[Paper Review] Vertex-disjoint properly edge-colored cycles in edge-colored complete graphs

Ruonan Li, Hajo Broersma+1 more|arXiv (Cornell University)|Aug 29, 2017

Limits and Structures in Graph Theory9 references3 citations

TL;DR

This paper confirms a conjecture on vertex-disjoint properly edge-colored cycles in edge-colored complete graphs for k=2, proving that if the maximum monochromatic degree Δ⁰⁰(G) ≤ n−5, then G contains 2 vertex-disjoint properly edge-colored cycles. It establishes a structural connection between such cycles and directed cycles in multipartite tournaments, using this link to derive partial solutions to the Bermond-Thomasson conjecture on disjoint cycles in directed graphs.

ABSTRACT

It is conjectured that every edge-colored complete graph $G$ on $n$ vertices satisfying $Δ^{mon}(G)\leq n-3k+1$ contains $k$ vertex-disjoint properly edge-colored cycles. We confirm this conjecture for $k=2$, prove several additional weaker results for general $k$, and we establish structural properties of possible minimum counterexamples to the conjecture. We also reveal a close relationship between properly edge-colored cycles in edge-colored complete graphs and directed cycles in multi-partite tournaments. Using this relationship and our results on edge-colored complete graphs, we obtain several partial solutions to a conjecture on disjoint cycles in directed graphs due to Bermond and Thomassen.

Motivation & Objective

• To confirm a conjecture stating that every edge-colored complete graph G with Δ⁰⁰(G) ≤ n−3k+1 contains k vertex-disjoint properly edge-colored cycles.
• To prove the conjecture specifically for k=2, showing that Δ⁰⁰(G) ≤ n−5 guarantees two vertex-disjoint properly edge-colored cycles.
• To establish structural properties of potential minimal counterexamples to the general conjecture.
• To reveal a deep connection between properly edge-colored cycles in edge-colored complete graphs and directed cycles in multipartite tournaments.
• To use this connection to obtain partial solutions to the Bermond-Thomasson conjecture on disjoint cycles in directed graphs.

Proposed method

• Prove that the existence of k vertex-disjoint properly edge-colored cycles in edge-colored complete graphs is equivalent to the existence of k vertex-disjoint dicycles in a related multipartite tournament.
• Construct a multipartite tournament MT from a given edge-colored complete graph G using color degree and edge-coloring rules.
• Use the minimum out-degree condition δ⁺(MT) ≥ f(k) to ensure structural richness in the tournament.
• Apply a known result on multipartite tournaments (Proposition 12) to guarantee k disjoint dicycles under certain conditions.
• Use contradiction and structural decomposition (via Lemma 20) to analyze vertices not in any PC cycle and partition the vertex set accordingly.
• Define a modified ℓ-partite tournament MT′ to preserve minimum out-degree and avoid dicycles of forbidden lengths, enabling application of existing theorems.

Experimental results

Research questions

• RQ1 Does every edge-colored complete graph G with Δ⁰⁰(G) ≤ n−5 contain two vertex-disjoint properly edge-colored cycles?
• RQ2 What structural properties must a minimal counterexample to the general k-cycle conjecture possess?
• RQ3 How are properly edge-colored cycles in edge-colored complete graphs related to directed cycles in multipartite tournaments?
• RQ4 Can the connection between edge-colored graphs and directed graphs be used to make progress on the Bermond-Thomasson conjecture?
• RQ5 Under what conditions does a multipartite tournament with minimum out-degree f(k) contain k disjoint dicycles?

Key findings

• The conjecture is confirmed for k=2: if Δ⁰⁰(G) ≤ n−5, then G contains two vertex-disjoint properly edge-colored cycles.
• The paper proves that a minimum counterexample to the general conjecture must have no monochromatic edge-cut and contain a vertex with color degree at most ℓ not lying in any PC cycle.
• A structural partition of the vertex set is constructed via Lemma 20, enabling the translation of the edge-colored graph problem into a directed graph problem.
• The existence of k disjoint properly edge-colored cycles in G is equivalent to the existence of k disjoint dicycles in a derived multipartite tournament MT.
• The construction ensures that if the tournament MT has minimum out-degree ≥ f(k) and no dicycles of length in I, then G contains no PC cycles of length in I.
• Using this equivalence, the paper derives that if the tournament MT′ has δ⁺(MT′) ≥ f(k) and no dicycles of length i ∈ I, then MT′ contains k disjoint dicycles, which correspond to k vertex-disjoint PC cycles in G.