Conjecture A.10 in Extremal Graph Theory

• Conjecture A.10 defines two key extremal problems: the Erdős–Gyárfás conjecture on power-of-two cycles and Thomassen’s pillar conjecture on complex cyclical graph structures.
• The article presents rigorous structural results using cycle-length restrictions, forbidden subgraphs, and expansion techniques to ensure specific cycle embeddings under high minimum degree.
• Advanced methodologies such as expander extraction, kraken constructions, and polynomial algorithms are employed to resolve these conjectures within special classes and, in the case of pillars, in full generality.

Conjecture A.10 refers to two prominent and distinct problems in extremal graph theory, depending on context: the Erdős–Gyárfás Conjecture regarding cycles of power-of-two length in graphs of minimum degree at least three, and Thomassen’s Pillar Conjecture concerning the existence of complex cyclical structures in graphs of sufficiently high minimum degree. This article details both, presenting precise statements, terminology, and the main structural and algorithmic results, most notably their recent resolutions in special classes and generality.

1. Formal Statements of Conjecture A.10

Erdős–Gyárfás Conjecture

Let $G$ be a finite simple graph with minimum degree $\delta(G)$. A cycle of $G$ is "power-of-two length" if its number of vertices is $2^k$ for some integer $k \ge 1$. The conjecture is formally stated as:

Conjecture A.10 (Erdős–Gyárfás, 1997): Every graph $G$ with $\delta(G) \ge 3$ contains at least one cycle whose length is a power of two.

Thomassen’s Pillar Conjecture

Let $s \ge 1$. A "pillar" of height $s$ consists of two vertex-disjoint cycles $C_1$ and $C_2$, both of length $s$, together with $s$ internally vertex-disjoint paths (the "rungs") of identical length connecting corresponding vertices in order. Thomassen’s conjecture is:

Conjecture A.10 (Thomassen’s Pillar Conjecture): Every graph $G$ with $\delta(G) \ge 10^{10^{10}}$ contains a pillar as a subgraph (Fernández et al., 2022).

2. Historical Context and Major Results

Both conjectures motivated extensive research on degree conditions guaranteeing complex cycle structures.

Erdős–Gyárfás Conjecture

• Lower Bound Progress: Markström (2004) proved that any 3-regular graph violating the conjecture must have at least 30 vertices; bipartite counterexamples must have at least 32 vertices (Nowbandegani & Esfandiari, 2011). For cubic claw-free graphs, the bound is at least 114 vertices (Nowbandegani et al., 2014).
• Special Classes: The conjecture has been confirmed for $K_{1,m}$-free graphs with $\delta \ge m+1$ or maximum degree at least $2m-1$ (Shauger, 1998), planar claw-free graphs (Daniel & Shauger, 2001), 3-connected cubic planar graphs (Heckman & Krakovski, 2013), $P_8$-free graphs (Gao & Shan, 2022), and certain Cayley graphs (Ghaffari & Mostaghim, 2018; Ghasemi & Varmazyar, 2021). Claw-free graphs with $\delta \ge 3$ have a cycle of length $2^k$ or $3 \cdot 2^k$ (Nowbandegani et al., 2014). Liu & Montgomery (2023) established that sufficiently large average degree implies the conjecture (Hu et al., 2023).

Thomassen’s Pillar Conjecture

Despite advances in embedding techniques for cycles and paths, Thomassen’s 1989 pillar conjecture remained unproven for decades. Recent work resolves the conjecture up to replacing the explicit bound with another (yet still astronomical) absolute constant, and extends to robust consequences for "near-regular" structures (Fernández et al., 2022).

3. Pillars: Definition and Structural Role

Given integer $s \ge 1$, a pillar is defined as follows:

• Two vertex-disjoint cycles $C_1 = v_1 v_2 \ldots v_s v_1$ and $C_2 = w_1 w_2 \ldots w_s w_1$.
• A collection of $s$ internally vertex-disjoint paths $\{Q_i : 1 \le i \le s\}$, each a $v_i$–$w_i$ path of the same length $\ell$, and all such paths are internally vertex-disjoint and have pairwise disjoint interiors.
• The pillar $P$ is the union $P = C_1 \cup C_2 \cup \bigcup_{i=1}^s Q_i$.

This structure generalizes to "K$_k$-pillars" (systems of $k$ equal cycles with all intercycle matchings subdivided equally), as shown in Theorem 1.2 of (Fernández et al., 2022).

