Henning–Yeo Graphs: Extremal Matching Benchmark

• Henning–Yeo graphs are an infinite two-parameter family of r-regular graphs characterized by a perfect fractional matching (n/2) alongside suboptimal integral matchings.
• Their construction splits into even-degree series with explicit integral-to-fractional ratios and odd-degree series that use global degree-balancing for structure.
• They serve as extremal test instances in matching theory, providing critical insights for evaluating APS conjecture bounds under the EPR model.

Searching arXiv for the cited paper and closely related work on Henning–Yeo graphs and the APS conjecture. Henning–Yeo graphs are an infinite two-parameter family of $r$-regular graphs whose maximum matchings realize the tight lower bounds of the classical Tutte–Berge theorem, while still admitting a perfect fractional matching of value $n/2$. In the setting of the EPR model, this combination makes them extremal test instances for comparing upper bounds based on maximum-weight fractional matching (MWFM) and maximum-weight matching (MWM): by construction they are as “unfavourable” as possible for integral matchings, yet they retain the full fractional benchmark (Tao et al., 14 Jul 2025).

1. Extremal position within matching theory

The 2025 study "Testing APS conjecture on regular graphs" presents Henning–Yeo graphs as a special class of regular graphs introduced by Henning and Yeo in 2007. Their defining property is that the maximum integral matching attains a tight lower bound, whereas the fractional matching remains perfect. In the terminology of the source, they lie farthest below the perfect-matching bound $n/2$ while still admitting a perfect fractional matching of value $n/2$ (Tao et al., 14 Jul 2025).

This places the family at a sharp interface between integral and fractional matching theory. The graphs are designed to saturate the Tutte–Berge barrier: fractional matching gives $n/2$, but every integral matching must leave at least a constant fraction of vertices unmatched. A plausible implication is that they are not merely extremal examples for classical matching bounds, but also adversarial instances for any framework that attempts to substitute fractional matching by integral matching without significant loss.

2. Even-degree series

For even degree $k \ge 4$ and integer parameter $p \ge 1$, the construction begins with any $k$-regular (multi)graph $X_p$ on $p$ vertices. One then forms the quasi-complete graph

$n/2$0

that is, the complete graph on $n/2$1 vertices minus a single edge $n/2$2. If the two open vertices $n/2$3 are each attached to one external neighbor, the resulting block becomes $n/2$4-regular. Each edge $n/2$5 of $n/2$6 is then replaced by a fresh copy of $n/2$7: the edge $n/2$8 is deleted, then $n/2$9 is joined to $n/2$0 and $n/2$1 to $n/2$2. The resulting graph is denoted $n/2$3 (Tao et al., 14 Jul 2025).

The source states that $n/2$4 is $n/2$5-regular and has

$n/2$6

Henning and Yeo prove, by a direct Tutte–Berge-type argument, that

$n/2$7

and that this bound is tight on $n/2$8 (Tao et al., 14 Jul 2025).

The significance of the even series is that the deficit of integral matching is explicit and formulaic. For this branch, the integral-to-fractional ratio is given in closed form, so the family provides a clean asymptotic and finite-$n/2$9 laboratory for testing conjectured matching-based bounds.

3. Odd-degree series

For odd degree $n/2$0, the construction uses a bipartite template $n/2$1 with parts

$n/2$2

where each $n/2$3 is joined to $n/2$4 consecutive vertices of $n/2$5 in a cyclic fashion. One then takes $n/2$6 on vertices $n/2$7, deletes a perfect-matching-like set of $n/2$8 edges in the first $n/2$9 vertices, and also deletes the two edges $n/2$0 and $n/2$1 to obtain a quasi-complete $n/2$2. For each $n/2$3 whose degree $n/2$4, exactly $n/2$5 fresh copies of this quasi-complete block are attached by identifying $n/2$6 with $n/2$7. The final graph is denoted $n/2$8 (Tao et al., 14 Jul 2025).

For this odd series, the graph has

$n/2$9

The corresponding tight lower bound is

$k \ge 4$0

and the source states that this inequality is attained on $k \ge 4$1 (Tao et al., 14 Jul 2025).

