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Various concepts in graph theory, including degree sequences, hamiltonian graphs, graph toughness, and the 4-color conjecture. Necessary conditions for a graph to be hamiltonian and introduces the concept of graph toughness. It also explains how the 4-color conjecture can be reduced to 3-connected cubic graphs and how these graphs are related to hamiltonicity.
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If G is a graph with n vertices and degrees
d 1 ≤ d 2 ≤... ≤ dn then the n-tuple (d 1 ,... , dn)
is called the degree sequence of G.
An arbitrary integer sequence (a 1 , a 2 ,... , an) is
called Hamiltonian, if every graph with n
vertices and a degree sequence pointwise
greater than (a 1 , a 2 ,... , an) is hamiltonian.
An integer sequence (a 1 , a 2 ,... , an) such that 0 ≤ a 1 ≤ a 2 ≤... an < n and n ≥ 3 is hamiltonian if and only if the following holds for every i < n/ 2 : ai ≤ i → an−i ≥ n − i.
Toghness Conjecture (Chv´atal 1973): There exists an integer t such that every t-tough graph has a Hamilton Cycle.
The 4-color conjecture can be reduced to simple 3 -connected maximal planar graphs, i.e to 3-connected triangulations. Considering the dual, we get that 4 -color conjecture is equivalent to the assertion that every 3 -connected cubic plane graph is 4 -face colorable.
If every 3 -connected cubic graph is hamiltonian the above assertion is true. Tutte demonstrated that this is not true.
Every 4 -connected planar graph has a hamiltonian cycle.
