scholarly journals The complexity of the four colour theorem

2010 ◽  
Vol 13 ◽  
pp. 414-425 ◽  
Author(s):  
Cristian S. Calude ◽  
Elena Calude
Keyword(s):  
Complex Systems ◽  
Planar Graph ◽  
Riemann Hypothesis ◽  
Logic Program ◽  
Formal Proof ◽  
Complexity Class ◽  
Computational Method ◽  
Equational Logic ◽  
Four Color Theorem ◽  
Mathematical Problems

AbstractThe four colour theorem states that the vertices of every planar graph can be coloured with at most four colours so that no two adjacent vertices receive the same colour. This theorem is famous for many reasons, including the fact that its original 1977 proof includes a non-trivial computer verification. Recently, a formal proof of the theorem was obtained with the equational logic program Coq [G. Gonthier, ‘Formal proof–the four color theorem’,Notices of Amer. Math. Soc.55 (2008) no. 11, 1382–1393]. In this paper we describe an implementation of the computational method introduced by C. S. Calude and co-workers [Evaluating the complexity of mathematical problems. Part 1’,Complex Systems18 (2009) 267–285; A new measure of the difficulty of problems’,J. Mult. Valued Logic Soft Comput.12 (2006) 285–307] to evaluate the complexity of the four colour theorem. Our method uses a Diophantine equational representation of the theorem. We show that the four colour theorem is in the complexity class ℭU,4. For comparison, the Riemann hypothesis is in class ℭU,3while Fermat’s last theorem is in class ℭU,1.

The mathematical problems of complex systems investigation under uncertainties

2017 ◽  
pp. 135-171
Author(s):  
K. Atoyev ◽  
P. Knopov ◽  
V. Pepeliaev ◽  
P. Kisała ◽  
R. Romaniuk ◽  
...  
Keyword(s):  
Complex Systems ◽  
Mathematical Problems

Visual Information and Valid Reasoning

1996 ◽  
Author(s):  
Jon Barwise ◽  
John Etchemendy
Keyword(s):  
Visual Information ◽  
Formal Proof ◽  
Theory And Practice ◽  
Visual Images ◽  
Standard View ◽  
Mathematical Problems ◽  
Mathematical Tradition ◽  
Cognitive Activities ◽  
The Relationship ◽  
Mathematical Discovery

Psychologists have long been interested in the relationship between visualization and the mechanisms of human reasoning. Mathematicians have been aware of the value of diagrams and other visual tools both for teaching and as heuristics for mathematical discovery. As the chapters in this volume show, such tools are gaining even greater value, thanks in large part to the graphical potential of modern computers. But despite the obvious importance of visual images in human cognitive activities, visual representation remains a second-class citizen in both the theory and practice of mathematics. In particular, we are all taught to look askance at proofs that make crucial use of diagrams, graphs, or other nonlinguistic forms of representation, and we pass on this disdain to our students. In this chapter, we claim that visual forms of representation can be important, not just as heuristic and pedagogic tools, but as legitimate elements of mathematical proofs. As logicians, we recognize that this is a heretical claim, running counter to centuries of logical and mathematical tradition. This tradition finds its roots in the use of diagrams in geometry. The modern attitude is that diagrams are at best a heuristic in aid of finding a real, formal proof of a theorem of geometry, and at worst a breeding ground for fallacious inferences. For example, in a recent article, the logician Neil Tennant endorses this standard view: . . . [The diagram] is only an heuristic to prompt certain trains of inference; . . . it is dispensable as a proof-theoretic device; indeed, . . . it has no proper place in the proof as such. For the proof is a syntactic object consisting only of sentences arranged in a finite and inspectable array (Tennant [1984]). . . . It is this dogma that we want to challenge. We are by no means the first to question, directly or indirectly, the logocentricity of mathematics arid logic. The mathematicians Euler and Venn are well known for their development of diagrammatic tools for solving mathematical problems, and the logician C. S. Peirce developed an extensive diagrammatic calculus, which he intended as a general reasoning tool.

