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A mathematical chess problem is a mathematical problem which is formulated using a chessboard and chess pieces. These problems belong to recreational mathematics. The most well-known problems of this kind are the eight queens puzzle and the knight's tour problem, which have connection to graph theory and combinatorics. Many famous ...
In graph theory, the degree diameter problem is the problem of finding the largest possible graph for a given maximum degree and diameter.The Moore bound sets limits on this, but for many years mathematicians in the field have been interested in a more precise answer.
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.
The size of G is bounded above by the Moore bound; for 1 < k and 2 < d, only the Petersen graph, the Hoffman-Singleton graph, and possibly graphs (not yet proven to exist) of diameter k = 2 and degree d = 57 attain the Moore bound. In general, the largest degree-diameter graphs are much smaller in size than the Moore bound.
The Pappus graph. The Levi graph of the Pappus configuration is known as the Pappus graph.It is a bipartite symmetric cubic graph with 18 vertices and 27 edges. [3]Adding three more parallel lines to the Pappus configuration, through each triple of points that are not already connected by lines of the configuration, produces the Hesse configuration.
A seven-coloring of the plane, and a four-chromatic unit distance graph in the plane (the Moser spindle), proving that the chromatic number of a plane is bounded above by 7 and below by 4 The Golomb graph, Solomon W. Golomb's ten-vertex four-chromatic unit distance graph. In geometric graph theory, the Hadwiger–Nelson problem, named after ...
The wheel graph can be realized as a strict unit distance graph with six of its vertices forming a unit regular hexagon and the seventh at the center of the hexagon. Removing one of the edges from the center vertex produces a subgraph that still has unit-length edges, but which is not a strict unit distance graph.
For a Hamiltonian decomposition to exist in an undirected graph, the graph must be connected and regular of even degree. A directed graph with such a decomposition must be strongly connected and all vertices must have the same in-degree and out-degree as each other, but this degree does not need to be even. [1]