Countable connected-homogeneous digraphs

A digraph is connected-homogeneous if every isomorphism between two finite connected induced subdigraphs extends to an automorphism of the whole digraph. In this paper, we completely classify the countable connected-homogeneous digraphs.Comment: 49 page

Distance magic-type and distance antimagic-type labelings of graphs

Generally speaking, a distance magic-type labeling of a graph G of order n is a bijection f from the vertex set of the graph to the first n natural numbers or to the elements of a group of order n, with the property that the weight of each vertex is the same. The weight of a vertex x is defined as the sum (or appropriate group operation) of all the labels of vertices adjacent to x. If instead we require that all weights differ, then we refer to the labeling as a distance antimagic-type labeling. This idea can be generalized for directed graphs; the weight will take into consideration the direction of the arcs. In this manuscript, we provide new results for d-handicap labeling, a distance antimagic-type labeling, and introduce a new distance magic-type labeling called orientable Gamma-distance magic labeling. A d-handicap distance antimagic labeling (or just d-handicap labeling for short) of a graph G=(V,E) of order n is a bijection f from V to {1,2,...,n} with induced weight function w(x_{i})=\underset{x_{j}\in N(x_{i})}{\sum}f(x_{j}) \] such that f(x_{i})=i and the sequence of weights w(x_{1}),w(x_{2}),...,w(x_{n}) forms an arithmetic sequence with constant difference d at least 1. If a graph G admits a d-handicap labeling, we say G is a d-handicap graph. A d-handicap incomplete tournament, H(n,k,d) is an incomplete tournament of n teams ranked with the first n natural numbers such that each team plays exactly k games and the strength of schedule of the ith ranked team is d more than the i+1st ranked team. That is, strength of schedule increases arithmetically with strength of team. Constructing an H(n,k,d) is equivalent to finding a d-handicap labeling of a k-regular graph of order n. In Chapter 2 we provide general constructions for every d at least 1 for large classes of both n and k, providing breadth and depth to the catalog of known H(n,k,d)\u27s. In Chapters 3 - 6, we introduce a new type of labeling called orientable Gamma-distance magic labeling. Let Gamma be an abelian group of order n. If for a graph G=(V,E) of order n there exists an orientation of G and a companion bijection f from V to Gamma with the property that there is an element mu in Gamma (called the magic constant) such that \[ w(x)=\sum_{y\in N_{G}^{+}(x)}\overrightarrow{f}(y)-\sum_{y\in N_{G}^{-}(x)}\overrightarrow{f}(y)=\mu for every x in V where w(x) is the weight of vertex x, we say that G is orientable Gamma-distance magic}. In addition to introducing the concept, we provide numerous results on orientable Z_n distance magic graphs, where Z_n is the cyclic group of order n. In Chapter 7, we summarize the results of this dissertation and provide suggestions for future work

The Shifted Turan Sieve Method on Tournaments

This article has been published in a revised form in the Canadian Mathematical Bulletin http://dx.doi.org/10.4153/S000843951900016X. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. Copyright © Canadian Mathematical Society 2019.Abstract. We construct a shi ed version of the Turán sieve method developed by R. Murty and the second author and apply it to counting problems on tournaments. More precisely, we obtain upper bounds for the number of tournaments which contain a fixed number of restricted r-cycles. These are the first concrete results which count the number of cycles over “all tournaments”.Research partially supported by NSERC Discovery Grants || CAPES and CSF/CNPQ, Brazil

Combinatorics

Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combiantorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session

Hamilton decompositions of regular expanders: a proof of Kelly's conjecture for large tournaments

A long-standing conjecture of Kelly states that every regular tournament on n vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles. We prove this conjecture for large n. In fact, we prove a far more general result, based on our recent concept of robust expansion and a new method for decomposing graphs. We show that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. This enables us to obtain numerous further results, e.g. as a special case we confirm a conjecture of Erdos on packing Hamilton cycles in random tournaments. As corollaries to the main result, we also obtain several results on packing Hamilton cycles in undirected graphs, giving e.g. the best known result on a conjecture of Nash-Williams. We also apply our result to solve a problem on the domination ratio of the Asymmetric Travelling Salesman problem, which was raised e.g. by Glover and Punnen as well as Alon, Gutin and Krivelevich.Comment: new version includes a standalone version of the `robust decomposition lemma' for application in subsequent paper