Hurwitz equivalence of braid monodromies and extremal elliptic surfaces

We discuss the equivalence between the categories of certain ribbon graphs and subgroups of the modular group $\Gamma$ and use it to construct exponentially large families of not Hurwitz equivalent simple braid monodromy factorizations of the same element. As an application, we also obtain exponentially large families of {\it topologically} distinct algebraic objects such as extremal elliptic surfaces, real trigonal curves, and real elliptic surfaces

Mapping tori of small dilatation irreducible train-track maps

We prove that for every P there is a bound B depending only on P so that the mapping torus of every P--small irreducible train-track map can be obtained by surgery from one of B mapping tori. We show that given an integer P>0 there is a bound $M$ depending only on P, so that there exists a presentation of the fundamental group of the mapping torus of a P--small irreducible train-track map with less than M generators and M relations.Comment: Some figures in colo

removable edges in connected graphs and the construction of connected graphs

连通图的构造是近二十年来图论的研究热点.由于它与网络模型和组合优化的密切联系,使得它具有重要的理论价值和应用价值.可收缩边和可去边是连通图的构造的有力工具,同时在使用归纳法证明连通图的一些性质中也起到重要作用.本文主要研究连通图的构造,连通图中可去边的存在性以及连通图中可收缩边和可去边在特定子图上的分布情况.下面是本文的一些主要结果:\vskip3mm1.利用边点割端片的性质给出4连通图的圈上存在至少两条可去边的充分条件;同时给出4连通图4圈上,边点割原子及分离对上,生成树上和生成树外的可去边的分布.\vskip3mm2.给出3连通图中可去边在完美匹配上的分布以及4连通图中可收缩边在完美匹配上...The construction of connected graphs is a hot topic in the research of graph theory in recent twenty years.Because of its close connection to network modelling and comibinatorial optimization,construction of connected graphs plays a significantly important role not only in theoretical respect,but also for practical applications.Contractible edges and removable edges in connected graphs are a power...学位：理学博士院系专业：数学系_基础数学学号：B20022300

Removing chambers in Bruhat-Tits buildings

We introduce and study a family of countable groups constructed from Euclidean buildings by "removing" suitably chosen subsets of chambers

Minimal Connectivity

A k-connected graph such that deleting any edge / deleting any vertex / contracting any edge results in a graph which is not k-connected is called minimally / critically / contraction-critically k-connected. These three classes play a prominent role in graph connectivity theory, and we give a brief introduction with a light emphasis on reduction- and construction theorems for classes of k-connected graphs.Comment: IMADA-preprint-math, 33 page

Some problems in combinatorial topology of flag complexes

In this work we study simplicial complexes associated to graphs and their homotopical and combinatorial properties. The main focus is on the family of flag complexes, which can be viewed as independence complexes and clique complexes of graphs. In the first part we study independence complexes of graphs using two cofibre sequences corresponding to vertex and edge removals. We give applications to the connectivity of independence complexes of chordal graphs and to extremal problems in topology and we answer open questions about the homotopy types of those spaces for particular families of graphs. We also study the independence complex as a space of configurations of particles in the so-called hard-core models on various lattices. We define, and investigate from an algorithmic perspective, a special family of combinatorially defined homology classes in independence complexes. This enables us to give algorithms as well as NP-hardness results for topological properties of some spaces. As a corollary we prove hardness of computing homology of simplicial complexes in general. We also view flag complexes as clique complexes of graphs. That leads to the study of various properties of Vietoris-Rips complexes of graphs. The last result is inspired by a problem in face enumeration. Using methods of extremal graph theory we classify flag triangulations of 3-manifolds with many edges. As a corollary we complete the classification of face vectors of flag simplicial homology 3-spheres

Smallest snarks with oddness 4 and cyclic connectivity 4 have order 44

The family of snarks -- connected bridgeless cubic graphs that cannot be 3-edge-coloured -- is well-known as a potential source of counterexamples to several important and long-standing conjectures in graph theory. These include the cycle double cover conjecture, Tutte's 5-flow conjecture, Fulkerson's conjecture, and several others. One way of approaching these conjectures is through the study of structural properties of snarks and construction of small examples with given properties. In this paper we deal with the problem of determining the smallest order of a nontrivial snark (that is, one which is cyclically 4-edge-connected and has girth at least 5) of oddness at least 4. Using a combination of structural analysis with extensive computations we prove that the smallest order of a snark with oddness at least 4 and cyclic connectivity 4 is 44. Formerly it was known that such a snark must have at least 38 vertices [J. Combin. Theory Ser. B 103 (2013), 468--488] and one such snark on 44 vertices was constructed by Lukot'ka et al. [Electron. J. Combin. 22 (2015), #P1.51]. The proof requires determining all cyclically 4-edge-connected snarks on 36 vertices, which extends the previously compiled list of all such snarks up to 34 vertices [J. Combin. Theory Ser. B, loc. cit.]. As a by-product, we use this new list to test the validity of several conjectures where snarks can be smallest counterexamples.Comment: 21 page