4. Proof Strategies and Key Techniques

Erdős–Gyárfás for $P_{10}$-Free Graphs

The main result in (Hu et al., 2023) is:

• Theorem 1.1: Every $P_{10}$-free graph $G$ with $\delta(G) \ge 3$ contains a 4-cycle or an 8-cycle.

Proof Outline:

• A "hole" is an induced cycle. A "good hole" is an induced cycle minimal in length (at least 5) and maximizing a set $I_C$ measuring triangle-edges.
• By structural lemmas, under the assumption of no $C_4$ or $C_8$, the length of a minimal good hole in $G$ is forced to $5$, and attaching prescribed paths to each vertex (as per Lemmas 2.5–2.6) necessarily leads to a forbidden $P_{10}$ or produces a $C_4$ or $C_8$.
• The proof iterates over subcases, exploiting precise restrictions on cycle attachments and induced subgraphs, ultimately resolving the conjecture for $P_{10}$-free graphs (Hu et al., 2023).

Thomassen’s Pillar Conjecture

(Fernández et al., 2022) proves Conjecture A.10 for an unspecified but absolute constant $C$.

Key Steps:

1. Reduction to Expanders: Apply a theorem to extract a large bipartite subexpander $H \subseteq G$ with strong expansion and minimum degree.
2. Krakens (Partial Pillars): Build "krakens," which are cycles with many disjoint short legs extending from designated vertices. Extract enough krakens so that two have equal cycle length.
3. Hand-in-Hand Lemma: For two krakens of the same cycle length, use an "adjuster" construction and expansion arguments to build $s$ vertex-disjoint paths of prescribed length joining corresponding legs.
4. Assembly: Combine two such krakens and $s$ paths to assemble the desired pillar, maintaining expansion and diameter bounds throughout.
5. Algorithmic Construction: Every step is presented as polynomial-time in $n$.

This approach also extends to $K_k$-pillars for each $k$.

5. Algorithmic and Quantitative Aspects

All steps outlined for constructing pillars in (Fernández et al., 2022) are algorithmic and polynomial-time:

• Expander extraction via iterative deletion based on expansion sets (using network flows).
• Kraken construction with BFS and greedy matchings.
• Disjoint kraken extraction and creation of small-diameter vertex libraries with greedy algorithms.
• Final path assembly using limited augmentations and BFS/DFS in expanders.

Quantitative bounds remain enormous and implicit; improving these (for example, reducing the required minimum degree from a value like $10^{10^{10}}$ to more feasible growth rates) represents a challenging open direction.

6. Extensions, Implications, and Open Problems

Key extensions and open problems include:

• K$_k$-Pillars: There exists a constant $C(k)$ so that $\delta(G) \ge C(k)$ forces a $K_k$-pillar (Theorem 1.2, (Fernández et al., 2022)).
• Forcibility Boundary: Some 3-regular subgraphs are not forced by any constant minimum degree (Pyber–Rödl–Szemerédi). It is open which "near-3-regular" families are so forced.
• Nested-Cycle Conjecture: Thomassen conjectured the forcibility of $k$ nested, non-crossing cycles for all fixed $k$; only the $k=2$ case has been settled.
• Parameter Improvement: Reducing the constant $C$ needed for pillar forcing, potentially approaching towers of logarithms, is a major open direction both for theory and practical extremal graph theory applications.
• Structural Expansion Methods: The introduction of sublinear expanders, krakens, and adjusters is anticipated to see further use in algorithmic and quantitative combinatorics (Fernández et al., 2022).

7. Conclusion and Significance

Conjecture A.10, in its Erdős–Gyárfás and Thomassen pillar formulations, marks foundational milestones in extremal graph theory. Recent advances have resolved both conjectures within substantial graph subclasses and, for the pillar problem, in full generality with large but explicit degree bounds. The core ideas—expansion, cycle linkage, forbidden induced subgraphs, and innovative algorithmic constructions—advance understanding of how local degree properties dictate rich, global cycle embeddings. The continuing challenge lies both in tightening quantitative thresholds and in extending these techniques to fuller classes of regular and near-regular subgraphs.

References:

• "Erdős–Gyárfás Conjecture for $P_{10}$-free Graphs" (Hu et al., 2023)
• "How to build a pillar: a proof of Thomassen's conjecture" (Fernández et al., 2022)