Pair-Trace Absorption Certificates for Regular Induced Subgraphs

Abstract: We study a fixed-core absorption problem for regular induced subgraphs. A set is q-modular if all induced degrees are congruent modulo q. Given a q-modular witness A and a retained core U subset A, we ask when deleting equal-trace q-tuples from A\U can make U into a 2q-modular witness. The main contribution is a finite absorption-or-obstruction certificate. We give an exact quotient formula for the deletion-tail obstruction in complement-orbit coordinates: the correct expression uses oriented differences n_B - n_{U\B}, not sums. Equal-trace q-tuples absorb exactly the span of their trace classes in F_2^U / 1_U. In particular, a connected graph of q-heavy two-point traces on U, together with one odd trace when |U| is even, absorbs every top-bit defect by deleting at most q(|U|-1) tail vertices. If fixed-core absorption fails, the obstruction is an explicit even parity cut of U. We also record the parity base, the terminal modular criterion, and a conditional modular-witness threshold theorem explaining the relevance to the Erdos-Fajtlowicz-Staton problem. The paper does not claim to solve that problem or to improve the general lower bound for F(n).

• The paper introduces a modular dyadic lifting framework using q-modular sets to align vertex degrees toward full regularity.
• It designs explicit linear-algebraic absorption certificates based on trace classes and parity-cut obstructions for effective verification.
• The method reveals critical structural insights for absorption in graphs, offering algorithmic clarity and guiding future progress on the Erdős–Fajtlowicz–Staton conjecture.

Formal Summary of "Pair-Trace Absorption Certificates for Regular Induced Subgraphs" (2604.23882)

Introduction and Context

The paper addresses the problem of identifying large regular induced subgraphs within arbitrary graphs, focusing on the function $F(n)$: the largest $k$ such that any $n$-vertex graph contains a regular induced subgraph with $k$ vertices. This is tightly connected to the Erdős–Fajtlowicz–Staton conjecture, which asks whether $F(n)/\log n \to \infty$ as $n\to\infty$. The challenge centers on the rigidity of exact regularity, contrasting with more flexible notions like repeated degrees or near-regularity.

The authors develop a modular "dyadic" lifting framework wherein regularity modulo $q$ is progressively lifted to higher moduli, with each absorption step seeking to eliminate "top-bit" modular defects. The contribution is not in establishing new lower bounds for $F(n)$, but rather in formalizing explicit, checkable algebraic and combinatorial certificates that witness the success or failure of such modular lifting steps.

Modular Absorption Framework

The methodology hinges on the notion of $q$-modular sets: subsets of vertices whose internal degrees are congruent modulo $q$. The dyadic program attempts to iteratively "lift" a $k$0-modular set to a $k$1-modular one, ultimately reaching genuine regularity when the retained set has size at most $k$2. The central technical objects are:

• Trace classes: The pattern, for a vertex outside a fixed "core," of which core vertices are adjacent to that vertex.
• Equal-trace $k$3-tuples (twin blocks): These are sets of $k$4 vertices outside the core all with the same trace, which can be deleted to manipulate the modular defect of the core in a controlled manner.

A fixed-core absorption step seeks to delete disjoint equal-trace $k$5-tuples so as to synchronize the degrees of a core $k$6 modulo $k$7. The obstruction and absorption criteria for such steps are distilled into precise linear-algebraic certificates.

Main Results and Certificate Characterizations

Conditional Dyadic Threshold Theorem

The paper proves that, under the assumption that $k$8-modular to $k$9-modular lifting can be performed with polynomial loss, i.e., if every sufficiently large $n$0-modular set contains a $n$1-modular subset of prescribed (exact) size, then $n$2. Thus, if these dyadic lifting steps can be successfully iterated, the desired superlogarithmic growth of $n$3 follows.

Tail Obstruction and Quotient Formula

The critical algebraic obstruction at each dyadic absorption step is the coset of the core’s tail-counting function modulo constant functions. The exact obstruction is shown to be governed by oriented differences in trace class multiplicities: for each pair of complementary traces $n$4, the obstruction to absorption is $n$5, not their sum.

Linear Algebraic Absorption Certificate

For a fixed core $n$6 of size $n$7 in a $n$8-modular witness $n$9, let $k$0 denote the "top-bit" degree label of $k$1. The absorption step (synchronizing $k$2 modulo $k$3) is possible if and only if $k$4 (the target defect vector) lies in the $k$5-span of the trace vectors $k$6 for traces $k$7 appearing at least $k$8 times in $k$9. This is efficiently checkable via Gaussian elimination.

Strongly, the result quantifies: full absorption of arbitrary top-bit defects can be guaranteed if the available traces span the codimension-one space $F(n)/\log n \to \infty$0, in which case $F(n)/\log n \to \infty$1 deletions suffice.

Graph-Theoretic Certificates and Obstruction Duality

When the pair-trace graph (on $F(n)/\log n \to \infty$2) formed by traces appearing at multiplicity at least $F(n)/\log n \to \infty$3 is connected (and, for even $F(n)/\log n \to \infty$4, some odd-cardinality trace is available), the space is spanned and absorption is possible. Conversely, failure to synchronize is witnessed by a parity-cut: an even subset $F(n)/\log n \to \infty$5 orthogonal to all available trace classes but detecting the top-bit defect. This duality yields short, concrete failure certificates.

Algorithmic Implications

The entire absorption criterion is formalized as a finite linear system over $F(n)/\log n \to \infty$6, facilitating polynomial-time verification for any fixed core. Additionally, the trace-reservoir model quantifies the probability of absorption in random settings, showing that with a sufficiently large and randomly distributed reservoir, absorption holds with high probability.

Practical and Theoretical Implications

The presented framework does not directly improve lower bounds for $F(n)/\log n \to \infty$7. Rather, its impact is twofold:

• Certifiability: Every dyadic absorption attempt has a short witness (algorithmic certificate) of either success (a trace class decomposition) or failure (an explicit parity-cut obstruction).
• Structural Clarity: The method reveals what structural properties a graph must have to resist lifting, viz., insufficient trace diversity or pair-trace graph connectivity. As demonstrated by calibrations (e.g., perfect graphs, bounded neighborhood diversity), graphs with large homogeneous sets or low trace complexity are not obstructions.

The framework thus isolates the "hard instance" regime: high trace complexity with no large clique/independent set/twin class. It suggests that any future progress toward the Erdős–Fajtlowicz–Staton conjecture via dyadic methods must overcome the challenge of guaranteeing trace-richness or similar structure in arbitrary graphs.

Future Directions

The key open direction is global trace availability: given an arbitrary $F(n)/\log n \to \infty$8-modular witness, can one force the existence of enough disjoint equal-trace $F(n)/\log n \to \infty$9-tuples to enable full absorption via density or exchange arguments? Alternatively, must an extremal construction instead admit a regularizing substructure as guaranteed by the parity-cut certificate? Bridging this gap, either by constructive or probabilistic means, is the main unresolved task.

Additionally, further sharpening or extending such certificate frameworks could make them amenable to computer-assisted enumeration or more specialized results in restricted graph classes.

Conclusion

The authors provide a comprehensive, algorithmically verifiable certificate framework for one-step fixed-core modular absorption in the search for large regular induced subgraphs. This decomposes the modular lifting problem into precise algebraic and combinatorial statements, clarifying both what is possible locally and what must fail globally in hard cases. While not resolving the superlogarithmic growth of $n\to\infty$0, these results delineate the technical frontiers and essential ingredients for any approach based on modular absorption and trace structure.