2018/07/31 by Damian Reding, Reding, Damian, Anusch Taraz +1
Computer Science · Mathematics · #Advanced Graph Theory Research #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #math.CO
paper · pdf · doi:10.48550/arxiv.1807.11890
17 pages
arxiv created 2018/07/31 · openalex publication_date 2018/07/31 · arxiv updated 2018/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study graphs with the property that every edge-colouring admits a monochromatic cycle (the length of which may depend freely on the colouring) and describe those graphs that are minimal with this property. We show that every member in this class reduces recursively to one of the base graphs K5-e or K4\vee K4 (two copies of K4 identified at an edge), which implies that an arbitrary n-vertex graph with e(G)≥ 2n-1 must contain one of those as a minor. We also describe three explicit constructions governing the reverse process. As an application we are able to establish Ramsey infiniteness for each of the three possible chromatic subclasses χ=2, 3, 4, the unboundedness of maximum degree within the class as well as Ramsey separability of the family of cycles of length ≤ l from any of its proper subfamilies.