From the Proof of the Four Color Theorem to P=NP

2021 ◽  
Author(s):  
Ibrahim Cahit
Keyword(s):  
Four Color Theorem

scholarly journals Toward a language theoretic proof of the four color theorem

2012 ◽  
Vol 48 (2) ◽  
pp. 414-431 ◽  
Author(s):  
Bobbe Cooper ◽  
Eric Rowland ◽  
Doron Zeilberger
Keyword(s):  
Theoretic Proof ◽  
Four Color Theorem

Classifying Graphs

2017 ◽  
Author(s):  
Arthur Benjamin ◽  
Gary Chartrand ◽  
Ping Zhang
Keyword(s):  
Graph Theory ◽  
Rational Number ◽  
Regular Graphs ◽  
Weighted Graphs ◽  
Reconstruction Problem ◽  
Regular Polyhedra ◽  
Four Color Theorem ◽  
Stanislaw Ulam ◽  
Irregular Graphs ◽  
Isomorphic Graphs

This chapter considers the richness of mathematics and mathematicians' responses to it, with a particular focus on various types of graphs. It begins with a discussion of theorems from many areas of mathematics that have been judged among the most beautiful, including the Euler Polyhedron Formula; the number of primes is infinite; there are five regular polyhedra; there is no rational number whose square is 2; and the Four Color Theorem. The chapter proceeds by describing regular graphs, irregular graphs, irregular multigraphs and weighted graphs, subgraphs, and isomorphic graphs. It also analyzes the degrees of the vertices of a graph, along with concepts and ideas concerning the structure of graphs. Finally, it revisits a rather mysterious problem in graph theory, introduced by Stanislaw Ulam and Paul J. Kelly, that no one has been able to solve: the Reconstruction Problem.

Adjacency constraints in harvest scheduling: an aggregation heuristic

1990 ◽  
Vol 20 (7) ◽  
pp. 978-986 ◽  
Author(s):  
Juan M. Torres-Rojo ◽  
J. Douglas Brodie
Keyword(s):  
Adjacency Matrix ◽  
Harvest Scheduling ◽  
Constraint Aggregation ◽  
Four Color Theorem ◽  
Surrounding Areas

An heuristic for adjacency constraint aggregation is proposed. The heuristic is composed of two procedures. Procedure 1 consists of identifying harvesting areas for which it is not necessary to write adjacency constraints. Procedure 2 consists of writing one adjacency constraint for each one of the harvesting areas not identified in procedure 1. Such adjacency constraints consider all the adjacency relations between the harvesting area and its surrounding areas. The heuristic is based on the concept of penalties and the four-color theorem. The aggregated constraints present fewer variables per constraint than the aggregator described by B.J. Meneghin, M.W. Kirby, and J.G. Jones (1988. USDA For. Serv. Rocky Mt. For. Range Exp. Stn. Gen. Tech. Rep. RM-161. pp. 46–53) and can easily be generated mechanically from the adjacency matrix. In addition, the proposed heuristic does not require the tedious task of identifying type 1 and 2 constraints as with Meneghin's algorithm. Hence the combinatorial work to compute the aggregated constraints is reduced significantly. Comparisons showed that the proposed heuristic requires about a third of the constraints required by the conventional adjacency constraint formulation and about the same number of constraints as the procedure suggested by B.J. Meneghin, M.W. Kirby, and J.G. Jones (1988).

Mathematical Proof of Four-Color Theorem

2019 ◽  
Vol 09 (03) ◽  
pp. 410-413
Author(s):  
山中 邹
Keyword(s):  
Mathematical Proof ◽  
Four Color Theorem

scholarly journals From the four-color theorem to a generalizing “four-letter theorem”: A sketch for “human proof” and the philosophical interpretation

2020 ◽  
Author(s):  
Vasil Dinev Penchev
Keyword(s):  
Mathematical Proof ◽  
General Theorem ◽  
Human Capabilities ◽  
Letter Alphabet ◽  
Philosophical Interpretation ◽  
Four Color Theorem ◽  
The Universe

The “four-color” theorem seems to be generalizable as follows. The four-letter alphabet is sufficient to encode unambiguously any set of well-orderings including a geographical map or the “map” of any logic and thus that of all logics or the DNA (RNA) plan(s) of any (all) alive being(s).Then the corresponding maximally generalizing conjecture would state: anything in the universe or mind can be encoded unambiguously by four letters.That admits to be formulated as a “four-letter theorem”, and thus one can search for a properly mathematical proof of the statement.It would imply the “four colour theorem”, the proof of which many philosophers and mathematicians believe not to be entirely satisfactory for it is not a “human proof”, but intermediated by computers unavoidably since the necessary calculations exceed the human capabilities fundamentally. It is furthermore rather unsatisfactory because it consists in enumerating and proving all cases one by one.Sometimes, a more general theorem turns out to be much easier for proving including a general “human” method, and the particular and too difficult for proving theorem to be implied as a corollary in certain simple conditions.The same approach will be followed as to the four colour theorem, i.e. to be deduced more or less trivially from the “four-letter theorem” if the latter is proved. References are only classical and thus very well-known papers: their complete bibliographic description is omitted.

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