A tight lower bound on the maximum genus of a simplicial graph

Remarks on the lower bounds for the average genus

“…Theorem 3.4 improve a result of Chen and Gross [2] that for a 2-connected simple graph G with minimum degree 3, the average genus is larger than clog(β(G)) for a positive constant c, as well as a result of Chen, Gross and Rieper [5] that the average genus of a simple graph G with minimum degree 3 is larger than β(G) 8 .…”

Linear Time Canonicalization and Enumeration of Non-Isomorphic 1-Face Embeddings

“…Furthermore, the finer partitionings of the data sets according to the number of edges, the size of the automorphism group, or the connectivity, yield data sets which still contain sufficiently-many graphs and are difficult enough to gain meaningful insights into the performance of the algorithms. We consider our choices of parameters to further partition the data sets natural, and in particular we focus on girth and connectivity due to their known relationships with the maximum genus [7,8,20,38].…”

Maximum genus, connectivity, and Nebeský's Theorem

“…The next case is when G is simple and connected, first done in [4]. As before, we need to show that ν(G, A) ≤ (β(G) − 2)/2 for every pair (G, A).…”

“…We also get analogous results for Betti number and vertexconnectivity. These classes have been examined before [2,4,5,7,8,11], nevertheless, we present a unified approach with new, much shorter proofs, providing additional information about the structure of the extremal graphs.…”