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.

Bollobás, Brightwell and Leader [2] showed that there are at most 2( n 2)+o(n 2) 2-SAT functions on n variables, and conjectured that in fact almost every 2-SAT function is unate: i.e., has a 2-SAT formula in which no variable’s positive and negative literals both appear. We prove their conjecture, finding the number of 2-SAT functions on n variables to be 2( n 2)+n(1 + o(1)). As a corollary of...

Trivially, a SOMA(k, n) is Trojan for k = 0 and k = 1. Moreover, R. A. Bailey proved that every SOMA(n − 1, n) is Trojan. Computational analysis of some examples suggested that a similar result held for k = n − 2. This led Bailey, Cameron and Soicher to ask (BCC Problem 16.19 in Discrete Math. Vol. 197/198) whether a SOMA(n − 2, n) must be Trojan. In this paper, we prove that this is indeed the...

We study reals with infinitely many incompressible prefixes. Call A ∈ 2 Kolmogorov random if (∃∞n) C(A n) > n − O(1), where C denotes plain Kolmogorov complexity. This property was suggested by Loveland and studied by Martin-Löf, Schnorr and Solovay. We prove that 2-random reals are Kolmogorov random.1 Together with the converse—proved by Nies, Stephan and Terwijn [11]—this provides a natural c...