The graph-embedding problem concerns the determination of surfaces in which a graph can be embedded and thereby generalizes the planarity problem.
It was not until the late 1960s that the embedding problem for the complete graphs four-colour map problem, which asks whether the countries on every map can be coloured by using just four colours in such a way that countries sharing an edge have different colours.
The knight’s tour ( number game: Chessboard problems) is another example of a recreational problem involving a Hamiltonian circuit.
Hamiltonian graphs have been more challenging to characterize than Eulerian graphs, since the necessary and sufficient conditions for the existence of a Hamiltonian circuit in a connected graph are still unknown.
The vertices and edges of a polyhedron form a graph on its surface, and this notion led to consideration of graphs on other surfaces such as a torus (the surface of a solid doughnut) and how they divide the surface into disklike faces.
Euler’s formula was soon generalized to surfaces as Euler characteristic).
If there is a path linking any two vertices in a graph, that graph is said to be connected.
A path that begins and ends at the same vertex without traversing any edge more than once is called a A graph is a collection of vertices, or nodes, and edges between some or all of the vertices.
in Indo-Hungarian Pre-Conference School of Conference on Algorithm and Discrete Applied Mathematics (CALDAM 2016) organized by Department of Future Studies, University of Kerala, Thiruvanathpuram, during Feb.18-20, 2016.
in ADMA Pre-Conference Workshop on Recent Advances in Signed Graphs and their Applications, organized by Department of Mathematics, Siddaganga Institute of Technology, Tumkur, Karnataka, during June 06-08, 2016., at BITS Pilani KK Birla Goa Campus, Goa, sponsored by National Board of Higher Mathematics NBHM in collaboration with School of Technology and Computer Science, Tata Institute of Fundamental Research(TIFR) Mumbai, during Jan.