Hello Math-Fun, (full text and illustrations on my personal blog page here: https://cinquantesignes.blogspot.com/2020/05/two-graphs-with-distinct-vertic...) The idea is to build a planar graph with distinct positive integers k at every vertex – this integer being the exact number of edges linking the node k to the graph. (Pardon my ignorance of the precise words but my knowledge in graph theory is close to –273.15°C.) Both (uncompleted) graphs showed here don't admit that an edge crosses another (a good example is known as the Three utilities problem): someone walking from a vertex to another must not meet any crossroad. The first graph is above and is called (by me) the Maxi-graph. We want indeed this graph to show the maximum number of edges compatible with the constraint. We have: 1 linked to 2, 2 linked to 1 and 3, 3 inked to 2, 4 and 5, 4 linked to 3, 6, 7 and 8, 5 linked to 3, 9, 10, 11 and 12, 6 linked to 4, 13, 14, 15, 16 and 17, 7 linked to 4, 18, 19, 20, 21, 22 and 23, 8 linked to 4, 24, 25, 26, 27, 28, 29 and 30, 9 linked to 5, ... etc. We could have drawn a few shortcuts (linking 5 to 8, for instance) – but no! this is precisely what the Maxi-graph is about: building as many roads as possible! The second graph is the Mini-graph – we take into account the environmental costs of asphalt and concrete; we want, as before, to build a graph starting with 1, than linking 1 to 2, then linking this 2 to another node (3), then link the node 3 to 4 and 5... with a first divergence here from the Maxi-graph: 4 will be linked to 3, as before, but also to 5! (and to 6 and 7). Here is how the Mini-graph looks (for the first 11 vertices) (...) Best, É.