Let F be a finite field of characteristic 2 and h be the element x3 + y3 + xyz of F [[x, y, z]]. In an earlier paper we made a precise conjecture as to the values of the colengths of the ideals (x, y, z, h) for q a power of 2. We also showed that if the conjecture holds then the Hilbert-Kunz series of H = uv+ h is algebraic (of degree 2) over Q(w), and that μ(h) is algebraic (explicitly, 4 3+ 5...

A hypergraph H is linear if no two distinct edges of H intersect in more than one vertex and loopless if no edge has size one. A q-edge-colouring of H is a colouring of the edges of H with q colours such that intersecting edges receive different colours. We use ∆H to denote the maximum degree of H. A well known conjecture of Erdös, Farber and Lovász is equivalent to the statement that every loo...

In 1971, McMullen and Walkup posed the following conjecture, which is called the generalized lower bound conjecture: If P is a simplicial d-polytope then its h-vector (h0, h1, . . . , hd) satisfies h0 ≤ h1 ≤ · · · ≤ h⌊ d2 ⌋. Moreover, if hr−1 = hr for some r ≤ d2 then P can be triangulated without introducing simplices of dimension ≤ d− r. The first part of the conjecture was solved by Stanley ...

The second part of this conjecture is analogous to Conjecture 2.2 (i) in [PV], which considers hypersurfaces in P. If C is a curve of genus 2 as above and P = (a : y : b) is a rational point on C (i.e., we have F (a, b) = y with a, b coprime integers), then we denote by H(P ) the height H(a : b) = max{|a|, |b|} of its x-coordinate. Conjecture 2. Let ε > 0. Then there is a constant Bε and a Zari...

In this paper we prove the following conjecture of Alon and Yuster. Let H be a graph with h vertices and chromatic number k. There exist constants c(H) and n0(H) such that if n¿n0(H) and G is a graph with hn vertices and minimum degree at least (1 − 1=k)hn + c(H), then G contains an H-factor. In fact, we show that if H has a k-coloring with color-class sizes h16h26 · · · 6h k , then the conject...

The bipartite case of the Bollob as and Koml os conjecture states that for every 0; > 0 there is an = ( 0; ) > 0 such that the following statement holds: If G is any graph with (G) n2 + n; then G contains as subgraphs all bipartite graphs, H; satisfying (H) 0 and b(H) n: Here b(H); the bandwidth of H, is the smallest b such that the vertices of H can be ordered as v1; : : : ; vn such that vi H ...

The Lescure–Meyniel conjecture is the analogue of Hadwiger’s for immersion order. It states that every graph G contains complete Kχ(G) as an immersion, and like its minor-order counterpart it open even graphs with independence number 2. We show α(G)≥2 no hole length between 4 2α(G) satisfies this conjecture. In particular, C4-free α(G)=2 give another generalisation corollary, follows. Let H be ...

The square G2 of a graph G is the graph defined on V (G) such that two vertices u and v are adjacent in G2 if the distance between u and v in G is at most 2. Let χ(H) and χl(H) be the chromatic number and the list chromatic number of H, respectively. A graph H is called chromatic-choosable if χl(H) = χ(H). It is an interesting problem to find graphs that are chromatic-choosable. Motivated by th...

We show that Ambro–Kawamata’s non-vanishing conjecture holds true for a quasi-smooth WCI X which is Fano or Calabi-Yau, i.e. we prove that, if H is an ample Cartier divisor on X , then |H | is not empty. If X is smooth, we further show that the general element of |H | is smooth. We then verify Ambro–Kawamata’s conjecture for any quasi-smooth weighted hypersurface. We also verify Fujita’s freene...

where {e1, · · · , en} (resp. {ξ1, · · · , ξm}) is an orthonormal basis of the tangent (resp. normal space) at the point x ∈ M , and R,R are the curvature tensors for the tangent and normal bundles, respectively. In the study of submanifold theory, De Smet, Dillen, Verstraelen, and Vrancken [5] made the following DDVV Conjecture: Conjecture 1. Let h be the second fundamental form, and let H = 1...