scholarly journals Mathematical Problems for Complex Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-2
Author(s):  
Haijun Jiang ◽  
Haibo He ◽  
Jianlong Qiu ◽  
Qiankun Song ◽  
Jianquan Lu
Keyword(s):  
Complex Systems ◽  
Mathematical Problems

scholarly journals Distributed Coloring in Sparse Graphs with Fewer Colors

10.37236/8395 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Pierre Aboulker ◽  
Marthe Bonamy ◽  
Nicolas Bousquet ◽  
Louis Esperet
Keyword(s):  
Planar Graph ◽  
Efficient Algorithm ◽  
Distributed Algorithm ◽  
Planar Graphs ◽  
List Coloring ◽  
Sparse Graphs ◽  
Four Color Theorem

This paper is concerned with efficiently coloring sparse graphs in the distributed setting with as few colors as possible. According to the celebrated Four Color Theorem, planar graphs can be colored with at most 4 colors, and the proof gives a (sequential) quadratic algorithm finding such a coloring. A natural problem is to improve this complexity in the distributed setting. Using the fact that planar graphs contain linearly many vertices of degree at most 6, Goldberg, Plotkin, and Shannon obtained a deterministic distributed algorithm coloring $n$-vertex planar graphs with 7 colors in $O(\log n)$ rounds. Here, we show how to color planar graphs with 6 colors in $\text{polylog}(n)$ rounds. Our algorithm indeed works more generally in the list-coloring setting and for sparse graphs (for such graphs we improve by at least one the number of colors resulting from an efficient algorithm of Barenboim and Elkin, at the expense of a slightly worst complexity). Our bounds on the number of colors turn out to be quite sharp in general. Among other results, we show that no distributed algorithm can color every $n$-vertex planar graph with 4 colors in $o(n)$ rounds.

scholarly journals A Linear-Time Algorithm for 4-Coloring Some Classes of Planar Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Zuosong Liang ◽  
Huandi Wei
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Linear Time ◽  
Time Algorithm ◽  
Vertex Coloring ◽  
Computer Assisted ◽  
Linear Time Algorithm ◽  
Four Color Theorem ◽  
Finite Set ◽  
Proper Vertex

Every graph G = V , E considered in this paper consists of a finite set V of vertices and a finite set E of edges, together with an incidence function that associates each edge e ∈ E of G with an unordered pair of vertices of G which are called the ends of the edge e . A graph is said to be a planar graph if it can be drawn in the plane so that its edges intersect only at their ends. A proper k -vertex-coloring of a graph G = V , E is a mapping c : V ⟶ S ( S is a set of k colors) such that no two adjacent vertices are assigned the same colors. The famous Four Color Theorem states that a planar graph has a proper vertex-coloring with four colors. However, the current known proof for the Four Color Theorem is computer assisted. In addition, the correctness of the proof is still lengthy and complicated. In 2010, a simple O n 2 time algorithm was provided to 4-color a 3-colorable planar graph. In this paper, we give an improved linear-time algorithm to either output a proper 4-coloring of G or conclude that G is not 3-colorable when an arbitrary planar graph G is given. Using this algorithm, we can get the proper 4-colorings of 3-colorable planar graphs, planar graphs with maximum degree at most five, and claw-free planar graphs.

scholarly journals Planar Ramsey Graphs

10.37236/8366 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Maria Axenovich ◽  
Ursula Schade ◽  
Carsten Thomassen ◽  
Torsten Ueckerdt
Keyword(s):  
Planar Graph ◽  
Four Color Theorem ◽  
Ramsey Graphs ◽  
Monochromatic Copy

We say that a graph $H$ is planar unavoidable if there is a planar graph $G$ such that any red/blue coloring of the edges of $G$ contains a monochromatic copy of $H$, otherwise we say that $H$ is planar avoidable. That is, $H$ is planar unavoidable if there is a Ramsey graph for $H$ that is planar. It follows from the Four-Color Theorem and a result of Gonçalves that if a graph is planar unavoidable then it is bipartite and outerplanar. We prove that the cycle on $4$ vertices and any path are planar unavoidable. In addition, we prove that all trees of radius at most $2$ are planar unavoidable and there are trees of radius $3$ that are planar avoidable. We also address the planar unavoidable notion in more than two colors.

scholarly journals The relationship of arithmetic as two twin Peano arithmetic(s) and set theory: A new glance from the theory of information

2020 ◽  
Author(s):  
Vasil Dinev Penchev
Keyword(s):  
Peano Arithmetic ◽  
Complex Hilbert Space ◽  
New Concepts ◽  
Four Color Theorem ◽  
The World ◽  
Mathematical Problems ◽  
Theory Of Information ◽  
Relationship Of ◽  
The Relationship ◽  
Mathematics And Physics

