# セミナー

## 組合せ論セミナー

タイトル Hamiltonian Properties of Polyhedra 2017年1月23日 16:30～ Carol Zamfirescu 氏(Ghent University, Belgium) 矢上キャンパス14-631B In this talk we will discuss several related problems concerning hamiltonian properties of planar 3-connected graphs, which we will call polyhedra. The presentation will be divided intofour parts, with emphasis on new results, the techniques behind them, and open research problems. The first topic deals with 3-(vertex-)cuts in polyhedra, where we present a strengthening of the　classic theorem of Tutte that 4-connected polyhedra are hamiltonian (joint work with Gunnar　Brinkmann). This result also strengthens a theorem of Jackson and Yu on triangulations. In order toprovide context, we end this first part by giving an overview of results on hamiltonian properties　of polyhedra with few 3-cuts (based on joint work with Kenta Ozeki and Nico Van Cleemput). In the　second part of the talk, we present a geometrically motivated generalisation of Halin graphs, and　study the hamiltonian properties of this family of polyhedra. We strengthen a result of Bondy. This　is based on joint work with Boris Schauerte and Tudor Zamfirescu. The third topic concerns new　bounds for the orders of the smallest $k$-regular polyhedra, $k \in \{3,4,5\}$, which are　non-hamiltonian or non-traceable. We improve bounds of Zaks and Owens. This is joint work with Nico　Van Cleemput. The talk ends with a study of 3-fragments (and in consequence cubic vertices) in planar hypohamiltonian graphs (which are necessarily 3-connected, and thus polyhedra). If time　permits, more details -- such as sketches of proofs -- will be given for each of the above topics. http://www.comb.math.keio.ac.jp/seminar/index.html