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Infinite Ramsey-Minimal Graphs for Star Forests

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Abstract

For graphs F, G, and H, we write \(F \rightarrow (G,H)\) if every red-blue coloring of the edges of F produces a red copy of G or a blue copy of H. The graph F is said to be (G, H)-minimal if it is subgraph-minimal with respect to this property. The characterization problem for Ramsey-minimal graphs is classically done for finite graphs. In 2021, Barrett and the second author generalized this problem to infinite graphs. They asked which pairs (G, H) admit a Ramsey-minimal graph and which ones do not. We show that any pair of star forests such that at least one of them involves an infinite-star component admits no Ramsey-minimal graph. Also, we construct a Ramsey-minimal graph for a finite star forest versus a subdivision graph. This paper builds upon the results of Burr et al. (Discrete Math 33:227–237, 1981) on Ramsey-minimal graphs for finite star forests.

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  1. Department of Mathematics, Universitas Indonesia, Depok, 16424, Indonesia

    Fawwaz Fakhrurrozi Hadiputra & Valentino Vito

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  1. Fawwaz Fakhrurrozi Hadiputra
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  2. Valentino Vito
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Correspondence to Valentino Vito.

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Hadiputra, F.F., Vito, V. Infinite Ramsey-Minimal Graphs for Star Forests. Graphs and Combinatorics 40, 24 (2024). https://doi.org/10.1007/s00373-024-02752-1

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