The paper introduces and utilizes a few new concepts: “nonstandard Peano arithmetic”, “complementary Peano arithmetic”, “Hilbert arithmetic”. They identify the foundations of both mathematics and physics demonstrating the equivalence of the newly introduced Hilbert arithmetic and the separable complex Hilbert space of quantum mechanics in turn underlying physics and all the world. That new both mathematical and physical ground can be recognized as information complemented and generalized by quantum information. A few fundamental mathematical problems of the present such as Fermat’s last theorem, four-color theorem as well as its new-formulated generalization as “four-letter theorem”, Poincaré’s conjecture, “P vs NP” are considered over again, from and within the new-founding conceptual reference frame of information, as illustrations. Simple or crucially simplifying solutions and proofs are demonstrated. The link between the consistent completeness of the system mathematics-physics on the ground of information and all the great mathematical problems of the present (rather than the enumerated ones) is suggested.

scholarly journals Mathematical Problems of Managing the Risks of Complex Systems under Targeted Attacks with Known Structures

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2468
Author(s):  
Alexander Shiroky ◽  
Andrey Kalashnikov
Keyword(s):  
Complex Systems ◽  
Optimal Allocation ◽  
Risk Function ◽  
Chain Structure ◽  
System Structure ◽  
Optimal Placement ◽  
Targeted Attacks ◽  
Mathematical Problems ◽  
Local Risk ◽  
Simple Chain

This paper deals with the problem of managing the risks of complex systems under targeted attacks. It is usually solved by using Defender–Attacker models or similar ones. However, such models do not consider the influence of the defending system structure on the expected attack outcome. Our goal was to study how the structure of an abstract system affects its integral risk. To achieve this, we considered a situation where the Defender knows the structure of the expected attack and can arrange the elements to achieve a minimum of integral risk. In this paper, we consider a particular case of a simple chain attack structure. We generalized the concept of a local risk function to account for structural effects and found an ordering criterion that ensures the optimal placement of the defending system’s elements inside a given simple chain structure. The obtained result is the first step to formulate the principles of optimally placing system elements within an arbitrarily complex network. Knowledge of these principles, in turn, will allow solving the problems of optimal allocation of resources to minimize the risks of a complex system, considering its structure.

scholarly journals On $t$-Common List-Colorings

10.37236/6738 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Hojin Choi ◽  
Young Soo Kwon
Keyword(s):  
Positive Integer ◽  
Planar Graph ◽  
Chromatic Number ◽  
Upper Bound ◽  
Nonnegative Integer ◽  
Planar Graphs ◽  
Four Color Theorem ◽  
List Colorings ◽  
List Chromatic Number ◽  
Positive Integers

In this paper, we introduce a new variation of list-colorings. For a graph $G$  and for a given nonnegative integer $t$, a $t$-common list assignment of $G$ is a mapping $L$ which assigns each vertex $v$ a set $L(v)$ of colors such that given set of $t$ colors belong to $L(v)$ for every $v\in V(G)$. The $t$-common list chromatic number of $G$ denoted by $ch_t(G)$ is defined as the minimum positive integer $k$ such that there exists an $L$-coloring of $G$ for every $t$-common list assignment $L$ of $G$, satisfying $|L(v)| \ge k$ for every vertex $v\in V(G)$. We show that for all positive integers $k, \ell$ with $2 \le k \le \ell$ and for any positive integers $i_1 , i_2, \ldots, i_{k-2}$ with $k \le i_{k-2} \le \cdots \le i_1 \le \ell$, there exists a graph $G$ such that $\chi(G)= k$, $ch(G) =  \ell$ and $ch_t(G) = i_t$ for every $t=1, \ldots, k-2$. Moreover, we consider the $t$-common list chromatic number of planar graphs. From the four color theorem and the result of Thomassen (1994), for any $t=1$ or $2$, the sharp upper bound of $t$-common list chromatic number of planar graphs is $4$ or $5$. Our first step on $t$-common list chromatic number of planar graphs is to find such a sharp upper bound. By constructing a planar graph $G$ such that $ch_1(G) =5$, we show that the sharp upper bound for $1$-common list chromatic number of planar graphs is $5$. The sharp upper bound of $2$-common list chromatic number of planar graphs is still open. We also suggest several questions related to $t$-common list chromatic number of planar graphs.

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