Nonorientable genus of nearly complete bipartite graphs

Nonorientable Genus of Nearly Complete Bipartite Graphs

The nonorientable genus of complete tripartite graphs

In 1976, Stahl and White conjectured that the nonorientable genus of Kl,m,n, where l ≥ m ≥ n, is (l−2)(m+n−2) 2 . The authors recently showed that the graphs K3,3,3 , K4,4,1, and K4,4,3 are counterexamples to this conjecture. Here we prove that apart from these three exceptions, the conjecture is true. In the course of the paper we introduce a construction called a transition graph, which is cl...

Classification of nonorientable regular embeddings of complete bipartite graphs

A 2-cell embedding of a graph G into a closed (orientable or nonorientable) surface is called regular if its automorphism group acts regularly on the flags mutually incident vertex-edge-face triples. In this paper, we classify the regular embeddings of complete bipartite graphs Kn,n into nonorientable surfaces. Such a regular embedding of Kn,n exists only when n = 2p a1 1 p a2 2 · · · p ak k (a...

META-HEURISTIC ALGORITHMS FOR MINIMIZING THE NUMBER OF CROSSING OF COMPLETE GRAPHS AND COMPLETE BIPARTITE GRAPHS

The minimum crossing number problem is among the oldest and most fundamental problems arising in the area of automatic graph drawing. In this paper, eight population-based meta-heuristic algorithms are utilized to tackle the minimum crossing number problem for two special types of graphs, namely complete graphs and complete bipartite graphs. A 2-page book drawing representation is employed for ...

Nearly bipartite graphs

We prove that if a nonbipartite graph G on n vertices has minimal degree δ ≥ n 4k + 2 + ck,m where ck,m does not depend on n and n is sufficiently large, if C2s+1 ⊂ G for some k ≤ s ≤ 4k + 1 then C2s+2j+1 ⊂ G for every j = 1, ...,m. We give a structural description of all graphs on n vertices with