No abstract

Abstract. Let R be a connected 2-manifold without boundary obtained from a (possibly infinite) collection of polygons by identifying them along edges of equal length. Let V be the set of vertices, and for every v ∈ V , let κ(v) denote the (Gaussian) curvature of v: 2π minus the sum of incident polygon angles. Descartes showed that v∈V κ(v) = 4π whenever R may be realized as the surface of a convex polytope in R 3 . More generally, if R is made of finitely many polygons, Euler's formula is equivalent to the equation v∈V κ(v) = 2πχ(R) where χ(R) is the Euler characteristic of R. Our main theorem shows that whenever v∈V :κ(v)<0 κ(v) converges and there is a positive lower bound on the distance between any pair of vertices in R, there exists a compact closed 2-manifold S and an integer t so that R is homeomorphic to S minus t points, and further v∈V κ(v) ≤ 2πχ(S) − 2πt.In the special case when every polygon is regular of side length one and κ(v) > 0 for every vertex v, we apply our main theorem to deduce that R is made of finitely many polygons and is homeomorphic to either the 2-sphere or to the projective plane. Further, we show that unless R is a prism, antiprism, or the projective planar analogue of one of these that |V | ≤ 3444. This resolves a recent conjecture of Higuchi.

An immersion of a graph H into a graph G is a one-to-one mapping f : V (H) → V (G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P uv corresponding to edge uv has endpoints f (u) and f (v). The immersion is strong if the paths P uv are internally disjoint from f (V (H)). It is proved that for every positive integer t, every simple graph of minimum degree at least 200t contains a strong immersion of the complete graph K t . For dense graphs one can say even more. If the graph has order n and has 2cn 2 edges, then there is a strong immersion of the complete graph on at least c 2 n vertices in G in which each path P uv is of length 2. As an application of these results, we resolve a problem raised by Paul Seymour by proving that the line graph of every simple graph with average degree d has a clique minor of order at least cd 3/2 , where c > 0 is an absolute constant. For small values of t, 1 ≤ t ≤ 7, every simple graph of minimum degree at least t − 1 contains an immersion of K t (Lescure and Meyniel [13], DeVos et al. [6]). We provide a general class of examples showing that this does not hold when t is large.

International audienc

This article proves the conjecture of Thomas that, for every graph G; there is an integer k such that every graph with no minor isomorphic to G has a 2-coloring of either its vertices or its edges where each color induces a graph of tree-width at most k: Some generalizations are also proved. r