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The edge-connectivity version of Menger's theorem is as follows: . Let G be a finite undirected graph and x and y two distinct vertices. Then the size of the minimum edge cut for x and y (the minimum number of edges whose removal disconnects x and y) is equal to the maximum number of pairwise edge-disjoint paths from x to y.
The maximum number of pieces from consecutive cuts are the numbers in the Lazy Caterer's Sequence. When a circle is cut n times to produce the maximum number of pieces, represented as p = f (n), the n th cut must be considered; the number of pieces before the last cut is f (n − 1), while the number of pieces added by the last cut is n.
A paper machine can produce an unlimited number of master (jumbo) rolls, each 5600 mm wide. The following 13 items must be cut, in the table below. The important thing about this kind of problem is that many different product units can be made from the same master roll, and the number of possible combinations is itself very large, in general ...
In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph.A path is called simple if it does not have any repeated vertices; the length of a path may either be measured by its number of edges, or (in weighted graphs) by the sum of the weights of its edges.
An alternative definition of the intersection number of a graph is that it is the smallest number of cliques in (complete subgraphs of ) that together cover all of the edges of . [ 1 ] [ 12 ] A set of cliques with this property is known as a clique edge cover or edge clique cover , and for this reason the intersection number is also sometimes ...
The same set of points can often be constructed using a smaller set of tools. For example, using a compass, straightedge, and a piece of paper on which we have the parabola y=x 2 together with the points (0,0) and (1,0), one can construct any complex number that has a solid construction. Likewise, a tool that can draw any ellipse with already ...
In January 2002, she folded a 4,000-foot-long (1,200 m) piece of toilet paper twelve times in the same direction, debunking a long-standing myth that paper cannot be folded in half more than eight times. [21] [22] The fold-and-cut problem asks what shapes can be obtained by folding a piece of paper flat, and making a single straight complete ...
A bipartite graph with 4 vertices on each side, 13 edges, and no , subgraph, and an equivalent set of 13 points in a 4 × 4 grid, showing that (;).. The number (;) asks for the maximum number of edges in a bipartite graph with vertices on each side that has no 4-cycle (its girth is six or more).