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Applying the Mycielskian repeatedly, starting with the one-edge graph, produces a sequence of graphs M i = μ(M i−1), sometimes called the Mycielski graphs. The first few graphs in this sequence are the graph M 2 = K 2 with two vertices connected by an edge, the cycle graph M 3 = C 5, and the Grötzsch graph M 4 with 11 vertices and 20 edges.
The art of crochet has been used to demonstrate hyperbolic planes, the first such demonstration having been made by Daina Taimiņa. [28] In 2000, Keith Henderson demonstrated a quick-to-make paper model dubbed the "hyperbolic soccerball" (more precisely, a truncated order-7 triangular tiling). [29] [30]
Traditionally needlepoint designs completely cover the canvas. [1] Although needlepoint may be worked in a variety of stitches, many needlepoint designs use only a simple tent stitch and rely upon color changes in the yarn to construct the pattern. Needlepoint is the oldest form of canvas work.
An independent set of ⌊ ⌋ vertices (where ⌊ ⌋ is the floor function) in an n-vertex triangle-free graph is easy to find: either there is a vertex with at least ⌊ ⌋ neighbors (in which case those neighbors are an independent set) or all vertices have strictly less than ⌊ ⌋ neighbors (in which case any maximal independent set must have at least ⌊ ⌋ vertices). [4]
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The first is "simple open crochet" (crochet simple ajour), a mesh of chain-stitch arches. The second (illustrated here) starts in a semi-open form ( demi jour ), where chain-stitch arches alternate with equally long segments of slip-stitch crochet, and closes with a star made with "double-crochet stitches" ( dubbelde hekelsteek : double-crochet ...
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