Why One More Step Can Shrink a Network’s Required Seed Set
A directed-graph puzzle reveals why one more step can make a tiny set reach every node—and why the two-step version remains open.
graph theory
How a Random Ordering Turns Average Degree Into a Guaranteed Tree
A six-decade graph theory puzzle gets an elegant answer: random vertex orderings show when average density must contain every tree.
How Pruning a Quantum Graph Can Make Its Dynamics Clearer
A graph-theory framework maps quantum dissipation, revealing which network links matter and helping predict quantum decay spectra.
Erase the Drawing, Keep the Order—and Count Every Crossing
A combinatorial method turns local edge orders into a sharp lower bound on crossings, settling a long-standing complete-graph conjecture.
The Graph-Theory Rule That Helps Proteins Build One Shape
A graph-theory framework explains how local binding rules and assembly order can turn many possible networks into one reliable structure.
Why Circular Order Beats Randomness at Counting Certain Cycles
A new proof shows why a balanced circular tournament beats randomness at counting cycles whose lengths are divisible by four.
A Local Rule Reveals the Covering Density of Enormous Random Trees
A canonical covering rule turns local subtree patterns into large-scale packing and covering densities across diverse random-tree models.