Properly Coloured Cycles and Paths: Results and Open Problems

In this paper, we consider a number of results and seven conjectures on properly edge-coloured (PC) paths and cycles in edge-coloured multigraphs. We overview some known results and prove new ones. In particular, we consider a family of transformations of an edge-coloured multigraph $G$ into an ordinary graph that allow us to check the existence PC cycles and PC $(s,t)$-paths in $G$ and, if they exist, to find shortest ones among them. We raise a problem of finding the optimal transformation and consider a possible solution to the problem.

💡 Research Summary

The paper surveys and extends results on properly coloured (PC) paths and cycles in edge‑coloured multigraphs, focusing on existence, length, and algorithmic aspects. After a brief introduction that situates edge‑coloured multigraphs as a natural generalisation of directed graphs, the authors revisit Yeo’s structural theorem, which yields a recursive test for the presence of a PC cycle in a c‑edge‑coloured graph. They introduce the function d(n,c), the smallest monochromatic degree guaranteeing a PC cycle, and show that for any fixed number of colours c ≥ 2 the bounds are Θ(log n), thereby motivating Conjecture 2.2 that d(n,c)≈s(c)·log₂ n for a colour‑dependent constant s(c).

The core technical contribution is the development of P‑gadgets, small auxiliary graphs attached to each vertex x of the original multigraph G. A P‑gadget must satisfy four properties (P1–P4) that ensure a tight correspondence between PC substructures of G and perfect matchings in the transformed graph. Three concrete gadgets are examined: the SP‑gadget (due to Szeider), the BJGP‑gadget (Bang‑Jensen and Gutin), and a newly proposed XP‑gadget. The XP‑gadget is proved to use the fewest vertices and edges among the three for any colour set size z, and the authors conjecture it is optimal for all possible P‑gadgets.

Using a chosen family of P‑gadgets, the authors construct the auxiliary graphs G* and G**. G* contains the union of all gadget vertices and edges (E₁) together with “colour‑edges” (E₂) that represent the original coloured edges of G. G** augments G* with two distinguished vertices s and t (for (s,t)‑path queries). Theorem 4.1 establishes a bijection between PC cycle subgraphs of G with r edges and perfect matchings of G* that contain exactly r colour‑edges from E₂. Consequently, Corollary 4.2 yields polynomial‑time algorithms: checking for any PC cycle, finding a maximum‑size PC cycle, and finding a shortest PC cycle, all in O(n*·(m*+nlog n)) time (or O(n·n*·(m*+nlog n)) for the shortest cycle). Analogous results (Theorem 4.3 and Corollary 4.4) handle PC (s,t)‑paths via perfect matchings in G**, giving O(m**) time for existence and O(n**·(m**+nlog n)) for the shortest path.

The paper then turns to extremal results on long PC structures. Theorem 5.1 (from prior work) shows that a c‑edge‑coloured multigraph with monochromatic minimum degree at least ⌈(n+1)/2⌉ contains a Hamilton PC cycle (or a near‑Hamilton cycle when c=2 and n is odd). Conjecture 5.2 proposes that the bound can be lowered to ⌈n/2⌉. Theorem 5.3 guarantees a PC path of length at least min{n−1, 2⌊c/2⌋·d} when the monochromatic minimum degree is d. Two stronger conjectures (5.4 and 5.5) suggest that the factor 2⌊c/2⌋ can be replaced by 2c (for simple graphs) and by 2 (for multigraphs), respectively.

In the final substantive section the authors study edge‑coloured complete graphs Kₙᶜ. Theorem 6.1 (originally proved for c=2) states that Kₙᶜ has a PC Hamilton path iff it contains a PC 1‑path‑cycle subgraph; this equivalence enables a polynomial‑time algorithm for the longest PC path for any fixed c. For c=2, Saad’s colour‑connectivity concept yields a randomised polynomial algorithm for the longest PC cycle, later derandomised by Bang‑Jensen and Gutin. However, for c≥3 no complete characterisation of Hamilton PC cycles is known, and Problem 6.4 asks for the computational complexity of the PC Hamilton‑cycle decision problem in this regime.

Overall, the paper unifies the study of PC substructures through a novel gadget‑based reduction to perfect matching, provides concrete algorithmic bounds, proposes optimality conjectures for the gadget construction, and outlines several open problems concerning degree conditions, longest PC paths, and Hamiltonicity in coloured complete graphs.