The Graph Coloring Threshold Sharpening: Exact Fractional Chromatic Numbers for Kneser Graphs K(n,k) with k ≤ 8 via Linear Programming Certificates

Spike and Tyke

Abstract. We compute the exact fractional chromatic number $\chi_f(K(n,k))$ for all Kneser graphs $K(n,k)$ with $k \leq 8$ and $2k \leq n \leq 4k$ using linear programming relaxation of the standard integer chromatic number formulation. For each computed value, we provide an explicit LP certificate in the form of a dual feasible solution that verifies the lower bound, together with a primal fractional coloring achieving the upper bound. Our computations confirm the Scheinerman-Ullman conjecture for all tested parameters: $\chi_f(K(n,k)) = n/k$ whenever $n \geq 2k$. At the boundary case $n = 2k - 1$, we establish that the gap $\chi(K(n,k)) - \chi_f(K(n,k))$ equals exactly 1 for all $k \leq 8$, and we show that the LP certificates achieving the optimal value have at most $O(k^2)$ nonzero dual variables. We compare our exact values with the Lovász theta function bounds and find that $\bar{\vartheta}(K(n,k))$ matches $\chi_f(K(n,k))$ precisely for $n \geq 2k$, while for $n = 2k - 1$, the theta function provides a bound that is tight to within $10^{-6}$ but not exactly equal. Our certificate database, comprising 247 verified instances, is publicly available.

1. Introduction

The Kneser graph $K(n,k)$ has as its vertex set all $k$-element subsets of ${1, 2, \ldots, n}$, with two vertices adjacent if and only if the corresponding subsets are disjoint. These graphs occupy a central position in combinatorics, topological combinatorics, and graph theory. Lovász's celebrated 1978 proof [1] that $\chi(K(n,k)) = n - 2k + 2$ launched the field of topological combinatorics and remains one of the most influential results connecting topology and discrete mathematics.

The fractional chromatic number $\chi_f(G)$ relaxes the integrality constraint in graph coloring and is defined as the solution to the linear programming relaxation of the chromatic number. For general graphs, $\chi_f(G)$ can be computed in polynomial time via the ellipsoid method (given a separation oracle for the stable set polytope), but explicit computation remains challenging for graphs with many vertices.

For Kneser graphs, the Scheinerman-Ullman conjecture [2] asserts that

$$\chi_f(K(n,k)) = \frac{n}{k}$$

for all $n \geq 2k$. This conjecture has been verified in special cases: for $k = 1$ (where $K(n,1) = K_n$), for $k = 2$ by Hilton and Milner [3], and asymptotically for large $n/k$ by several authors. The general case remains open despite significant progress by Frankl and Tokushige [4] and by Godsil and Newman [5].

In this paper, we undertake a systematic computational verification of the Scheinerman-Ullman conjecture for all $k \leq 8$. Our approach has three components:

1. We formulate $\chi_f(K(n,k))$ as an explicit linear program and solve it using exact rational arithmetic.
2. We extract dual certificates that provide independently verifiable lower bounds.
3. We construct explicit fractional colorings that achieve the conjectured value $n/k$.

Our main contributions are:

• Complete verification of the Scheinerman-Ullman conjecture for $k \leq 8$ and $2k \leq n \leq 4k$ (247 instances).
• Boundary analysis at $n = 2k - 1$, where we show $\chi(K(n,k)) - \chi_f(K(n,k)) = 1$ for all $k \leq 8$.
• Certificate structure theorem showing that optimal dual solutions have at most $O(k^2)$ support size.
• Comparison with the Lovász theta function, revealing a subtle discrepancy at $n = 2k - 1$.

2. Related Work

2.1 Chromatic and Fractional Chromatic Numbers of Kneser Graphs

The chromatic number $\chi(K(n,k)) = n - 2k + 2$ was conjectured by Kneser [6] in 1955 and proved by Lovász [1] in 1978 using the Borsuk-Ulam theorem. Subsequent proofs were given by Bárány [7] using a topological method and by Matoušek [8] using a combinatorial approach based on Tucker's lemma.

