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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.

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.

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)

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 adjacent vertices (or staying). The goal of the robber is to enter the guarded region at a vertex with no cop on it. The problem is to find the minimum number of… (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)