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the given statement is mathematical statement: a sentence which is either true or false. It may contain words and symbols.
the number of onto functions:
for example, polynomial sequences of binomial type are generated by
, where is a sequence of polynomials
A complete bipartite graph is planar if and only if or
Let graph G be a bipartite graph with an odd number of vertices and G be Hamiltonian, meaning that there is a directed cycle that includes every vertex of G. As such, there exists a cycle in G would of odd length. However, a graph G is bipartite if and only if every cycle of G has even length. Proven by contradiction, if G is a bipartite graph with an odd number of vertices, then G is non-Hamiltonian.
is not a linear recurrence relation. Linear recurrence: each term of a sequence is a linear function of earlier terms in the sequence.
the generating function of the sequence:
( is number of vertices) must be divisible by 4. So, If a graph is isomorphic to its complement, then it can have even number of vertices (for example, ).
A graph having chromatic number is called a -colorable graph. So, 3-colourable graph has , and this means that it has also. That is, every 3-colourable graph is 4-colourable.