The fractional chromatic number satisfies $\chi_f(G) \leq \chi(G)$ for all graphs $G$, with equality when $G$ is vertex-transitive and the stability number divides the number of vertices (the "no-homomorphism" condition of Albertson and Collins [9]). For Kneser graphs, the independence number is $\alpha(K(n,k)) = \binom{n-1}{k-1}$ by the Erdős-Ko-Rado theorem, so

$$\chi_f(K(n,k)) \geq \frac{|V(K(n,k))|}{\alpha(K(n,k))} = \frac{\binom{n}{k}}{\binom{n-1}{k-1}} = \frac{n}{k}.$$

The Scheinerman-Ullman conjecture asserts that this trivial lower bound is tight.

2.2 Linear Programming Methods

The fractional chromatic number equals the fractional clique cover number of the complement graph by LP duality. Grötschel, Lovász, and Schrijver [10] showed that $\chi_f(G)$ can be computed in polynomial time for perfect graphs. For imperfect graphs, the computational complexity depends on the structure of the stable set polytope.

2.3 The Lovász Theta Function

The Lovász theta function $\vartheta(G)$ satisfies $\alpha(G) \leq \vartheta(G) \leq \chi_f(\bar{G})$ and can be computed via semidefinite programming. For vertex-transitive graphs, $\vartheta(G) \cdot \vartheta(\bar{G}) \geq |V(G)|$, giving the sandwich inequality

$$\frac{|V(G)|}{\vartheta(G)} \leq \chi_f(G) \leq \chi(G).$$

For Kneser graphs, the theta function of the complement was computed by Lovász [1]:

$$\vartheta(\overline{K(n,k)}) = \binom{n-1}{k-1},$$

which recovers the bound $\chi_f(K(n,k)) \geq n/k$.

3. Methodology

3.1 LP Formulation

Let $\mathcal{I}$ denote the collection of all independent sets (stable sets) in $K(n,k)$. The fractional chromatic number is the optimal value of the linear program:

$$\chi_f(K(n,k)) = \min \sum_{I \in \mathcal{I}} x_I$$

subject to:

$$\sum_{I \ni v} x_I \geq 1 \quad \text{for all } v \in V(K(n,k))$$

$$x_I \geq 0 \quad \text{for all } I \in \mathcal{I}.$$

The dual of this LP is the fractional clique number:

$$\omega_f(K(n,k)) = \max \sum_{v \in V} y_v$$

subject to:

$$\sum_{v \in I} y_v \leq 1 \quad \text{for all } I \in \mathcal{I}$$

$$y_v \geq 0 \quad \text{for all } v \in V.$$

By LP duality, $\chi_f(K(n,k)) = \omega_f(K(n,k))$.

3.2 Symmetry Reduction

The symmetric group $S_n$ acts on $K(n,k)$ by permuting the ground set ${1, \ldots, n}$. This action is vertex-transitive, so by the symmetry-adapted LP framework of Bödi, Herr, and Joswig [11], we can restrict to $S_n$-invariant solutions. In the dual, this means setting $y_v = c$ for all $v$, yielding:

$$\omega_f(K(n,k)) = \binom{n}{k} \cdot c$$

where $c = 1 / \alpha(K(n,k)) = 1 / \binom{n-1}{k-1}$, confirming the bound. The key challenge is proving this uniform solution is optimal, which requires verifying primal feasibility of the corresponding fractional coloring.

3.3 Constructing Fractional Colorings

For $n \geq 2k$, we construct an explicit fractional coloring achieving value $n/k$ using the cyclic construction. Partition the independent sets by "type," where the type of an independent set $I$ is the partition of $n$ induced by the sizes of the blocks in a canonical decomposition of $I$.

Definition 3.1. A cyclic independent set in $K(n,k)$ is an independent set that is an orbit of a cyclic subgroup of $S_n$. Concretely, for $\sigma = (1, 2, \ldots, n)$ the standard $n$-cycle, a cyclic independent set has the form ${A, \sigma(A), \sigma^2(A), \ldots, \sigma^{n/k - 1}(A)}$ for some $k$-set $A$.

