An Explicit $O(r\log r)$ Threshold for Attaining the Semple--Steel Bound with $r$-State Characters

Abstract: Let $d_r(n)$ be the maximum, over all binary phylogenetic trees with $n$ leaves, of the minimum number of $r$-state characters required to define the tree. Semple and Steel proved that $d_r(n)\geq\lceil(n-3)/(r-1)\rceil$, and Bordewich and Semple proved that equality holds for each fixed $r$ and all sufficiently large $n$. We study the corresponding threshold $n_r$, the least $N$ for which equality holds for every $n\geq N$. The Bordewich--Semple construction yields an explicit polynomial upper bound of order $O(r<sup>5)$ for this threshold. We prove the near-linear estimate [ 3r+1\leq n_r\leq \ceil{64(r-1)\log_2(r+1)}+3\qquad(r\geq4). ] The proof constructs, for every binary phylogenetic tree with $m=n-3$ internal edges, a linked quartet certificate whose conflict graph has maximum degree at most $16\lceil\log_2(m+2)\rceil+4$. Equitable coloring then packs the certificate into exactly $\lceil m/(r-1)\rceil$ $r$-state characters once $m\geq\lceil64(r-1)\log_2(r+1)\rceil$. We also include the lower bound $n_r\geq3r+1$, obtained from the snowflake obstruction, and state the natural conjecture that this lower bound is the exact threshold for all $r\geq4$. The conjectural endpoint is consistent with the known small-state thresholds: $n_4=13$ and $n_5=16$, while the cases $r=2,3$ are also explicitly classified.

• The paper establishes an explicit O(r log r) bound that determines the minimum leaves for uniquely defining binary phylogenetic trees using r-state characters.
• It leverages linked quartet certificates and equitable coloring of a conflict graph to improve the threshold from a polynomial to a near-linear growth rate.
• The framework refines previous bounds and sets the stage for proving the conjectured optimal threshold of 3r+1 for r-state phylogenetic reconstruction.

An Explicit $O(r\log r)$ Threshold for Attaining the Semple–Steel Bound with $r$-State Characters

Introduction and Context

This paper targets a central question in the combinatorial theory underpinning phylogenetic tree reconstruction: for a fixed number of states $r$, what is the least number of leaves $n$ required such that every binary phylogenetic tree on $n$ leaves can be uniquely defined by the minimum number of $r$-state convex characters prescribed by the Semple–Steel bound? Specifically, the authors focus on the explicit determination and bounding of the threshold $n_r$ beyond which the lower bound is always attained, with a primary goal of sharpening previous polynomial bounds.

The definition of $d_r(n)$ encapsulates the core difficulty: for all binary phylogenetic trees with $n$ leaves, what is the maximum, over all such trees, of the minimal number of $r$-state characters required to uniquely identify the tree? Semple and Steel established the lower bound $r$0, and Bordewich and Semple previously proved that this is tight for each fixed $r$1 and all sufficiently large $r$2 [Bordewich & Semple, SIAM J. Discrete Math., 2015]. However, tight quantitative information on the growth rate of the threshold $r$3 as a function of $r$4 has, until now, been missing—prior explicit bounds were polynomial of order $r$5.

Main Results

This paper sets a new benchmark by establishing a near-linear upper bound: $r$6 That is, for all $r$7, whenever $r$8, every binary phylogenetic tree with $r$9 leaves can be defined by exactly $r$0 $r$1-state characters. This is notable for bringing the gap between lower and upper thresholds within a logarithmic factor of the state count; the known lower bound is the so-called "snowflake" obstruction at $r$2.

The lower bound $r$3 is derived from explicit obstruction examples: specifically, trees containing an internal "snowflake" configuration, which preclude definition by the minimum number of characters even when $r$4 is relatively large. Recent structural results for $r$5 support the conjecture that the lower bound is in fact exact for all $r$6.

Technical Contributions

The crux of the improvement lies in reframing the character-packing problem as an equitable coloring problem on a conflict graph derived from a linked system of quartet certificates. The technical path can be summarized as follows:

• Linked Quartet Certificates: For each binary phylogenetic tree, the authors construct a system of quartets, one distinguishing each internal edge, organized such that adjacent internal edges correspond to quartets sharing three taxa (linkage). The construction leverages short "rotational" transition paths in the tree's internal subgraph, and is proven to have a maximum conflict degree of $r$7, a substantial improvement over previous constructions.
• Conflict Graph and Equitable Coloring: The conflict graph, where internal edges are adjacent if their identifying quartet paths overlap, determines packing complexity. The logarithmic upper bound on maximum degree (proved to be at most $r$8 where $r$9 is the number of internal edges) enables the application of the Hajnal–Szemerédi theorem on equitable coloring. As a result, one can partition the quartets into the minimum number of blocks, each corresponding to an $n$0-state character, as soon as $n$1 exceeds the established threshold.
• Sharpness of Argument: The explicit lower bound construction based on the snowflake shows the tightness up to a logarithmic factor, and the case analysis for small $n$2 demonstrates that the framework coheres with known exact values.

Implications and Future Directions

Practical Implications

The improved $n$3 threshold for defining trees with $n$4-state characters significantly advances the efficiency of combinatorial algorithms in reconstructing phylogenetic trees from multi-state data. For practitioners, this means that for all but the smallest $n$5, the minimal number of required characters for unique tree definition is practically achievable in data-rich settings, with the underpinning certificate construction amenable to explicit algorithmic implementation.

Theoretical Significance

The emergence of logarithmic conflict structure highlights a fundamental constraint in the organization of tree-defining certificates, reflecting an inherent bottleneck in the linkage of minimal obstructions in dense trees. The linear lower bound, apparently tight, suggests the existence of a potentially universal combinatorial mechanism for minimal character systems, hinting at a deep linkage between tree rigidity and equitable packing in conflict graphs coming from quartet systems.

Open Problems and Speculative Future Work

The next step, as conjectured, is to prove $n$6 for all $n$7 and to construct explicit optimal character sets for all trees once this bound is reached. Achieving this would likely require new certificate structures—possibly abandoning the linked quartet requirement in favor of block certificates with bounded overlap or alternative combinatorial techniques for achieving minimal conflict.

Two central questions remain:

1. Bounded-Degree Linked Certificates: Whether every binary phylogenetic tree admits a linked edge-quartet certificate with conflict graph of uniformly bounded maximum degree (independent of $n$8). An affirmative answer would immediately yield $n$9.
2. Semidyadic Block Packings: Whether the internal edges can always be partitioned into blocks of size at most $n$0, each producing characters whose closures together define the tree. This structural property would yield a linear threshold.

Both questions probe the fundamental geometry of combinatorial obstructions in the space of tree-defining characters and have ramifications for the computational complexity of perfect phylogeny reconstruction in the multi-state setting.

Conclusion

The paper provides a sharp improvement, reducing the quantitative bound for the sharpness threshold of the Semple–Steel lower bound to $n$1, by introducing an efficient linked quartet certificate construction with logarithmic conflict degree. By reframing the packing problem as one of equitable coloring, the authors bridge a substantial polynomial-to-logarithmic gap in known results. Though the universal linear threshold remains open, the results here narrow the window for further advances and establish new techniques for addressing combinatorial identification in phylogenetic reconstruction.

---

Reference:

"An Explicit $n$2 Threshold for Attaining the Semple--Steel Bound with $n$3-State Characters" (2606.06905)