For a graph G = (V,E) and x : E â†’ < satisfying âˆ‘ e3v xe = 1 for each v âˆˆ V , set h(x) = âˆ‘ e xe log(1/xe) (with log = log2). We show that for any n-vertex G, random (not necessarily uniform) perfectâ€¦ (More)

We show that every regular tournament on n vertices has at least n!/(2+o(1)) Hamiltonian cycles, thus answering a question of Thomassen [17] and providing a partial answer to a question of Friedgutâ€¦ (More)

Let G be a simple graph of order n with no isolated vertices and no isolated edges. For a positive integer w, an assignment f on G is a function f : E(G) â†’ {1, 2, . . . , w}. For a vertex v, f(v) isâ€¦ (More)