Graph Colouring: The process of assigning colours to elements of a graph, typically vertices, so that adjacent elements receive different colours.

Planar graph: A graph that can be embedded in the plane without any of its edges crossing. Distance query: A computation request that involves determining the shortest path or minimum distance ...

It is proved that every finite graph with n≧7 vertices and at least $\frac {9} {2}\mathrm {n}-12$ edges which is not a 4-cockade composed of complete 6-graphs and/or complete 8-graphs with four ...