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In graph theory, the hypercube graph Q n is the graph formed from the vertices and edges of an n-dimensional hypercube. For instance, the cube graph Q 3 is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. Q n has 2 n vertices, 2 n – 1 n edges, and is a regular graph with n edges touching each vertex.
An induced path is sometimes called a snake, and the problem of finding long induced paths in hypercube graphs is known as the snake-in-the-box problem. Similarly, an induced cycle is a cycle that is an induced subgraph of G ; induced cycles are also called chordless cycles or (when the length of the cycle is four or more) holes .
In graph theory terminology, this is called finding the longest possible induced path in a hypercube; it can be viewed as a special case of the induced subgraph isomorphism problem. There is a similar problem of finding long induced cycles in hypercubes, called the coil-in-the-box problem.
This can be generalized to any number of dimensions. This process of sweeping out volumes can be formalized mathematically as a Minkowski sum: the d-dimensional hypercube is the Minkowski sum of d mutually perpendicular unit-length line segments, and is therefore an example of a zonotope. The 1-skeleton of a hypercube is a hypercube graph.
Hypercube graphs exhibit a similar phenomenon to cycle graphs. The two- and three-dimensional hypercube graphs (the 4-cycle and the graph of a cube, respectively) have distinguishing number three. However, every hypercube graph of higher dimension has distinguishing number only two. [4] The Petersen graph has distinguishing number 3.
Goldbach’s Conjecture. One of the greatest unsolved mysteries in math is also very easy to write. Goldbach’s Conjecture is, “Every even number (greater than two) is the sum of two primes ...
The resulting graph is a bipartite Kneser graph; the graph formed in this way with n = 2 has 20 vertices and 30 edges, and is called the Desargues graph. All median graphs are partial cubes. [3] The trees and hypercube graphs are examples of median graphs. Since the median graphs include the squaregraphs, simplex graphs, and Fibonacci cubes, as ...
A college student just solved a seemingly paradoxical math problem—and the answer came from an incredibly unlikely place. Skip to main content. 24/7 Help. For premium support please call: 800 ...
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