A graph G is called H-Ramsey if any two-coloring of the edges of G contains a monochromatic copy of H. An H-Ramsey graph is called H-minimal if no proper subgraph of it is H-Ramsey. We investigate the minimum degree of H-minimal graphs, a problem initiated by Burr, Erdos, and Lovasz. We determine the smallest possible minimum degree of H-minimal graphs for numerous bipartite graphs H, including bi-regular bipartite graphs and forests. We also make initial progress for graphs of larger chromatic number. This represents joint work with Philipp Zumstein and Stefanie Zurcher.
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