Z2-genus of graphs and minimum rank of partial symmetric matrices

Abstrakt

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.

Klíčová slova

• graph genus
• minimum rank of a partial matrix
• Hanani–Tutte theorem
• graph amalgamation