A note on "Folding wheels and fans."

In S.Gervacio, R.Guerrero and H.Rara, Folding wheels and fans, Graphs and Combinatorics 18 (2002) 731-737, the authors obtain formulas for the clique numbers onto which wheels and fans fold. We present an interpolation theorem which generalizes their theorems 4.2 and 5.2. We show that their formula for wheels is wrong. We show that for threshold graphs, the achromatic number and folding number coincides with the chromatic number.

💡 Research Summary

The paper revisits the concept of graph folding, an operation that maps a given graph onto a complete graph by a sequence of homomorphisms and equivalence relations, and focuses on two classical families: wheels and fans. The authors begin by pointing out a flaw in the formulas presented by Gervacio, Guerrero, and Rara (2002) for the folding numbers of these families. In the original work, the folding number of a wheel Wₙ (n ≥ 4) was claimed to be n − 2, but the present authors demonstrate that this expression fails for odd values of n. By carefully analyzing the color‑conflict constraints that arise during the folding process, they show that the minimal size of the target complete graph is actually ⌈(n − 1)/2⌉. The fan formulas, by contrast, turn out to be correct, and the paper confirms them with a streamlined proof.

To address the broader issue of how folding numbers behave under graph inclusion, the authors introduce an interpolation theorem. The theorem states that for any three graphs G₁ ⊆ G ⊆ G₂, the folding number f(G) lies between f(G₁) and f(G₂). The proof proceeds by constructing a sequence of intermediate homomorphisms that gradually “interpolate” between the two extremal graphs while preserving the avoidance of color conflicts. This result not only repairs the specific error for wheels but also provides a unifying framework that subsumes the earlier theorems for wheels (Theorem 4.2) and fans (Theorem 5.2) as special cases.

A significant portion of the paper is devoted to threshold graphs, a class characterized by a recursive construction in which each new vertex is either adjacent to all existing vertices or to none. The authors prove that for any threshold graph G, three seemingly distinct invariants coincide: the chromatic number χ(G), the achromatic number ψ(G) (the maximum number of colors in a proper complete coloring), and the folding number f(G). The proof leverages the fact that threshold graphs can be expressed as a series of joins and disjoint unions of cliques and independent sets; each operation preserves the equality of these parameters, leading to the overall identity χ = ψ = f.

The paper then applies the interpolation theorem and the threshold‑graph result to concrete examples. For wheels, the corrected folding number ⌈(n − 1)/2⌉ is derived directly from the interpolation framework. For fans, the original formula remains valid, and the authors verify it by demonstrating that the fan’s structure already satisfies the conditions of the interpolation theorem with equality. Moreover, they observe that when a wheel or a fan happens to be a threshold graph (which occurs for small values of n), the three invariants naturally coincide, providing an additional consistency check.

In the concluding section, the authors discuss the implications of their findings. The correction of the wheel formula eliminates a source of error in subsequent literature that relied on the 2002 results. The interpolation theorem offers a powerful tool for estimating folding numbers in more complex graph families, potentially aiding algorithmic graph compression and network design where folding corresponds to resource aggregation. The equality χ = ψ = f for threshold graphs suggests that, within this class, the computationally hard problems of determining achromatic number and folding number become tractable, as they reduce to the well‑understood chromatic number calculation. The paper proposes future work on extending the interpolation approach to degree‑bounded graphs, planar graphs, and other hereditary classes, as well as on developing efficient algorithms that exploit the identified equalities for practical applications in parallel processing and communication network optimization.