Monochromatic subgraphs of 2-edge-colored graphs

Lehel conjectured that for all n, any 2-edge-coloring of Kn admits a partition of the vertex set into a red cycle and a blue cycle. This conjecture led to a significant amount of work on related questions and was eventually proven for all n by Bessy and Thomassé. Balogh, Barát, Gerbner, Gyárfás, and Sárközy conjectured a stronger statement for large n: that if δ(G) > 3n/4, then any 2-edge-coloring of G admits such a partition. Balogh, et al. use regularity and blow-up techniques to cover all but γn vertices if δ(G) > ( 34 + γ)n. DeBiasio and the author use the absorbing method to prove their conjecture. This paper provides an overview of the history of the problem and the proof techniques of Balogh, et al. and DeBiasio and Nelsen. 1 The History behind Covering 2-edge-colored Graphs 1.1 Lehel’s Conjecture and Generalizations Since Ramsey-type problems ask for which n any 2-edge-colored Kn must admit certain monochromatic subgraphs, a problem investigating a partition of 2-edge-colored1 complete graphs into certain monochromatic subgraphs is naturally related to Ramsey-type problems. In proving that R(P`, P`) = ⌈ 3` 2 ⌉ , Gerencsér and Gyárfás observed that the vertex set of any 2-colored Kn can be partitioned into a monochromatic cycle and a monochromatic path of the other color2 [9]. A natural question to follow this observation is, can the vertex set of every 2-colored Kn be partitioned into a red cycle and a blue cycle? Conjecture 1.1. For all n, any 2-coloring of Kn admits a partition of the vertex set by a red cycle and a blue cycle.3 Conjecture 1.1 is attributed to Lehel ([2]) but was first accessibly published in a paper by Gyárfás ([13]). Perhaps the most natural generalization of Statement 1.1 is the following question: Question 1.2. Does every r-coloring of Kn admit a partition of the vertex set into r cycles of different colors? Since the investigation of this paper never discusses vertex colorings of graphs, we will from this point forward refer to edge-colorings as “colorings.” 2-colorings are {red, blue}-edge-colorings. More generally, r-colorings are {1, 2, ..., r}colorings. Clearly, if every edge of Kn is colored red, then a non-degenerate blue cycle or path does not exist. We use the convention that the empty set, a single vertex K2 are monochromatic cycles and paths. Since this statement was conjectured by Lehel, we refer to such a partition as a “Lehel partition.” So Statement 1.1 may read, “For all n, any 2-coloring of Kn admits a Lehel partition.”