A biased graph consists of a graph T and a subclass B of the polygons of T, such that no theta subgraph of T contains exactly two members of B. A subgraph is balanced when all its polygons belong to B. A vertex is a balancing vertex if deleting it leaves a balanced graph. We give a construction for unbalanced biased graphs having a balancing vertex and we show that an unbalanced biased graph ha...

Blbzsik, Z., M. Hujter, A. Pluhir and Z. Tuza, Graphs with no induced C4 and 2K,, Discrete Mathematics 115 (1993) 51-55. We characterize the structure of graphs containing neither the 4-cycle nor its complement as an induced subgraph. This self-complementary class B of graphs includes split graphs, which are graphs whose vertex set is the union of a clique and an independent set. In the members...

Let r be a positive integer. Consider r-regular graphs in which no induced subgraph on four vertices is an independent pair of edges. The number v of vertices in such a graph does not exceed 5r/2; this proves a conjecture of Bermond. More generally, it is conjectured that if v > 2r, then the ratio v/r must be a rational number of the form 2 + 1/(2k). This is proved for v/r > 21 _ ~ . The extrem...

We study the length Lk of the shortest permutation containing all patterns of length k. We establish the bounds e−2k2 < Lk ≤ (2/3 + o(1))k2 . We also prove that as k → ∞, there are permutations of length (1/4+o(1))k2 containing almost all patterns of length k.

Several renowned open conjectures in combinatorics and number theory involve arithmetic progressions. Van der Waerden famously proved in 1927 that for each positive integer k there exists a least positive integer w(k) such that any 2-coloring of 1, . . . , w(k) produces a monochromatic k-term arithmetic progression. The best known upper bound for w(k) is due to Gowers and is quite large. Ron Gr...

In the analysis of algebraic equations in discrete statistical systems, it is often necessary to find the complete sets of non-negative integer solutions k1,k2,k3, . . . to the partition–distribution equations: k1 þ k2 þ k3 þ . . . 1⁄4 m and k1þ 2k2 þ 3k3 þ . . . 1⁄4 n. In this paper, two efficient methods, indirect and recurrent, are derived to find the desired results for any given positive i...