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The n th crossing number of a graph G, denoted cr n (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 for which cr 0 (G) = a, cr 1 (G) = b, and cr 2 (G) = 0. This provides support for a conjecture of Archdeacon et al. and resolves a problem of Salazar.

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. This partially answers questions of Erd˝ os, Saks, and Sós and of Pultr.

Eberhard proved that for every sequence (p k), 3 ≤ k ≤ r, k = 5, 7 of non-negative integers satisfying Euler's formula P k≥3 (6 − k)p k = 12, there are infinitely many values p6 such that there exists a simple convex polyhedron having precisely p k faces of length k for every k ≥ 3, where p k = 0 if k > r. In this paper we prove a similar statement when… (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 on the class of graphs, which seems to be an essential extension of the homomor-phism order (studied extensively, see [10]). In this paper we study… (More)

Tension-continuous (shortly T T) mappings are mappings between the edge sets of graphs. They generalize graph homomorphisms. From another perspective , tension-continuous mappings are dual to the notion of flow-continuous mappings and the context of nowhere-zero flows motivates several questions considered in this paper. Extending our earlier research we… (More)