Some Problems in Combinatorics

Let In = {1, 2, . . . , n} and x : In 7→ R be a map such that ∑ i∈In x(i) ≥ 0. (For any i, its image is denoted by x(i).) Let F = {J ⊂ In : |J | = k, and ∑ j∈J x(j) ≥ 0}. In [25] Manickam and Singhi have conjectured that |F| ≥ ( n−1 k−1 ) whenever n ≥ 4k and showed that the conclusion of the conjecture holds when k divides n. For any two integers r and ` let [r]` denote the smallest positive integer congruent to r (mod `). In [11] Bier and Manickam have shown that if k > 3 and n ≥ k(k − 1)(k − 2) + k(k − 1)(k − 2) + k[n]k then the conjecture holds. In Chapter 2 we give a short proof to show that the conjecture holds when n ≥ 2ek. Let Φ be an irreducible root system and ∆ be a base for Φ; it is well known that any root in Φ is an integral combination of the roots in ∆. In comparison to this fact, we establish the following result in Chapter 3: Any indecomposable subset T of Φ is contained in the Z-span of an indecomposable linearly independent subset of T . With every graph one can associate a certain convex polyhedral cone, called its alternating cone (because integral vectors in the cone correspond to closed alternating walks (edge, nonedge, edge, nonedge,...) in the graph). This definition is motivated by the notion of threshold graphs. In Chapter 4 we study the alternating cone and related questions.