Publication at Faculty of Mathematics and Physics, Central Library of Charles University |
2019
Abstract
The genus g(G) of a graph G is the minimum g such that G has an embedding on the orientable surface M_g of genus g. A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times.
The Z_2-genus of a graph G, denoted by g_0(G), is the minimum g such that G has an independently even drawing on M_g. We prove the following.
If G is a union of G_1 and G_2 where G_1 and G_2 intersect in two vertices u and, and G-u-v has k connected components (among which we count the edge uv if present, then |g_0(G)-(g_0(G_1)+g_0(G_2))|= m>= 3, we prove that g_0(K_{m,n})/g(K_{m,n})=1-O(1/n). Similar results are proved also for the Euler genus.