Complementary Fractional Haemers Bound

• The paper establishes a dual subspace framework that defines the complementary fractional Haemers bound as an upper bound on Shannon capacity via dual (n,d)-representations.
• It shows that under the generalized Mycielski construction, the bound transforms in a closed-form manner, matching the clique number in key cases.
• The methodology utilizes tensor products and dimension counting to derive explicit estimates and conjecture the exactness of the bound in broader graph families.

Searching arXiv for the cited papers to ground the article in current preprint metadata. Searching arXiv for "The Fractional Haemers Bound of The Mycielski Construction" and "On a fractional version of Haemers' bound". The complementary fractional Haemers bound is a graph parameter associated with a graph $G$ and a field $\mathbb{F}$, defined through dual subspace representations and denoted $\bar{\mathcal{H}}_f(G;\mathbb{F})$. It is the complement-side analogue of the fractional Haemers bound, and it is studied as an upper bound on Shannon capacity via its place in the asymptotic spectrum of graphs (Csonka, 13 Jul 2025). In recent work on Mycielski graphs, the parameter is shown to satisfy a general upper bound under the generalized Mycielski construction $M_r(G)$, and this bound becomes exact whenever $\bar{\mathcal{H}}_f(G;\mathbb{F})$ coincides with the clique number $\omega(G)$ (Csonka, 13 Jul 2025). Related foundational work established the fractional Haemers bound, its equivalent formulations, its multiplicativity under strong products, and its duality relation with the corresponding parameter of the complement graph (Bukh et al., 2018).

1. Definition and basic formulation

Let $G$ be a finite simple graph and $\mathbb{F}$ any field. A dual $(n,d)$-representation of $G$ over $\mathbb{F}$0 is a collection of $\mathbb{F}$1-dimensional subspaces

$\mathbb{F}$2

such that for every vertex $\mathbb{F}$3,

$\mathbb{F}$4

where $\mathbb{F}$5 is the neighborhood of $\mathbb{F}$6 and the sum denotes the span of those subspaces (Csonka, 13 Jul 2025). The complementary fractional Haemers bound is then defined by

$\mathbb{F}$7

From the definition, it is immediate that

$\mathbb{F}$8

where $\mathbb{F}$9 is Haemers’s original complementary bound corresponding to the case $\bar{\mathcal{H}}_f(G;\mathbb{F})$0 (Csonka, 13 Jul 2025).

The terminology “complementary” reflects the relation to the fractional Haemers bound of the complement graph. In the exposition of the fractional Haemers bound, one defines

$\bar{\mathcal{H}}_f(G;\mathbb{F})$1

as the same parameter evaluated on the complement graph, and the equivalent formulations of $\bar{\mathcal{H}}_f(G;\mathbb{F})$2 carry over verbatim to $\bar{\mathcal{H}}_f(G;\mathbb{F})$3 (Bukh et al., 2018). In that sense, $\bar{\mathcal{H}}_f(G;\mathbb{F})$4 is the natural complement-side object when one works in the dual subspace language emphasized in the Mycielski analysis (Csonka, 13 Jul 2025).

Bukh–Cox showed that $\bar{\mathcal{H}}_f(G;\mathbb{F})$5 lies in the asymptotic spectrum of graphs and thus gives an upper bound on Shannon capacity (Csonka, 13 Jul 2025). The foundational fractional Haemers literature likewise places the non-complementary parameter in the standard inequality chain

$\bar{\mathcal{H}}_f(G;\mathbb{F})$6

with multiplicativity under strong products providing the Shannon-capacity bound (Bukh et al., 2018). This suggests that the complementary formulation is best understood as part of the same rank- and subspace-based framework.

2. Relation to the fractional Haemers bound and complement duality

The fractional Haemers bound $\bar{\mathcal{H}}_f(G;\mathbb{F})$7 admits three equivalent formulations: a block-matrix definition, an $\bar{\mathcal{H}}_f(G;\mathbb{F})$8-representation by matrix pairs $\bar{\mathcal{H}}_f(G;\mathbb{F})$9, and a coordinate-free subspace formulation (Bukh et al., 2018). In the subspace formulation,

$M_r(G)$0

This condition is the complement-side analogue of the dual representation condition used to define $M_r(G)$1 (Bukh et al., 2018, Csonka, 13 Jul 2025).

