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The coarseness xi(G) of a graph G is the maximum number of edge-disjoint nonplanar subgraphs contained in a given graph G. The coarseness of a planar graph G is therefore ...

Arrange copies of the n digits 1, ..., n such that there is one digit between the 1s, two digits between the 2s, etc. For example, the unique (modulo reversal) n=3 solution ...

In the mice problem, also called the beetle problem, n mice start at the corners of a regular n-gon of unit side length, each heading towards its closest neighboring mouse in ...

The nth cubic number n^3 is a sum of n consecutive odd numbers, for example 1^3 = 1 (1) 2^3 = 3+5 (2) 3^3 = 7+9+11 (3) 4^3 = 13+15+17+19, (4) etc. This identity follows from ...

The Pappus graph is a cubic symmetric distance-regular graph on 18 vertices, illustrated above in three embeddings. It is Hamiltonian and can be represented in LCF notation ...

A quartic graph is a graph which is 4-regular. The unique quartic graph on five nodes is the complete graph K_5, and the unique quartic graph on six nodes is the octahedral ...

A quintic graph is a graph which is 5-regular. The only quintic graph on n<=7 nodes is the complete graph K_6. Quintic graphs exist only on even numbers of nodes, and the ...

A radical integer is a number obtained by closing the integers under addition, multiplication, subtraction, and root extraction. An example of such a number is RadicalBox[7, ...

The hyperbolic sine integral, often called the "Shi function" for short, is defined by Shi(z)=int_0^z(sinht)/tdt. (1) The function is implemented in the Wolfram Language as ...

A Lie algebra is said to be simple if it is not Abelian and has no nonzero proper ideals. Over an algebraically closed field of field characteristic 0, every simple Lie ...