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Alternatively, If A is an adjacency matrix for the graph, modified to have nonzero entries on its main diagonal, then the nonzero entries of A k give the adjacency matrix of the k th power of the graph, [14] from which it follows that constructing k th powers may be performed in an amount of time that is within a logarithmic factor of the time ...
The works of Ramsey on colorations and more specially the results obtained by Turán in 1941 was at the origin of another branch of graph theory, extremal graph theory. The four color problem remained unsolved for more than a century. In 1969 Heinrich Heesch published a method for solving the problem using computers. [29]
A subdivision of a graph is a graph formed by subdividing its edges into paths of one or more edges. Kuratowski's theorem states that a finite graph G {\displaystyle G} is planar if it is not possible to subdivide the edges of K 5 {\displaystyle K_{5}} or K 3 , 3 {\displaystyle K_{3,3}} , and then possibly add additional edges and vertices, to ...
Pages in category "Unsolved problems in graph theory" The following 32 pages are in this category, out of 32 total. This list may not reflect recent changes. A.
When the concept of "scale-free" was initially introduced in the context of networks, [5] it primarily referred to a specific trait: a power-law distribution for a given variable , expressed as (). This property maintains its form when subjected to a continuous scale transformation k → k + ϵ k {\displaystyle k\to k+\epsilon k} , evoking ...
The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: The problem instance has a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-task assignment.
If P is any graph property which is monotone with respect to the subgraph ordering (meaning that if A is a subgraph of B and B satisfies P, then A will satisfy P as well), then the statements "P holds for almost all graphs in G(n, p)" and "P holds for almost all graphs in (, ())" are equivalent (provided pn 2 → ∞).
In this model, each process is modeled as a node of a graph. Each communication channel between two processes is an edge of the graph. Two commonly used algorithms for the classical minimum spanning tree problem are Prim's algorithm and Kruskal's algorithm. However, it is difficult to apply these two algorithms in the distributed message ...