TY - JOUR
AB - Van der Holst and Pendavingh introduced a graph parameter σ, which coincides with the more famous Colin de Verdière graph parameter μ for small values. However, the definition of a is much more geometric/topological directly reflecting embeddability properties of the graph. They proved μ(G) ≤ σ(G) + 2 and conjectured σ(G) ≤ σ(G) for any graph G. We confirm this conjecture. As far as we know, this is the first topological upper bound on σ(G) which is, in general, tight.
Equality between μ and σ does not hold in general as van der Holst and Pendavingh showed that there is a graph G with μ(G) ≤ 18 and σ(G) ≥ 20. We show that the gap appears at much smaller values, namely, we exhibit a graph H for which μ(H) ≥ 7 and σ(H) ≥ 8. We also prove that, in general, the gap can be large: The incidence graphs Hq of finite projective planes of order q satisfy μ(Hq) ∈ O(q3/2) and σ(Hq) ≥ q2.
AU - Kaluza, Vojtech
AU - Tancer, Martin
ID - 10335
JF - Combinatorica
SN - 0209-9683
TI - Even maps, the Colin de Verdière number and representations of graphs
ER -
TY - JOUR
AB - The problem of finding dense induced bipartite subgraphs in H-free graphs has a long history, and was posed 30 years ago by Erdős, Faudree, Pach and Spencer. In this paper, we obtain several results in this direction. First we prove that any H-free graph with minimum degree at least d contains an induced bipartite subgraph of minimum degree at least cH log d/log log d, thus nearly confirming one and proving another conjecture of Esperet, Kang and Thomassé. Complementing this result, we further obtain optimal bounds for this problem in the case of dense triangle-free graphs, and we also answer a question of Erdœs, Janson, Łuczak and Spencer.
AU - Kwan, Matthew Alan
AU - Letzter, Shoham
AU - Sudakov, Benny
AU - Tran, Tuan
ID - 9582
IS - 2
JF - Combinatorica
SN - 0209-9683
TI - Dense induced bipartite subgraphs in triangle-free graphs
VL - 40
ER -
TY - JOUR
AB - We find a graph of genus 5 and its drawing on the orientable surface of genus 4 with every pair of independent edges crossing an even number of times. This shows that the strong Hanani–Tutte theorem cannot be extended to the orientable surface of genus 4. As a base step in the construction we use a counterexample to an extension of the unified Hanani–Tutte theorem on the torus.
AU - Fulek, Radoslav
AU - Kynčl, Jan
ID - 7034
IS - 6
JF - Combinatorica
SN - 0209-9683
TI - Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4
VL - 39
ER -
TY - JOUR
AB - We show that the maximum number of edges bounding m faces in an arrangement of n line segments in the plane is O(m2/3n2/3+nα(n)+nlog m). This improves a previous upper bound of Edelsbrunner et al. [5] and almost matches the best known lower bound which is Ω(m2/3n2/3+nα(n)). In addition, we show that the number of edges bounding any m faces in an arrangement of n line segments with a total of t intersecting pairs is O(m2/3t1/3+nα(t/n)+nmin{log m,log t/n}), almost matching the lower bound of Ω(m2/3t1/3+nα(t/n)) demonstrated in this paper.
AU - Aronov, Boris
AU - Edelsbrunner, Herbert
AU - Guibas, Leonidas
AU - Sharir, Micha
ID - 4053
IS - 3
JF - Combinatorica
SN - 0209-9683
TI - The number of edges of many faces in a line segment arrangement
VL - 12
ER -
TY - JOUR
AB - Let C be a cell complex in d-dimensional Euclidean space whose faces are obtained by orthogonal projection of the faces of a convex polytope in d + 1 dimensions. For example, the Delaunay triangulation of a finite point set is such a cell complex. This paper shows that the in front/behind relation defined for the faces of C with respect to any fixed viewpoint x is acyclic. This result has applications to hidden line/surface removal and other problems in computational geometry.
AU - Edelsbrunner, Herbert
ID - 4069
IS - 3
JF - Combinatorica
SN - 0209-9683
TI - An acyclicity theorem for cell complexes in d dimension
VL - 10
ER -