Unknotting numbers and triple point cancelling numbers of torus-covering knots

It is known that any surface knot can be transformed to an unknotted surface knot or a surface knot which has a diagram with no triple points by a finite number of 1-handle additions. The minimum number of such 1-handles is called the unknotting number or the triple point cancelling number, respectively. In this paper, we give upper bounds and lower bounds of unknotting numbers and triple point cancelling numbers of torus-covering knots, which are surface knots in the form of coverings over the standard torus $T$. Upper bounds are given by using $m$-charts on $T$ presenting torus-covering knots, and lower bounds are given by using quandle colorings and quandle cocycle invariants.Comment: 26 pages, 14 figures, added Corollary 1.7, to appear in J. Knot Theory Ramification

Rainbow Turan Methods for Trees

The rainbow Turan number, a natural extension of the well-studied traditionalTuran number, was introduced in 2007 by Keevash, Mubayi, Sudakov and Verstraete. The rainbow Tur ́an number of a graph F , ex*(n, F ), is the largest number of edges for an n vertex graph G that can be properly edge colored with no rainbow F subgraph. Chapter 1 of this dissertation gives relevant definitions and a brief history of extremal graph theory. Chapter 2 defines k-unique colorings and the related k-unique Turan number and provides preliminary results on this new variant. In Chapter 3, we explore the reduction method for finding upper bounds on rainbow Turan numbers and use this to inform results for the rainbow Turan numbers of specific families of trees. These results are used in Chapter 4 to prove that the rainbow Turan numbers of all trees are linear in n, which correlates to a well-known property of the traditional Turan numbers of trees. We discuss improvements to the constant term in Chapters 4 and 5, and conclude with a discussion on avenues for future work

The First Classical Ramsey Number for Hypergraphs is Computed

With the help of the computer, we have shown that in any coloring with two colors of the triangles on a set of 13 points there must exist a monochromatic tetrahedron. This proves the new upper bound R (4,4;3) \u3c = 13. The previous best upper bound of 15 was derived independently by Giraud (1969 [2]), Schwenk (1978 [5]) and Sidorenko (1980 [6]). The first construction of a R (4,4;3)-good hypergraph on 12 points was presented by Isbell (1969 [3]), and the same one again more elegantly by Sidorenko (1980 [6]). We have constructed more than 200,000 R (4,4;3)-good hypergraphs on 12 points, but probably not the full set. R (4,4;3)=13 is the first known exact value of a classical Ramsey number for hypergraphs. The solution was achieved with the help of a variety of algorithms relying on a strong connection between the colorings with two colors of the triangles on n points and the so-called Tura´n set systems T(n ,5,4). The main criterion used to prune the search space for R (4,4;3)-good hypergraphs was to count the number of 4-sets containing two triangles of each color; such families of 4-sets are known to form Tura´n systems and their cardinalities must be minorized by the corresponding Tura´n numbers T(n ,5,4). We used an innovative method for generating large families of set systems which efficiently prevents isomorphic copies of set systems being produced. This method has many potential applications to other general computer searches for elusive combinatorial configurations. As a check on the correctness of the algorithms, many of the intermediate subfamilies of R (4,4;3)-good hypergraphs were generated by two different methods: from colorings of triangles on a smaller number of points and independently via Tura´n systems. An important component of the software used was a general set-system automorphism group program [4]