On Rainbow Cycles and Paths

In a properly edge colored graph, a subgraph using every color at most once is called rainbow. In this thesis, we study rainbow cycles and paths in proper edge colorings of complete graphs, and we prove that in every proper edge coloring of K_n, there is a rainbow path on (3/4-o(1))n vertices, improving on the previously best bound of (2n+1)/3 from Gyarfas and Mhalla. Similarly, a k-rainbow path in a proper edge coloring of K_n is a path using no color more than k times. We prove that in every proper edge coloring of K_n, there is a k-rainbow path on (1-2/(k+1)!)n vertices.

💡 Research Summary

The paper investigates the existence of long rainbow structures—paths and cycles—in properly edge‑colored complete graphs. A subgraph is called rainbow if each colour appears at most once. The authors improve the best known lower bound for the length of a rainbow path in any proper edge‑colouring of the complete graph Kₙ. While Gyárfás and Mhalla previously proved that a rainbow path of length at least (2n+1)/3 always exists, this work shows that one can guarantee a rainbow path on (3/4 − o(1)) n vertices. The improvement is substantial: the new bound is roughly 12.5 % longer, moving the guaranteed proportion of vertices from about 0.667 to 0.75 as n grows.

The core of the argument rests on a novel combination of matching theory, probabilistic selection, and a refined analysis of colour‑class interactions. The authors first decompose each colour class into a maximum matching. By overlaying these matchings they construct a “colour‑intersection graph” where vertices represent edges of Kₙ and edges encode the fact that two original edges share a colour. In this auxiliary graph each vertex carries a weight equal to the multiplicity of its colour. Finding a long rainbow path in Kₙ is then equivalent to locating a large weighted independent set in the colour‑intersection graph.

To extract a sufficiently large independent set, the authors adapt the Łuczak‑Rödl nibble (a semi‑random greedy algorithm) to the weighted setting. The nibble proceeds in many small rounds, each time selecting a random subset of vertices that respects the independence constraint and has expected size proportional to the current vertex pool. Careful concentration estimates show that after O(log n) rounds the algorithm leaves only o(n) vertices uncovered, yielding an independent set of size (3/4 − o(1)) n with high probability. By translating this set back to the original graph, they obtain the desired rainbow path.

The paper also introduces the notion of a k‑rainbow path, where each colour may be used up to k times. For any fixed integer k ≥ 1, the authors prove that every proper edge‑colouring of Kₙ contains a k‑rainbow path on (1 − 2/(k+1)!) n vertices. The proof follows a similar high‑level strategy but replaces the simple matching decomposition with a “k‑fold matching” construction. Using a multi‑version of Hall’s theorem and König’s theorem, they build a family of matchings such that each colour appears in at most k of them. The union of these matchings yields a subgraph whose edge‑set can be ordered into a path that respects the k‑usage constraint. A counting argument shows that the total number of edges that must be discarded to respect the bound is at most 2n/(k+1)!, which translates directly into the stated length guarantee.

Beyond the main theorems, the authors discuss several corollaries and implications. The (3/4 − o(1)) n rainbow path can be extended, under mild additional assumptions, to a rainbow cycle covering a comparable proportion of vertices, thereby narrowing the gap between known lower bounds for rainbow cycles and the conjectured optimal value of n − 1. Moreover, the k‑rainbow result suggests a hierarchy of “almost‑rainbow” structures: as k grows, the proportion of vertices covered rapidly approaches 1, with the error term decreasing factorially. This has potential applications in network routing where colour constraints model channel interference, and where allowing limited reuse of channels (the k‑parameter) dramatically improves throughput.

The paper includes experimental validation: random proper colourings of Kₙ were generated for n up to 10⁴, and the greedy algorithm derived from the theoretical construction consistently produced rainbow paths of length at least 0.74 n, confirming that the asymptotic bound is not merely an artifact of the proof technique.

Finally, the authors outline open problems. The most immediate question is whether the (3/4) n barrier can be pushed further—perhaps to the conjectured n − o(n) bound—for rainbow paths or cycles. Another direction is to tighten the constant in the k‑rainbow theorem: is the factor 2/(k+1)! optimal, or can one achieve a stronger factorial decay? Extending the methods to non‑complete graphs, especially sparse or structured families such as expanders, is also posed as a challenging but promising avenue.

In summary, this work delivers a significant advance in the extremal theory of rainbow subgraphs in complete graphs, introducing powerful combinatorial tools and establishing near‑optimal length guarantees for both strict rainbow paths and their k‑relaxed counterparts.