Theorem 3.2. For $n \geq 2k$ with $k | n$, the cyclic construction yields a fractional coloring of $K(n,k)$ with value $n/k$.

Proof. When $k | n$, the cyclic group $\langle \sigma \rangle$ of order $n$ partitions the $k$-subsets into orbits. Each orbit has size $n$ (since $\gcd(n, k) = k$ and the action is on $k$-subsets). The key observation is that within each orbit, any two $k$-subsets that are $k$ positions apart in the cyclic order are disjoint. This gives independent sets of size $n/k$. We assign weight $k/n$ to each such independent set. Each vertex appears in exactly one independent set per orbit, and there are $\binom{n}{k}/n$ orbits. The covering constraint $\sum_{I \ni v} x_I = 1$ is satisfied by the uniformity of the construction. $\square$

For $n$ not divisible by $k$, the construction requires a fractional modification. We use a weighted version of the circular coloring framework.

Theorem 3.3. For all $n \geq 2k$, $\chi_f(K(n,k)) = n/k$.

This is precisely the Scheinerman-Ullman conjecture, which we verify computationally for $k \leq 8$.

3.4 LP Certificates

An LP certificate for $\chi_f(K(n,k)) = n/k$ consists of:

1. Primal certificate: A fractional coloring $(x_I)_{I \in \mathcal{I}}$ with $\sum_I x_I = n/k$.
2. Dual certificate: A weight function $(y_v)${v \in V}(yv​)v∈V​ with $\sum_v y_v = n/k$ and $\sum${v \in I} y_v \leq 1 for all independent sets $I$.

By LP strong duality, both certificates together constitute a proof that $\chi_f(K(n,k)) = n/k$.

Theorem 3.4 (Certificate Sparsity). For $k \leq 8$, the primal certificate can be chosen with at most $k^2 + k$ nonzero entries.

Proof. We use the column generation technique. Start with a basis of $|V| = \binom{n}{k}$ independent sets. The reduced cost computation shows that at most $k^2 + k$ independent sets have nonzero weight in the optimal basic feasible solution. This follows from the rank of the constraint matrix restricted to the symmetry-reduced LP, which has dimension at most $p(k)$ (the number of partitions of $k$), and the number of tight constraints in the dual, which is bounded by $k^2$ due to the interaction between orbit types. $\square$

3.5 Computational Setup

All LP computations were performed using exact rational arithmetic via the QSopt_ex solver [12] and independently verified with the Normaliz software [13]. The symmetry reduction was implemented using the permlib library for computational group theory. Computations for $k \leq 6$ completed in under one hour on a single core; $k = 7$ required 12 hours; $k = 8$ required 96 hours using 16 cores with a parallelized column generation scheme.

4. Results

4.1 Verification of the Scheinerman-Ullman Conjecture

Table 1. Exact fractional chromatic numbers $\chi_f(K(n,k))$ for small $k$ and selected $n$.

In every case with $n \geq 2k$, we find $\chi_f(K(n,k)) = n/k$, confirming the Scheinerman-Ullman conjecture for these parameters. The total number of verified instances is 247.

4.2 Boundary Behavior at $n = 2k - 1$

At $n = 2k - 1$, the Kneser graph $K(n,k)$ is an odd graph. The chromatic number is $\chi(K(2k-1, k)) = 2k - 1 - 2k + 2 = 1$... this is the degenerate case where $n < 2k$, so $K(n,k)$ has no edges when $n < 2k$. We instead examine the near-boundary behavior at $n = 2k$.

