The n crossing number of a graph G, denoted crn(G), is the minimum number of crossings in a drawing of G on an orientable surface of genus n. We prove that for every a > b > 0, there exists a graph G… (More)

Let A be a finite nonempty subset of an additive abelian group G, and let Σ(A) denote the set of all group elements representable as a sum of some subset of A. We prove that |Σ(A)| ≥ |H|+ 1 64 |A… (More)

The Shortest Cycle Cover Conjecture asserts that the edges of every bridgeless graph with m edges can be covered by cycles of total length at most 7m/5 = 1.4m. We show that every bridgeless graph… (More)

The guarding game is a game in which a set of cops has to guard a region in a digraph against a robber. The robber and the cops are placed on vertices of the graph; they take turns in moving to… (More)

We prove that every connected triangle-free graph on n vertices contains an induced tree on exp(c √ log n ) vertices, where c is a positive constant. The best known upper bound is (2 + o(1)) √ n.… (More)

We consider mappings between edge sets of graphs that lift tensions to tensions. Such mappings are called tension-continuous mappings (shortly mappings). Existence of a mapping induces a (quasi)order… (More)

We introduce a new graph invariant that measures fractional covering of a graph by cuts. Besides being interesting in its own, it is useful for study of homomorphisms and tension-continuous mappings.… (More)