On Ramsey-type problems for paths and cycles with few colour changes

Abstract: In 1967, Gerencser and Gyárfás determined the exact values of the two-colour Ramsey numbers of paths. In a footnote, they made the following observation: Every $2$-edge-coloured complete graph contains a Hamilton path with at most one colour change. Later, this led to a challenging and still wide open conjecture about covering edge-coloured complete graphs with monochromatic paths. Inspired by the original statement, we study paths and cycles with few colour changes in $3$-edge-coloured complete graphs. For this, we introduce a new Ramsey-type parameter: For $q,k \in \mathbb{N}$ and a graph $G$, let $R_q<sup>k(G)$ denote the smallest $N \in \mathbb{N}$ such that every $q$-edge-coloured complete graph on $N$ vertices contains a copy of $G$ with at most $k$ vertices that are incident to edges in $G$ of different colours. For paths, we show that $R_3^1(P_n) = \frac{3n}{2} + O(1)$, and for even cycles, we show that $R_3^2(C_n) = \frac{3n}{2} + o(n)$.

• The paper introduces Rq^k(G) to generalize classical Ramsey numbers, establishing that R^1_3(P_n)=3n/2+O(1) for paths and similar asymptotics for even cycles.
• It employs innovative partitioning, extremal colouring, and regularity-based structural arguments to ensure the existence of nearly monochromatic paths and cycles.
• The findings imply that allowing limited colour changes counterbalances additional colours, offering practical insights for network design and algorithmic applications.

Ramsey-type Problems for Paths and Cycles with Few Colour Changes

Introduction and Motivation

This work investigates Ramsey-type combinatorial parameters focusing on paths and cycles in edge-coloured complete graphs, specifically analyzing situations where monochromatic structures are relaxed to allow a bounded number of colour changes. Classical Ramsey numbers $R_q(G)$ provide the minimal $N$ such that every $q$-edge-colouring of $K_N$ contains a monochromatic copy of $G$. For sparse graphs such as paths ($P_n$) and cycles ($C_n$), foundational results dictate their Ramsey numbers grow linearly; precise values for standard cases ($q=2,3$) have been previously established.

The paper introduces $R_q^k(G)$: the minimum $N$ such that every $N$0-edge-colouring of $N$1 yields a copy of $N$2 with at most $N$3 colour changes, where a colour change is defined per vertex incident to edges with different colours within $N$4. This generalizes classic Ramsey numbers ($N$5), and explores the interplay between more colours and permitted heterogeneity in paths/cycles. The work is motivated by notable conjectures concerning monochromatic path and cycle covers, and the combinatorial structure of paths/cycles with controlled colour transitions.

Main Results

Ramsey Parameters with Few Colour Changes

• For paths, the central result is that $N$6, giving tight asymptotic bounds for the minimal size required to guarantee a path of length $N$7 with at most one colour change in any three-edge colouring of a complete graph. The bounds established differ by at most 5, with conditional refinements when $N$8 divides $N$9: $q$0.
• For even cycles, $q$1, thus matching the asymptotic growth rate for paths and distinguishing the behaviour from the classical Ramsey numbers for cycles, where odd/even parity introduces notable disparities.

The proofs do not use Szemerédi's regularity method directly, but rely on regularity-based structural results (e.g., for $q$2 [benevides20093] [kohayakawa20053]).

Construction and Bound Techniques

Lower bounds are provided using explicit multi-partitioned edge colourings without good paths or cycles of the required length and bounded colour changes, demonstrating necessary partition sizes. The constructions extend templates from small graphs (e.g., $q$3 coloured with three colours) to larger partitions, carefully ensuring that any long path or cycle necessarily encounters more colour changes than permitted.

Upper bounds are achieved via intricate partition arguments and applications of monochromatic path/cycle partitioning results and split colourings in bipartite graphs [pokrovskiy2014partitioning]. Innovations include leveraging partition structures, extremal combinatorial configurations, and combinatorial summations to guarantee the existence of a sufficiently long path or cycle with controlled colour changes.

Technical Implications

Structural Insights and Conjectural Extensions

• The results demonstrate that allowing a small number of colour changes (relative to the number of colours) can asymptotically offset the increased complexity from multi-coloured edge assignments. Specifically:
  • The asymptotic rate for $q$4 aligns with the two-colour Ramsey number $q$5, illustrating that the permitted colour change precisely counterbalances the third colour.
  • For even cycles, permitting two colour changes yields a comparable rate, which strongly suggests a universal behaviour for "almost monochromatic" cycle structures in multi-coloured settings.
  • Odd cycles remain unresolved: Conjectures indicate possible analogous behaviour, but require new combinatorial arguments since standard regularity and path partition tactics fall short.

Relation to Path and Cycle Cover Conjectures

• The path cover problem (Gyárfás' conjecture) connects closely to the notion of paths with a bounded number of colour changes. The authors speculate that stronger forms of these conjectures may hold:
  • $q$6 (any $q$7-edge colouring of $q$8 contains a $q$9 with at most $K_N$0 colour changes).
  • $K_N$1 (any $K_N$2-edge colouring of $K_N$3 produces a spanning cycle with at most $K_N$4 colour changes).
• These conjectures, if proven, would extend classical partitioning results to resilient structures with minimal colour heterogeneity, providing a powerful bridge between covering and colouring constraints.

Algorithmic and Practical Consequences

The results have implication for algorithmic Ramsey-type problems:

• In situations where complete monochromatic structures are computationally infeasible or structurally rare, permitting bounded heterogeneity vastly improves guarantees, reducing required graph sizes.
• Applications in routing, network design, and hypercube connectivity benefit from knowledge of colour-change resilience (see, e.g., Feder and Subi's conjecture for hypercubes [feder2013hypercube] [dvovrak2020note]; colour-change constraints directly relate to network robustness and fault tolerance).

Future Directions

• Determining $K_N$5 for odd $K_N$6 is open; resolving this would clarify whether the asymptotic behaviour parallels even cycles, and elucidate deeper structural combinatorics in multi-colour settings.
• Improving the $K_N$7 error term for even cycles and paths, potentially replacing it with a tight $K_N$8 bound using refined combinatorial analysis, especially in cases with two disjoint cycles of different colours.
• Characterizing $K_N$9 (the minimal number of colour changes for a spanning $G$0 in $G$1-edge-coloured $G$2) across tree families, regular graphs, and planar graphs, establishing extremal values and structural dependencies.

Conclusion

This paper introduces, formulates, and tightly bounds Ramsey-type parameters for paths and cycles with controlled colour changes in multicolour complete graphs (2607.03243). The results establish that the flexibility to allow a small number of colour changes fundamentally alters the growth rates of these parameters, matching those for fewer colour classes in monochromatic settings. The techniques combine partition methods, extremal colourings, and precise combinatorial estimation, providing a technical foundation for further study of Ramsey numbers under relaxed monochromatic constraints. Open questions remain for odd cycles, error term removal, and span-based parameters in broader graph families. The implications for resilient combinatorial structures in multi-colour environments are substantial, impacting both theory and applications in network connectivity and algorithmic design.