I Simposio de Postgrado 2023. Ingeniería, ciencias e innovación

MÓDULO_ 07 Matemáticas aplicadas y Modelación matemática 134 MINIMUM DEGREE CONDITIONS FOR MONOCHROMATIC CYCLE PARTITIONING IN BIPARTITE GRAPHS ABSTRACT If we colour with r colours the edges of a graph with minimum degree n/2 + 1200r log(n) it is possible to construct a partition of the vertex set, which only contains monochromatic cycles, of size O(r^2). This result, proved by Korándi, Lang, Letzter, and Pokrovskiy, is the motivation for the study of this thesis. The result presented here is an adaptation of the minimum degree condition, conditional on the fact that now the studied graph is balanced bipartite. More precisely, for every η > 0 and for any sufficiently large balanced bipartite r-edge-coloured graph with minimum degree (1/4 + η)n, it is possible to ensure the existence of a vertex cover of size O(r^2) composed only of vertex-disjoint monochromatic cycles. For the proof of the result, we present the concept of birobustly matchable graphs and use the regularity lemma, in its r-colour version. Subsequently, we use a method proposed by Łuczak to cover almost the whole graph. We finish by using the “blow-up lemma” to cover the remaining vertices. 1 Departamento de Ingeniería Matemática, Universidad de Chile. *Email: matias.azocar@uchile.cl Matías Azócar 1* REFERENCIAS Dániel Korándi, Richard Lang, Shoham Letzter, and Alexey Pokrovskiy. Minimum degree conditions for monochromatic cycle partitioning. Journal of Combinatorial Theory, Series B, 146:96–123, 2021. doi: 10.1016/j.jctb.2020.07.005.