Orientations of hamiltonian cycles in large digraphs

We prove that, with some exceptions, every digraph with n ≥ 9 vertices and at least (n - 1) (n - 2) + 2 arcs contains all orientations of a Hamiltonian cycle

EXTENDING POTOČNIK AND ŠAJNA'S CONDITIONS ON VERTEX-TRANSITIVE SELF-COMPEMENTARY k-HYPERGRAPHS

Abstract : Let l be a positive integer, k = 2l or k = 2l + 1 and let n be a positive integer with $n \equiv 1$ (mod $2^{l+1}$). Potocnik and Sajna showed that if there exists a vertex-transitive self-complementary k-hypergraph of order n, then for every prime p we have $p^{n_{(p)}} \equiv 1 \pmod {2^{l+1}}$ (where $n_{(p)}$ denotes the largest integer $i$ for which $p^i$ divides $n$). Here we extend their result to any integer k and a larger class of integers n

Self-complementing permutations of k-uniform hypergraphs

Graphs and Algorithm

Self-complementing permutations of k-uniform hypergraphs

Graphs and AlgorithmsA k-uniform hypergraph H = ( V; E) is said to be self-complementary whenever it is isomorphic with its complement (H) over bar = ( V; ((V)(k)) - E). Every permutation sigma of the set V such that sigma(e) is an edge of (H) over bar if and only if e is an element of E is called self-complementing. 2-self-comlementary hypergraphs are exactly self complementary graphs introduced independently by Ringel ( 1963) and Sachs ( 1962). For any positive integer n we denote by lambda(n) the unique integer such that n = 2(lambda(n)) c, where c is odd. In the paper we prove that a permutation sigma of [1, n] with orbits O-1,..., O-m O m is a self-complementing permutation of a k-uniform hypergraph of order n if and only if there is an integer l >= 0 such that k = a2(l) + s, a is odd, 0 (i)n = b2(l+1) + r,r is an element of {0,..., 2(l) - 1 + s}, and (ii) Sigma(i:lambda(vertical bar Oi vertical bar) For k = 2 this result is the very well known characterization of self-complementing permutation of graphs given by Ringel and Sachs

On minimum (Kq; k) stable graphs

A graph G is a (K_q; k) vertex stable graph (q >= 3) if it contains a clique K_q after deleting any subset of k vertices (k >= 0). We are interested by the (K_q; kappa(q)) vertex stable graphs of minimum size where kappa(q) is the maximum value for which for every nonnegative integer k =3) s'il contient une clique K_q après destruction de n'importe quel sous-ensemble de k sommets (k>=0). Nous nous intéressons aux graphes (K_q; kappa(q)) stables ayant un nombre d'arêtes minimum, où kappa(q) est la plus grande valeur possible pour laquelle pour tout entier k < kappa(q) le seul graphe (Kq; k) stable minimum est la clique K_{q+k}