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A subgraph of an edge-colored graph is rainbow if all of its edges have different colors. For a graph H and a positive integer n, the anti-Ramsey number f (n, H) is the maximum number of colors in an edge-coloring of K n with no rainbow copy of H. The rainbow number rb(n, H) is the minimum number of colors such that any edge-coloring of K n with rb(n, H)… (More)
For two given graphs G and H the planar Ramsey number PR(G, H) is the smallest integer n such that every planar graph F on n vertices either contains a copy of G or its complement contains a copy H. By studying the existence of subhamiltonian cycles in complements of sparse graphs, we determine all planar Ramsey numbers for pairs of cycles.
The planar Ramsey number PR(G, H) is defined as the smallest integer n for which any 2-colouring of edges of K n with red and blue, where red edges induce a planar graph, leads to either a red copy of G, or a blue H. In this note we study the weak induced version of the planar Ramsey number in the case when the second graph is complete.