Apple improved the Math Notes feature in iOS 26 and iPadOS 26, allowing users to input equations to create graphs in three ...

And knowing that a subgraph is even or odd can give additional information about the structure of the graph. For example, an even graph that is “connected” — meaning you can always find a path between ...

The study of such graphs is called graph theory. Engineers need to find planarity in a graph when, for example, they are designing a computer chip without a crossed wire.

Long monochromatic paths in random graphs presented by Andrzej Dudek, Ph.D., Department of Mathematics, Western Michigan University. Abstract: Recall that the size-Ramsey number of F, r^(F, r), is the ...

As with our original example, this result about triangle-free graphs is related to the Erdős-Hajnal conjecture. If a graph is triangle-free, then it can’t have a clique larger than size 2, since a ...