A 32-leaf tree requiring six coordinates for an isometric $\ell_\infty$ embedding

Abstract: We disprove the conjecture that every tree with t leaves embeds isometrically into $\ell_\infty^{\lceil \log_2 t\rceil}$. We construct a 32-leaf tree whose least isometric $\ell_\infty$-dimension is six rather than five, and prove that every tree with at most 31 leaves attains the conjectured bound; Brigham et al. had recorded equality through 21 leaves. Thus 32 is the first failure, and the example answers affirmatively a question of Fitzpatrick and Nowakowski from 2000. The same topology has dimension six under every assignment of positive edge lengths, and therefore also disproves the later sharp leaf-threshold conjecture for weighted metric trees.

• The paper constructs a cubic tree with 32 leaves whose path metric requires exactly six coordinates for an isometric embedding into ℓ∞, exceeding the five-coordinate leaf bound.
• It characterizes embedding dimension through orientation covers, proving that topology—not positive edge lengths—determines the dimension for both weighted realizations and vertex metrics.
• An exhaustive computer-assisted census shows that every tree with at most 31 leaves meets the logarithmic lower bound, while open questions concern the long-term growth of the dimension gap.

Overview

This paper resolves a long-standing question about isometric embeddings of finite trees into $\ell_\infty$ spaces. For a finite tree $T$, $\dim_\infty(T)$ denotes the least $m$ such that its path metric embeds isometrically into $(\mathbb{R}^m, \|\cdot\|_\infty)$. The elementary leaf lower bound $\dim_\infty(T)\ge\lceil\log_2 t\rceil$, where $t$ is the number of leaves, was conjectured to be sharp for all $t$: Brigham, Chartrand, Dutton and Zhang conjectured $D(t)=\lceil\log_2 t\rceil$ (with $D(t)$ the maximum over all $T$0-leaf trees), having verified equality through 21 leaves (2608.16288). Chalmers disproves this by constructing a cubic 32-leaf tree with $T$1, while simultaneously proving that every tree with at most 31 leaves attains the bound. Thus 32 is exactly the first failure of the conjectured dimension.

The counterexample also refutes the weighted version of the conjecture due to Aksoy, Kılıç and Koçak, because the paper's central structural result shows that the embedding dimension depends only on the topology, not on positive edge lengths.

Orientation covers

The technical core is a three-way characterization of embedding dimension via orientation covers. An orientation cover of size $T$2 is a family of $T$3 orientations of the edge set such that every pair of leaves has its unique path directed in at least one orientation. The main proposition establishes that the following are equal: (i) the least $T$4 for an isometric embedding of the weighted metric realization into $T$5; (ii) the same for the vertex set alone under the induced path-length metric; and (iii) the least size of an orientation cover.

The proof directions are short but informative. Given a cover, each coordinate is built as a signed path integral from a root — a 1-Lipschitz function whose slope has absolute value one on every edge — and any pair of points lies on a leaf-to-leaf geodesic directed in some coordinate, so the sup-norm realizes the distance. Conversely, given an isometric embedding of the vertex metric, a coordinate achieving the diameter between two leaves must attain full magnitude with consistent sign along their path, which orients that path; leftover edges are oriented arbitrarily. A consequence recorded in a remark is that, for unit lengths, the constructed coordinates are integers, so the invariant coincides with the Chebyshev-lattice dimension studied earlier, and the counterexample applies equally to the vertex-set reading of the weighted conjecture.

An important corollary of this characterization is weight-independence: since $T$6 ignores edge lengths entirely, a single topology witnesses failure across all positive weightings at once.

The counterexample

The tree $T$7 is given explicitly by a parent map on vertices $T$8: it is cubic, with 32 leaves and 30 degree-three vertices, and its three branches at vertex 0 contain 4, 13 and 15 leaves. Six orientations covering all leaf pairs are supplied as hexadecimal masks over the lexicographically ordered edges, giving $T$9.

