[[alternative]]The Study of Decomposition, Covering and Packing of Complete Multi-Partite Graphs

計畫編號：NSC89-2115-M032-016研究期間：200008~200107研究經費：386,000[[sponsorship]]行政院國家科學委員

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

Random Embeddings of Graphs: The Expected Number of Faces in Most Graphs is Logarithmic

A random 2-cell embedding of a connected graph $G$ in some orientable surface is obtained by choosing a random local rotation around each vertex. Under this setup, the number of faces or the genus of the corresponding 2-cell embedding becomes a random variable. Random embeddings of two particular graph classes -- those of a bouquet of $n$ loops and those of $n$ parallel edges connecting two vertices -- have been extensively studied and are well-understood. However, little is known about more general graphs despite their important connections with central problems in mainstream mathematics and in theoretical physics (see [Lando & Zvonkin, Springer 2004]). There are also tight connections with problems in computing (random generation, approximation algorithms). The results of this paper, in particular, explain why Monte Carlo methods (see, e.g., [Gross & Tucker, Ann. NY Acad. Sci 1979] and [Gross & Rieper, JGT 1991]) cannot work for approximating the minimum genus of graphs. In his breakthrough work ([Stahl, JCTB 1991] and a series of other papers), Stahl developed the foundation of "random topological graph theory". Most of his results have been unsurpassed until today. In our work, we analyze the expected number of faces of random embeddings (equivalently, the average genus) of a graph $G$. It was very recently shown [Campion Loth & Mohar, arXiv 2022] that for any graph $G$, the expected number of faces is at most linear. We show that the actual expected number of faces is usually much smaller. In particular, we prove the following results: 1) $\frac{1}{2}\ln n - 2 < \mathbb{E}[F(K_n)] \le 3.65\ln n$, for $n$ sufficiently large. This greatly improves Stahl's $n+\ln n$ upper bound for this case. 2) For random models $B(n,\Delta)$ containing only graphs, whose maximum degree is at most $\Delta$, we show that the expected number of faces is $\Theta(\ln n)$.Comment: 44 pages, 6 figure

3-Maps And Their Generalizations

A 3-map is a 3-region colorable map. They have been studied by Craft and White in their paper 3-maps. This thesis introduces topological graph theory and then investigates 3-maps in detail, including examples, special types of 3-maps, the use of 3-maps to find the genus of special graphs, and a generalization known as n-maps