An extremal problem of d permutations containing every permutation of every t elements

An extremal problem for a graphic sequence to have a realization containing every 2-tree with prescribed size

A graph G is a 2-tree if G = K3, or G has a vertex v of degree 2, whose neighbors are adjacent, and G − v is a 2-tree. Clearly, if G is a 2-tree on n vertices, then |E(G)| = 2n − 3. A non-increasing sequence π = (d1, . . . , dn) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. Yin and Li (Acta Mathematica Sinica, English Series, 25(2009)795–80...

Graphs Containing Every 2-Factor

For a graph G, let σ2(G) = min{d(u) + d(v) : uv / ∈ E(G)}. We prove that every n-vertex graph G with σ2(G) ≥ 4n/3−1 contains each 2-regular n-vertex graph. This extends a theorem due to Aigner and Brandt and to Alon and Fisher.

Infinite words containing squares at every position

Richomme asked the following question: what is the infimum of the real numbers α > 2 such that there exists an infinite word that avoids α-powers but contains arbitrarily large squares beginning at every position? We resolve this question in the case of a binary alphabet by showing that the answer is α = 7/3.

Say every word on every slide

A Variant of the Euclid-Mullin Sequence Containing Every Prime

We consider a generalization of Euclid’s proof of the infinitude of primes and show that it leads to variants of the Euclid-Mullin sequence that provably contain every prime number.