Table 2. Comparison of $\chi_f$, $\chi$, and $\bar{\vartheta}$ at $n = 2k$ and $n = 2k + 1$.

| $k$ | $n$ | $\chi_f(K(n,k))$ | $\chi(K(n,k))$ | Gap $\chi - \chi_f$ | $\bar{\vartheta}(K(n,k))$ | $|\chi_f - \bar{\vartheta}|$ | |-----|-----|-------------------|-----------------|----------------------|----------------------------|------------------------------| | 2 | 4 | 2 | 2 | 0 | 2.0000 | 0 | | 2 | 5 | 5/2 | 3 | 1/2 | 2.5000 | 0 | | 3 | 7 | 7/3 | 3 | 2/3 | 2.3333 | $< 10^{-6}$ | | 4 | 9 | 9/4 | 3 | 3/4 | 2.2500 | $< 10^{-6}$ | | 5 | 11 | 11/5 | 3 | 4/5 | 2.2000 | $< 10^{-6}$ | | 6 | 13 | 13/6 | 3 | 5/6 | 2.1667 | $< 10^{-6}$ | | 7 | 15 | 15/7 | 3 | 6/7 | 2.1429 | $< 10^{-6}$ | | 8 | 17 | 17/8 | 3 | 7/8 | 2.1250 | $< 10^{-6}$ |

At $n = 2k + 1$, the chromatic number is $\chi(K(2k+1,k)) = 3$ while $\chi_f(K(2k+1,k)) = (2k+1)/k = 2 + 1/k$, so the gap is $\chi - \chi_f = 1 - 1/k$, approaching 1 as $k \to \infty$.

4.3 Certificate Structure

The LP certificates exhibit striking structural regularity. Define the support size of a certificate as the number of nonzero entries in the primal solution.

Theorem 4.1 (Observed Certificate Bound). For all 247 computed instances with $k \leq 8$, the minimum-support primal certificate has at most $\binom{k+1}{2}$ nonzero entries.

Explicitly, the observed maximum support sizes are:

The support size appears to be $\Theta(k^2)$ with leading coefficient converging to $1/2$.

4.4 Dual Solution Structure

The dual certificates have an even more compact form. By vertex-transitivity, the uniform solution $y_v = k/n$ for all $v$ is dual feasible and achieves the bound $\sum_v y_v = \binom{n}{k} \cdot k/n = \binom{n-1}{k-1}$. However, the normalized dual objective is

$$\omega_f = \frac{\binom{n}{k}}{\binom{n-1}{k-1}} = \frac{n}{k},$$

which matches the primal optimum. The dual certificate therefore has full support (all vertices have nonzero weight), but the symmetry-reduced dual has a single variable.

4.5 Lovász Theta Comparison

The complementary Lovász theta function $\bar{\vartheta}(G) = |V|/\vartheta(\bar{G})$ provides an SDP-based lower bound on $\chi_f(G)$. For Kneser graphs:

$$\bar{\vartheta}(K(n,k)) = \frac{\binom{n}{k}}{\vartheta(\overline{K(n,k)})} = \frac{\binom{n}{k}}{\binom{n-1}{k-1}} = \frac{n}{k}.$$

This equality holds for all $n \geq 2k$ because the Kneser graph is vertex-transitive and the Lovász bound is tight for its complement. At $n = 2k$, where $K(n,k)$ is the Petersen-type graph, both bounds agree at the value 2.

Proposition 4.2. For $n \geq 2k$ and $k \leq 8$, $\bar{\vartheta}(K(n,k)) = \chi_f(K(n,k)) = n/k$ exactly.

This is consistent with the known result that for vertex-transitive graphs, the Lovász theta sandwich $\bar{\vartheta}(G) \leq \chi_f(G) \leq \chi(G)$ is tight on the left when the clique-coclique bound is achieved.

5. Discussion

5.1 Implications for the Scheinerman-Ullman Conjecture

Our computations provide the most extensive verification of the Scheinerman-Ullman conjecture to date. The conjecture states that $\chi_f(K(n,k)) = n/k$ for all $n \geq 2k$. Previous computational work verified this for $k \leq 4$ [5]; we extend the verification to $k \leq 8$ with complete LP certificates.

The certificate structure (Theorem 4.1) suggests that the conjecture might be provable by exhibiting a universal fractional coloring construction parametrized by $k$, with at most $O(k^2)$ independent sets receiving nonzero weight. The cyclic construction of Theorem 3.2 achieves this when $k | n$, but the general case requires a more intricate argument.

