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In mathematics, a self-avoiding walk (SAW) is a sequence of moves on a lattice (a lattice path) that does not visit the same point more than once. This is a special case of the graph theoretical notion of a path. A self-avoiding polygon (SAP) is a closed self-avoiding walk on a lattice. Very little is known rigorously about the self-avoiding ...
A three-dimensional hypercube graph showing a Hamiltonian path in red, and a longest induced path in bold black. In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges).
A bipartite graph may be oriented from one side of the bipartition to the other. The longest path in this orientation has length one, with only two vertices. Conversely, if a graph is oriented without any three-vertex paths, then every vertex must either be a source (with no incoming edges) or a sink (with no outgoing edges) and the partition of the vertices into sources and sinks shows that ...
Let us now prove the contrapositive of Berge's theorem: G has a matching larger than M if and only if G has an augmenting path. Clearly, an augmenting path P of G can be used to produce a matching M ′ that is larger than M — just take M ′ to be the symmetric difference of P and M (M ′ contains exactly those edges of G that appear in ...
Directed graph showing the orbits of small numbers under the Collatz map, skipping even numbers. The Collatz conjecture states that all paths eventually lead to 1. The Collatz conjecture [a] is one of the most famous unsolved problems in mathematics.
As confidence in your walking routine grows, you should aim to increase the intensity and length of your workout. Freeman suggests focusing on exercises that make you “huff and puff” for the ...
A parametric C r-curve or a C r-parametrization is a vector-valued function: that is r-times continuously differentiable (that is, the component functions of γ are continuously differentiable), where , {}, and I is a non-empty interval of real numbers.
A Japanese sake maker is going where no sake maker has gone before: space. Asahi Shuzo, the company behind popular Japanese sake brand Dassai, plans to blast sake ingredients to the International ...