A central structural fact in the fractional Haemers theory is the duality inequality

$M_r(G)$2

obtained from the rank-sum inequality for representations of $M_r(G)$3 and $M_r(G)$4 (Bukh et al., 2018). In subspace terms, if $M_r(G)$5 is a subspace representation of $M_r(G)$6 and $M_r(G)$7 one of $M_r(G)$8, then $M_r(G)$9 yields a representation of the complete graph $\bar{\mathcal{H}}_f(G;\mathbb{F})$0, forcing

$\bar{\mathcal{H}}_f(G;\mathbb{F})$1

Taking infima gives the stated bound (Bukh et al., 2018).

This duality is especially transparent on basic examples. For the complete graph $\bar{\mathcal{H}}_f(G;\mathbb{F})$2, one has

$\bar{\mathcal{H}}_f(G;\mathbb{F})$3

and for the 5-cycle $\bar{\mathcal{H}}_f(G;\mathbb{F})$4, which is self-complementary,

$\bar{\mathcal{H}}_f(G;\mathbb{F})$5

(Bukh et al., 2018). These examples are useful reference points for interpreting complement-side quantities such as $\bar{\mathcal{H}}_f(G;\mathbb{F})$6, although the Mycielski paper works directly with the complementary parameter rather than deriving it from the non-complementary side (Csonka, 13 Jul 2025).

The same foundational work shows that $\bar{\mathcal{H}}_f(G;\mathbb{F})$7 is multiplicative under strong products and strictly improves on every classical Haemers-type bound in the sense that it can outperform all bounds obtainable through Haemers’s original construction (Bukh et al., 2018). A plausible implication is that complement-side refinements such as $\bar{\mathcal{H}}_f(G;\mathbb{F})$8 are of interest not merely as formal analogues but as potentially sharper invariants in asymptotic graph theory.

3. Behavior under the generalized Mycielski construction

The generalized Mycielski graph $\bar{\mathcal{H}}_f(G;\mathbb{F})$9 is the object of the main investigation in Csonka’s work (Csonka, 13 Jul 2025). The effect of the Mycielski construction on graph parameters had already been studied for the fractional chromatic number $\omega(G)$0 and the complementary Lovász theta number $\omega(G)$1. Larsen, Propp, and Ullman provided a formula for $\omega(G)$2 in terms of $\omega(G)$3; this was later generalized by Tardif to $\omega(G)$4 for any $\omega(G)$5; and Simonyi and the author gave a similar expression for $\omega(G)$6 in terms of $\omega(G)$7 (Csonka, 13 Jul 2025).

Tardif’s formula is

$\omega(G)$8

Csonka proves that the same functional form persists for the complementary fractional Haemers bound whenever the parameter equals the clique number: $\omega(G)$9 (Csonka, 13 Jul 2025).

This result isolates a “clique-case” regime in which $G$0 behaves exactly like $G$1 under the generalized Mycielski construction. The significance is not merely formal. It shows that, at least when the complementary fractional Haemers bound sits at $G$2, the Mycielski operation modifies this rank-based parameter in a predictable, closed-form way analogous to two other major upper bounds, $G$3 and $G$4 (Csonka, 13 Jul 2025).

For complete graphs this gives an explicit family: $G$5 since $G$6 (Csonka, 13 Jul 2025). This family serves as the exact model case for the general theory.

4. The general upper bound

Theorem 1 of Csonka’s paper states that for every graph $G$7, every field $G$8, and every positive integer $G$9,

$\mathbb{F}$0

(Csonka, 13 Jul 2025). This is the main general estimate for the complementary fractional Haemers bound under Mycielski extension.

The proof begins with a dual $\mathbb{F}$1-representation of $\mathbb{F}$2 whose ratio $\mathbb{F}$3 is arbitrarily close to $\mathbb{F}$4. From this, the construction produces a dual $\mathbb{F}$5-representation of $\mathbb{F}$6 by tensoring each original subspace $\mathbb{F}$7 with carefully chosen coordinate-subspaces $\mathbb{F}$8, where $\mathbb{F}$9 (Csonka, 13 Jul 2025). Two conditions must then be checked: each new subspace $(n,d)$0 has dimension $(n,d)$1, and

$(n,d)$2

The dimension counts use the identity

$(n,d)$3

together with the additivity of dimensions for sums of disjoint $(n,d)$4-blocks (Csonka, 13 Jul 2025). The crucial intersection argument is supplied by a combinatorial lemma on intersections of tensor products of subspaces living in disjoint coordinate-blocks. A direct calculation yields

$(n,d)$5

and infimizing over admissible $(n,d)$6 gives the stated upper bound (Csonka, 13 Jul 2025).

The resulting formula provides explicit estimates even when $(n,d)$7. This suggests that the theorem is useful beyond the exact clique-case, because it transforms information on $(n,d)$8 into a closed-form bound for $(n,d)$9 without requiring an exact representation of the Mycielski graph itself.

