDC: Oriented perfect path double covers require each directed edge in the symmetric orientation $G_s$ to appear exactly once in a path, and each vertex to be unique as a start/end across $\mathcal{P}$.
• SOCDC: A collection of at most $n-1$ cycles in $G_s$ covering each arc exactly once. The existence of an OPPDC in $G$ is equivalent to an SOCDC in $G \vee K_1$.

Both conjectures exclude small complete graphs ($K_3$, $K_5$ for OPPDC; $K_4$, $K_6$ for SOCDC) and prove that any minimal counterexample must be highly connected: 2-connected and 3-edge-connected for OPPDC; 3-connected and 3-edge-connected for SOCDC (Gh. et al., 2012, Gh. et al., 2012).

Significance: These results imply that any obstruction to SPDCC must have rich connectivity (no cut-vertices or small edge-cuts), limiting the search space for counterexamples and guiding decomposition strategies.

4. Cycle Double Cover, Flow-Tension Duality, and Combinatorial Generation Techniques

• The cycle double cover conjecture (CDC)—proven for bridgeless graphs (Shen, 2017)—constructs covers using Eulerian circuits in a two-sheet lift, with “segments” corresponding to symmetric paths that could underpin path double covers if stopped prior to recombination.
  • The definition of a segment:

  $s = v_1 v_2 \ldots v_k v_{k-1} \ldots v_2 v_1$

  Each “half” corresponds to a path; recombination yields cycles. Avoiding recombination is potentially an effective approach for SPDCC.

• Jaeger's directed CDC is approached via hexagon graphs and perfect matching theory (Jiménez et al., 2013). The combinatorial generation of hexagon graphs using McCuaig’s augmentations offers a structured model for incrementally constructing covers, suggesting an analogous approach for matching-based generation of SPDCCs.
• Flow-tension duality and Cayley graph homomorphisms (Hušek et al., 2019) connect the existence of oriented cycle double covers with group-theoretic flow properties. When a strong homomorphism property (SHP) holds, it is possible to construct an oriented CDC with at most $|M|$ cycles for a group $M$, suggesting possible “small” covers for SPDCC via similar group-based techniques.

5. Enumerative, Decomposition, and Coloring Approaches

Recent results show that in certain structural classes, not only do small CDCs exist but they are numerous (Jooken et al., 12 Jun 2025):

• In planar 4-connected graphs with $n$ vertices and $h$ Hamiltonian cycles,

  $$\Sigma_{k \le n-1} \mathfrak{c}(k) \ge \frac{h}{22}$$

  so there are linearly many small CDCs.

• For planar 4-connected triangulations or graphs with few 4-cuts, the number of small CDCs can be quadratic or exponential in $n$.

Parallel decomposition arguments (e.g., joining triangulations along facial triangles, controlling CDC size by the sum minus 3 cycles) allow for precise enumeration and tight bounds in cubic graphs (maximum CDC size $n/2$ for planar 2-connected cubic graphs).

Coloring methods, notably 4-face coloring or entropy inequalities, partition the edge set into even subgraphs, which can be further decomposed or adjusted toward path double cover properties.

6. Implications, Prospects for SPDCC, and Research Directions

The techniques developed for CDC, OPPDC, SOCDC, and path decomposition problems provide a rich toolkit for approaching SPDCC:

• By leveraging spanning tree decompositions, basic path covers, and systematic “controlled breaking” of cycles, one can attempt to construct path covers that double cover every edge, keeping the path count “small”—often linear in $n$.
• Enumeration and entropy arguments guarantee not just existence but multiplicity of small covers in highly connected or appropriately colored graphs.
• The minimal counterexample analyses (connectivity requirements, forbidden edge-cuts) narrow the path for obstruction to SPDCC, raising its plausibility for broad graph classes.
• Combinatorial generation and augmentation schemes (hexagon graphs, ladder-based construction) are promising for both structural insight and algorithmic generation of SPDCCs.

Future research directions include:

• Refining the upper bounds for path double covers (e.g., minimizing from $n-1$ toward $n/2$ or better).
• Developing decomposition and joining schemes tuned for paths rather than cycles.
• Formulating and proving enumerative analogues for SPDCC (proving exponential multiplicity in triangulations, etc.).
• Investigating group-theoretic and flow-based mechanisms for constructing SPDCCs with few paths.
• Extending coloring and entropy-based partition arguments specifically to the path context.

Key LaTeX formulas:

Gallai bound:

$$|\mathcal{P}| \le \left\lfloor\frac{n+1}{2}\right\rfloor.$$

Edge-based pseudo-tree bound:

$|\mathcal{P}| \le 2(e + 2).$

Hamiltonian path cover for $K_{2n}$:

$K_{2n} = \bigcup_{i=1}^{n} P_i,$

Enumerative bound for small CDCs:

$$\Sigma_{k \le n-1} \mathfrak{c}(k) \ge \frac{h}{22}$$

In summary, the small path double cover conjecture stands at the intersection of several powerful graph-theoretic concepts and techniques. Its resolution is intimately connected to longstanding results concerning cycle covers, path decomposition, and group flows, and ongoing research continues to make the existence and multiplicity of “small” path double covers in highly connected graphs increasingly plausible.