Compared with the even series, the odd construction is structurally more elaborate because it mixes a cyclic bipartite template with quasi-complete attachments. This suggests that the odd-degree extremality is less reducible to a single local gadget and instead depends on a global degree-balancing mechanism.

4. Enumerative formulas and matching ratios

Across both series, the source summarizes the order-size relation as

$k \ge 4$2

The tight lower bound on maximum-matching size is

$k \ge 4$3

The perfect fractional matching size is

$k \ge 4$4

The integral-to-fractional ratio is defined by

$k \ge 4$5

and for even $k \ge 4$6 the source gives

$k \ge 4$7

For comparison with the APS bound, the shifted ratio is

$k \ge 4$8

These quantities fall strictly below $k \ge 4$9 for small $p \ge 1$0, while $p \ge 1$1 and $p \ge 1$2 as $p \ge 1$3 (Tao et al., 14 Jul 2025).

The formulas separate two notions of near-perfectness. Fractionally, the graph always behaves like a regular graph with a perfect matching surrogate. Integrally, the realized matching can be significantly smaller. The ratios $p \ge 1$4 and $p \ge 1$5 quantify that separation directly and therefore control how difficult it is to distinguish MWFM-based and MWM-based bounds.

5. Structural characteristics and low-degree cases

Each $p \ge 1$6 and $p \ge 1$7 is exactly $p \ge 1$8-regular and highly symmetric inside each quasi-complete block. The family may contain multi-edges when realized as weighted graphs, but can be unweighted. The graphs are not bipartite in general, because the even blocks $p \ge 1$9 introduce odd cycles. By design, they saturate the Tutte–Berge barrier: fractional matching gives $k$0, but every integral matching must leave at least a constant fraction of vertices unmatched (Tao et al., 14 Jul 2025).

The low-degree examples emphasize where the gap is largest. For $k$1 in the odd series,

$k$2

and

$k$3

so

$k$4

For $k$5 in the even series,

$k$6

and

$k$7

hence

$k$8

while the source lists

$k$9

and separately states that, for large $X_p$0, numerically $X_p$1 once one accounts for rounding of $X_p$2. For $X_p$3, the source reports

$X_p$4

The largest relative gap therefore occurs at small $X_p$5 (Tao et al., 14 Jul 2025).

The low-$X_p$6 regime is the most informative for extremal testing. This suggests that the family is especially useful not because the asymptotic gap persists indefinitely, but because finite low-degree instances maximize the distinction between integral and fractional behavior.

6. Role in the APS conjecture and EPR-model testing

The APS conjecture asserts for the EPR model on any weighted graph $X_p$7 that

$X_p$8

strengthening the known bound

$X_p$9

On a regular graph of degree $p$0, one always has a perfect fractional matching with

$p$1

Henning–Yeo graphs force every integral matching to achieve only

$p$2

so the gap between the two conjectured upper bounds is as large as possible. For this reason, the source identifies them as ideal graphs on which to test conjectures that replace MWFM by MWM (Tao et al., 14 Jul 2025).

The same study reports a new algorithm, Fractional Entanglement Distribution (FED), based on quasi-homogeneous fractional matchings, and uses it to obtain high-accuracy energy estimates for the EPR model on Henning–Yeo graphs. The logic is explicit: if an approximate quantum-state algorithm could return

$p$3

then it would strictly exceed $p$4 on these graphs and thereby falsify the APS conjecture, unless the approximation ratio were pushed above $p$5. In practice, however, even the FED algorithm of Tao–Zuo (2025) yields energy ratios

$p$6

so a persistent gap remains, and the reported numerical results do not show any evidence that the APS conjecture could be violated (Tao et al., 14 Jul 2025).

Within this testing framework, Henning–Yeo graphs function as worst-case regular instances for distinguishing two upper-bound paradigms. Their importance is therefore dual: they are extremal objects in matching theory and calibrated stress tests for conjectural quantum-energy bounds.