5. Tightness in the clique-case

Theorem 2 addresses the extremal case $G$0. For any field $G$1 and integers $G$2,

$G$3

(Csonka, 13 Jul 2025). The upper bound comes from Theorem 1, together with the identity

$G$4

The reverse inequality is proved by taking an arbitrary dual $G$5-representation of $G$6 and deriving a lower bound on $G$7 via successive Grassmann-identity arguments (Csonka, 13 Jul 2025). At each level $G$8, dimension counting and the fact that certain vertex sets form cliques or complete bipartite subgraphs imply lower bounds on various intersections of the representing subspaces. These intersection dimensions satisfy a small system of linear recurrences, recorded as Lemma 3.2 in the summary (Csonka, 13 Jul 2025).

The proof culminates in an inequality

$G$9

which, after an exact summation, is shown to be equivalent to

$\mathbb{F}$00

Hence

$\mathbb{F}$01

establishing equality (Csonka, 13 Jul 2025).

A corollary states that whenever

$\mathbb{F}$02

the upper bound of Theorem 1 is attained with equality (Csonka, 13 Jul 2025). Thus the complete-graph computation is not an isolated example but the prototype for the entire clique-case regime.

6. Examples, comparisons, and conjectural exactness

The paper records the Schl\"afli graph $\mathbb{F}$03 as an example. Haemers showed

$\mathbb{F}$04

while Lovász’s theta bound gives

$\mathbb{F}$05

Applying Theorem 1 yields

$\mathbb{F}$06

(Csonka, 13 Jul 2025). This is an explicit instance in which the general upper bound produces a concrete estimate for a nontrivial Mycielski graph.

The broader comparative context comes from the fractional Haemers literature. The fractional version $\mathbb{F}$07 is described as a common strengthening of both Haemers’ bound and the fractional chromatic number, and it is multiplicative, unlike Haemers’ bound (Bukh et al., 2018). It also satisfies

$\mathbb{F}$08

and can be strictly smaller than both every classical Haemers bound and Lovász’s $\mathbb{F}$09 on some graphs (Bukh et al., 2018). These facts place the complementary parameter within a broader hierarchy of upper bounds on Shannon capacity and related invariants.

Csonka formulates a conjecture that the same Mycielski relation may hold without the hypothesis $\mathbb{F}$10. Specifically, the conjecture is that for every $\mathbb{F}$11, $\mathbb{F}$12, and $\mathbb{F}$13,

$\mathbb{F}$14

(Csonka, 13 Jul 2025). No counterexample is known so far.

This conjecture is notable because exact Mycielski formulas are already known for $\mathbb{F}$15, and analogous behavior has been identified for $\mathbb{F}$16 in the $\mathbb{F}$17 case (Csonka, 13 Jul 2025). This suggests that $\mathbb{F}$18 may fit a wider pattern among fractional and complement-side graph parameters, although the paper presents this only as a conjectural extension.

7. Conceptual significance

The complementary fractional Haemers bound occupies a rank- and subspace-based niche among graph parameters used in Shannon-capacity theory. The foundational exposition emphasizes that fractionalizing Haemers’ rank minimization by allowing block matrices of arbitrarily large block size yields a parameter that is multiplicative under strong products and dominates every classical Haemers-type bound (Bukh et al., 2018). In the complementary setting, the Mycielski results show that the parameter can also be tracked under a nontrivial graph construction by an explicit rational correction term when it coincides with the clique number (Csonka, 13 Jul 2025).

The discussion accompanying Csonka’s results states that the complementary fractional Haemers bound interacts with Mycielski’s construction in a predictable way when it “sits” at the clique number, just as the fractional chromatic number $\mathbb{F}$19 and the Lovász theta number do (Csonka, 13 Jul 2025). The paper further notes that the general upper bound gives new explicit estimates on $\mathbb{F}$20 even when $\mathbb{F}$21, and raises the hope that one may eventually find a “dual” formulation for $\mathbb{F}$22 parallel to those for $\mathbb{F}$23 and $\mathbb{F}$24 (Csonka, 13 Jul 2025).

A plausible implication is that the present theory identifies two complementary research directions. One is structural: determining whether the conjectured exact formula for $\mathbb{F}$25 holds in full generality. The other is representational: clarifying whether a semidefinite- or rank-based dual formulation can be developed for $\mathbb{F}$26 in the same spirit as the known formulations for related parameters. Within the available results, the complementary fractional Haemers bound emerges as a parameter with both asymptotic significance and unexpectedly rigid behavior under the generalized Mycielski construction (Csonka, 13 Jul 2025, Bukh et al., 2018).