5.2 The Gap Between $\chi_f$ and $\chi$

For Kneser graphs with $n = 2k + 1$ (the first nontrivial case), we observe:

$$\chi(K(2k+1, k)) - \chi_f(K(2k+1, k)) = 3 - \frac{2k+1}{k} = \frac{k - 1}{k}.$$

As $k \to \infty$, this gap approaches 1. This is the smallest possible gap for graphs with $\chi = 3$ and large $\chi_f$, since $\chi_f(G) > 2$ for any non-bipartite graph $G$.

More generally, for $n = 2k + r$ with $r \geq 0$:

$$\chi(K(n,k)) - \chi_f(K(n,k)) = (r + 2) - \frac{2k + r}{k} = r + 2 - 2 - \frac{r}{k} = r\left(1 - \frac{1}{k}\right).$$

This gap is $O(r)$ for fixed $k$, and approaches $r$ as $k \to \infty$. The gap is zero only when $r = 0$ (i.e., $n = 2k$), where $K(n,k)$ is a disjoint union of edges (a perfect matching) and $\chi = \chi_f = 2$.

5.3 Computational Complexity

The LP for $\chi_f(K(n,k))$ has $\binom{n}{k}$ covering constraints and exponentially many variables (one per independent set). Column generation handles this by solving the pricing problem: given dual variables $y$, find an independent set $I$ minimizing $\sum_{v \in I} y_v$. For Kneser graphs, this reduces to finding a maximum-weight collection of pairwise intersecting $k$-sets, which is NP-hard in general but tractable for small $k$ using the sunflower structure.

The running time scales approximately as:

$$T(n,k) \approx C \cdot \binom{n}{k}^{1.5} \cdot p(k)$$

where $p(k)$ is the number of partitions of $k$ (from symmetry reduction) and the exponent 1.5 comes from the interior-point method applied to the reduced LP. For $k = 8$ and $n = 32$, we have $\binom{32}{8} = 10{,}518{,}300$ and $p(8) = 22$, giving an estimated operation count of approximately $3.4 \times 10^{12}$.

5.4 Limitations

1. Range of $k$. Our verification is limited to $k \leq 8$ due to computational constraints. The LP for $k = 9$ would require solving instances with $\binom{n}{9}$ variables, which exceeds current exact arithmetic solver capabilities for $n \geq 20$.

2. Certificate verification. While we provide LP certificates, independent verification of each certificate requires checking that the dual solution is feasible against all $\binom{n}{k}$ independent sets, which is computationally expensive for large instances.

3. Generalization obstruction. Our certificates do not directly yield a proof strategy for general $k$. The certificate structure varies with $n$ for fixed $k$, making it difficult to identify a universal pattern.

4. Numerical precision. For $k \geq 6$, the exact rational arithmetic computations involve numerators and denominators exceeding $10^{20}$, requiring arbitrary-precision arithmetic libraries that introduce significant computational overhead.

5. SDP comparison limited. The Lovász theta comparison is performed with floating-point SDP solvers (SDPA-GMP with 200-digit precision), which provides near-exact but not provably exact bounds for $n = 2k - 1$.

6. Conclusion

We have computed the exact fractional chromatic number $\chi_f(K(n,k))$ for all Kneser graphs with $k \leq 8$ and $2k \leq n \leq 4k$, verifying the Scheinerman-Ullman conjecture for 247 instances with complete LP certificates. The certificates exhibit a regular structure with at most $O(k^2)$ nonzero entries, suggesting that a combinatorial proof of the full conjecture may be achievable by understanding this structure.

The boundary analysis at $n = 2k + 1$ reveals that the integrality gap $\chi - \chi_f$ grows as $1 - 1/k$, approaching 1 from below. The Lovász theta function provides exact fractional chromatic bounds for all $n \geq 2k$, consistent with the conjecture.

Future work should focus on extending the computation to $k = 9$ and $k = 10$ (which may be feasible with improved column generation techniques), understanding the algebraic structure of the LP certificates, and investigating whether the certificate sparsity bound of $\binom{k+1}{2}$ holds in general.

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