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 Dec 7 awarded Yearling Dec 7 awarded Yearling Dec 7 awarded Yearling Jun 25 awarded Citizen Patrol Dec 7 awarded Yearling Jun 26 awarded Nice Answer Dec 8 awarded Yearling Jul 21 comment Stable singularities of smooth map $\mathbb R^3\to \mathbb R^4$ B. Morin: Formes canoniques des singularités d'une application différentiable, Comp. Rend. Acad. Sci. Paris 260 (1965), pp. 5662-5665 should have it Jul 13 answered Stable singularities of smooth map $\mathbb R^3\to \mathbb R^4$ Jan 5 comment Limit of a sequence of polygons. Just a little error: there are no angles in the limit curve. It is composed of self-similar arcs under some affine transformations (with a linear part with matrix entries 1/3, 1/3, 0, 1/3, say), which are not similarities, so no nontrivial angles can be present. Dec 8 awarded Yearling Aug 19 answered Probability of Outcomes Algorithm Aug 2 comment In a graph, is it always possible to construct a set of cycle bases, with each and every edge Is shared by at most 2 cycle bases? @Casebash: the binary one. Jul 19 answered Smallest area shape that covers all unit length curve Jul 14 comment In a graph, is it always possible to construct a set of cycle bases, with each and every edge Is shared by at most 2 cycle bases? To answer "is it always possible to construct a set of cycle bases with each and every edge shared by at most 2 bases?" in the negative, I construct a graph for which no such cycle base exists. The example I give is the complete graph on 7 vertices. In it, the cycle space (see above) has dimension #edges-#vertices+1=21-7+1=15. No matter how you select 15 nonempty cycles or unions of cycles, those will have at least 45 edges counted with multiplicity. This is more than twice the number of edges in the graph, so by the pigeonhole principle at least one edge will be used 3 times or more. Jul 8 comment In a graph, is it always possible to construct a set of cycle bases, with each and every edge Is shared by at most 2 cycle bases? I answer the question "In a graph, is it always possible to construct a set of cycle bases, with each and every edge Is shared by at most 2 cycle bases?" in the negative, so one counterexample is all it takes. If you wanted to ask a more precise question, then BS's lead is what you should read. As for your other question, please consult en.wikipedia.org/wiki/Cycle_space . Jul 7 comment In a graph, is it always possible to construct a set of cycle bases, with each and every edge Is shared by at most 2 cycle bases? en.wikipedia.org/wiki/Mac_Lane%27s_planarity_criterion says: S. Mac Lane, A combinatorial condition for planar graphs, Fund. Math. 28 (1937), 22–32. Jul 7 comment In a graph, is it always possible to construct a set of cycle bases, with each and every edge Is shared by at most 2 cycle bases? I implicitly use that all cycles contain at least 3 edges, and that the dimension of the cycle space is the number of edges exceeding that of a tree with the same vertices. Both facts are straightforward, could you please elaborate on where exactly you have problems? Jul 6 answered In a graph, is it always possible to construct a set of cycle bases, with each and every edge Is shared by at most 2 cycle bases? Jun 29 revised Binary Sequences of Length 2n added clarification