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Mathematics > Combinatorics

arXiv:1807.11890 (math)
[Submitted on 31 Jul 2018]

Title:Minimal Ramsey graphs for cyclicity

Authors:Damian Reding, Anusch Taraz
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Abstract: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 $K_5-e$ or $K_4\vee K_4$ (two copies of $K_4$ identified at an edge), which implies that an arbitrary $n$-vertex graph with $e(G)\geq 2n-1$ 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 $\chi=2, 3, 4$, the unboundedness of maximum degree within the class as well as Ramsey separability of the family of cycles of length $\leq l$ from any of its proper subfamilies.
Comments: 17 pages
Subjects: Combinatorics (math.CO)
Cite as: arXiv:1807.11890 [math.CO]
  (or arXiv:1807.11890v1 [math.CO] for this version)
  https://doi.org/10.48550/arXiv.1807.11890

Submission history

From: Damian Reding [view email]
[v1] Tue, 31 Jul 2018 16:06:50 UTC (21 KB)
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