The lower bound $\dim_\infty(T)$0 rules out five orientations by an exact finite computation. Rooting at vertex 0, each branch carries a state: the antichain of inclusion-minimal coordinate sets $\dim_\infty(T)$1 over which the attachment-to-leaf path is consistently directed, taken up to permutation of coordinates. Internal branches combine child states through a recurrence governed by two parameters — the sign-difference set $\dim_\infty(T)$2 between child attachment edges and the agreement set $\dim_\infty(T)$3 with the parent edge — subject to the cross-child coverage condition that $\dim_\infty(T)$4 for all child masks. Propagated masks are reduced to minimal antichains, and dominated states are pruned: if state $\dim_\infty(T)$5 is dominated by $\dim_\infty(T)$6, every completion covering $\dim_\infty(T)$7's masks covers $\dim_\infty(T)$8's, since all subsequent tests are nonempty intersections preserved by the propagation rule. A lemma proves completeness of this pruning: any genuine five-orientation cover survives to the root compatibility check, which requires pairwise-compatible branch states with appropriate sign-difference sets. Exhausting the retained states (1, 6 and 9 nondominated states for the three root branches) and their relative coordinate permutations yields no compatible triple, so five orientations cannot cover $\dim_\infty(T)$9. The computation is exact and reproducible from archived ancillary material, including an optional rerun without dominance pruning.

Combining both bounds gives $m$0, hence $m$1 for every positive weighting $m$2.

The census up to 31 leaves

The second theorem pins down the transition point. For $m$3, the author enumerates all unlabelled cubic topologies via rooted unordered binary branch codes, retaining each topology once at its least edge-rooting; all 26,049,854 topologies admit an $m$4-orientation cover with $m$5. For $m$6, uniqueness of the leaf centroid (the centroid cannot be an edge when the leaf count is odd) reduces enumeration to unordered triples of branches at that vertex; all 75,021,750 triples admit five orientations. The case $m$7 follows from monotonicity $m$8 together with the leaf lower bound, and small cases are immediate. Reductions from Fitzpatrick–Nowakowski justify restricting attention to cubic trees throughout.

For the upper bound at $m$9: deleting a leaf from a cubic 32-leaf tree leaves a 31-leaf tree admitting a five-orientation cover, which extends arbitrarily over the deleted edge; a sixth orientation directing every edge away from the deleted leaf covers all paths involving it. Hence $(\mathbb{R}^m, \|\cdot\|_\infty)$0 exactly.

Consequences and open questions

The results determine $(\mathbb{R}^m, \|\cdot\|_\infty)$1 completely through 32: the leaf bound is exact through 31 and first fails at 32, where $(\mathbb{R}^m, \|\cdot\|_\infty)$2. Writing $(\mathbb{R}^m, \|\cdot\|_\infty)$3, we have $(\mathbb{R}^m, \|\cdot\|_\infty)$4 through 31 and $(\mathbb{R}^m, \|\cdot\|_\infty)$5. The paper poses whether $(\mathbb{R}^m, \|\cdot\|_\infty)$6 for all $(\mathbb{R}^m, \|\cdot\|_\infty)$7, and notes that it remains unknown whether $(\mathbb{R}^m, \|\cdot\|_\infty)$8 is bounded at all, or whether $(\mathbb{R}^m, \|\cdot\|_\infty)$9. These questions frame the gap between the known upper bounds (constant below 1.4405 times $\dim_\infty(T)\ge\lceil\log_2 t\rceil$0 from Brigham et al., $\dim_\infty(T)\ge\lceil\log_2 t\rceil$1 from Fitzpatrick–Nowakowski) and the now-failed constant-one conjecture.

Two limitations deserve plain statement. First, the nonexistence of a five-orientation cover rests on a computer-assisted exhaustive search; its reliability depends on the correctness of the completeness lemma for the pruning recurrence and on the archived verification code rather than on an independently checkable hand proof. Second, the census arguments are enumerative and provide no structural explanation for why dimension must exceed the leaf bound at 32 leaves specifically; the mechanism forcing the extra coordinate in this particular topology is not isolated analytically.

Conclusion

The paper settles a 25-year-old problem negatively but sharply: the conjectured identity $\dim_\infty(T)\ge\lceil\log_2 t\rceil$2 holds precisely for $\dim_\infty(T)\ge\lceil\log_2 t\rceil$3 and fails at $\dim_\infty(T)\ge\lceil\log_2 t\rceil$4, witnessed by an explicit cubic tree whose dimension is six under every positive edge weighting. The orientation-cover characterization does the double duty of proving weight-independence and reducing both the construction and the exhaustive disproof to finite combinatorial checks. What remains open is the asymptotic behavior of the excess $\dim_\infty(T)\ge\lceil\log_2 t\rceil$5 beyond the first failure.