A Survey of Recursive Combinatorics

The Combinatorics of Flat Folds: a Survey∗

We survey results on the foldability of flat origami models. The main topics are the question of when a given crease pattern can fold flat, the combinatorics of mountain and valley creases, and counting how many ways a given crease pattern can be folded. In particular, we explore generalizations of Maekawa’s and Kawasaki’s Theorems, develop a necessary and sufficient condition for a given assig...

On the centroid of recursive trees

It follows from our results that as n → ∞ the average distance between the root and the (nearer) centroid node of a recursive tree Tn tends to 1; and the average value of the label of the (nearer) centroid node tends to 5/2.

Recursive constructions for large sets of resolvable hybrid triple systems

Concerning cyclic group divisible designs with block size three

We determine a necessary and sufficient condition for the existence of a cyclic {3}-GDD with a uniform group size 6n. This provides a fundamental class of ingredients for some recursive constructions which settle existence of k-rotational Steiner triple systems completely.

A survey of the studies on Gallai and anti-Gallai graphs

The Gallai graph and the anti-Gallai graph of a graph G are edge disjoint spanning subgraphs of the line graph L(G). The vertices in the Gallai graph are adjacent if two of the end vertices of the corresponding edges in G coincide and the other two end vertices are nonadjacent in G. The anti-Gallai graph of G is the complement of its Gallai graph in L(G). Attributed to Gallai (1967), the study ...