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Tag Archives: Trees
A Proof by Induction with a Difficulty
The time has come to prove that the number of edges in every finite tree is one less than the number of vertices (a tree is a connected graph with no cycle). The proof is by induction, but first you need … Continue reading
A Beautiful Garden of Hypertrees
We had a series of posts (1,2,3,4) “from Helly to Cayley” on weighted enumeration of Q-acyclic simplicial complexes. The simplest case beyond Cayley’s theorem were Q-acyclic complexes with vertices, edges, and triangles. One example is the six-vertex triangulation of the … Continue reading
Helly’s Theorem, “Hypertrees”, and Strange Enumeration I
1. Helly’s theorem and Cayley’s formula Helly’s theorem asserts: For a family of n convex sets in , n > d, if every d+1 sets in the family have a point in common then all members in the family have a point in common. … Continue reading
Posted in Combinatorics, Convexity
Tagged Cayley theorem, Helly Theorem, Simplicial complexes, Topological combinatorics, Trees
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