Some Triangulated Surfaces without Balanced Splitting

Let G be the graph of a triangulated surface $\Sigma$ of genus $g\geq 2$. A cycle of G is splitting if it cuts $\Sigma$ into two components, neither of which is homeomorphic to a disk. A splitting cycle has type k if the corresponding components have genera k and g-k. It was conjectured that G contains a splitting cycle (Barnette '1982). We confirm this conjecture for an infinite family of triangulations by complete graphs but give counter-examples to a stronger conjecture (Mohar and Thomassen '2001) claiming that G should contain splitting cycles of every possible type.Comment: 15 pages, 7 figure

The complement of proper power graphs of finite groups

For a finite group $G$, the proper power graph $\mathscr{P}^*(G)$ of $G$ is the graph whose vertices are non-trivial elements of $G$ and two vertices $u$ and $v$ are adjacent if and only if $u \neq v$ and $u^m=v$ or $v^m=u$ for some positive integer $m$. In this paper, we consider the complement of $\mathscr{P}^*(G)$, denoted by ${\overline{\mathscr{P}^*(G)}}$. We classify all finite groups whose complement of proper power graphs is complete, bipartite, a path, a cycle, a star, claw-free, triangle-free, disconnected, planar, outer-planar, toroidal, or projective. Among the other results, we also determine the diameter and girth of the complement of proper power graphs of finite groups.Comment: 29 pages, 14 figures, Lemma 4.1 has been added and consequent changes have been mad

Jungerman ladders and index 2 constructions for genus embeddings of dense regular graphs

We construct several families of genus embeddings of near-complete graphs using index 2 current graphs. In particular, we give the first known minimum genus embeddings of certain families of octahedral graphs, solving a longstanding conjecture of Jungerman and Ringel, and Hamiltonian cycle complements, making partial progress on a problem of White. Index 2 current graphs are also applied to various cases of the Map Color Theorem, in some cases yielding significantly simpler solutions, e.g., the nonorientable genus of $K_{12s+8}-K_2$. We give a complete description of the method, originally due to Jungerman, from which all these results were obtained.Comment: 23 pages, 21 figures; fixed 2 figures from previous versio