On random k-out sub-graphs of large graphs

We consider random sub-graphs of a fixed graph $G=(V,E)$ with large minimum degree. We fix a positive integer $k$ and let $G_k$ be the random sub-graph where each $v\in V$ independently chooses $k$ random neighbors, making $kn$ edges in all. When the minimum degree $\delta(G)\geq (\frac12+\epsilon)n,\,n=|V|$ then $G_k$ is $k$-connected w.h.p. for $k=O(1)$; Hamiltonian for $k$ sufficiently large. When $\delta(G) \geq m$, then $G_k$ has a cycle of length $(1-\epsilon)m$ for $k\geq k_\epsilon$. By w.h.p. we mean that the probability of non-occurrence can be bounded by a function $\phi(n)$ (or $\phi(m)$) where $\lim_{n\to\infty}\phi(n)=0$

Proof of Koml\'os's conjecture on Hamiltonian subsets

Koml\'os conjectured in 1981 that among all graphs with minimum degree at least $d$, the complete graph $K_{d+1}$ minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this conjecture when $d$ is sufficiently large. In fact we prove a stronger result: for large $d$, any graph $G$ with average degree at least $d$ contains almost twice as many Hamiltonian subsets as $K_{d+1}$, unless $G$ is isomorphic to $K_{d+1}$ or a certain other graph which we specify.Comment: 33 pages, to appear in Proceedings of the London Mathematical Societ

Degree Conditions for Hamiltonian Properties of Claw-free Graphs

This thesis contains many new contributions to the field of hamiltonian graph theory, a very active subfield of graph theory. In particular, we have obtained new sufficient minimum degree and degree sum conditions to guarantee that the graphs satisfying these conditions, or their line graphs, admit a Hamilton cycle (or a Hamilton path), unless they have a small order or they belong to well-defined classes of exceptional graphs. Here, a Hamilton cycle corresponds to traversing the vertices and edges of the graph in such a way that all their vertices are visited exactly once, and we return to our starting vertex (similarly, a Hamilton path reflects a similar way of traversing the graph, but without the last restriction, so we might terminate at a different vertex). In Chapter 1, we presented an introduction to the topics of this thesis together with Ryjáček’s closure for claw-free graphs, Catlin’s reduction method, and the reduction of the core of a graph. In Chapter 2, we found the best possible bounds for the minimum degree condition and the minimum degree sums condition of adjacent vertices for traceability of 2-connected claw-free graph, respectively. In addition, we decreased these lower bounds with one family of well characterized exceptional graphs. In Chapter 3, we extended recent results about the conjecture of Benhocine et al. and results about the conjecture of Z.-H Chen and H.-J Lai. In Chapters 4, 5 and 6, we have successfully tried to unify and extend several existing results involving the degree and neighborhood conditions for the hamiltonicity and traceability of 2-connected claw-free graphs. Throughout this thesis, we have investigated the existence of Hamilton cycles and Hamilton paths under different types of degree and neighborhood conditions, including minimum degree conditions, minimum degree sum conditions on adjacent pairs of vertices, minimum degree sum conditions over all independent sets of t vertices of a graph, minimum cardinality conditions on the neighborhood union over all independent sets of t vertices of a graph, as well minimum cardinality conditions on the neighborhood union over all t vertex sets of a graph. Despite our new contributions, many problems and conjectures remain unsolved

The Set Cover Conjecture and Subgraph Isomorphism with a Tree Pattern

In the Set Cover problem, the input is a ground set of n elements and a collection of m sets, and the goal is to find the smallest sub-collection of sets whose union is the entire ground set. The fastest algorithm known runs in time O(mn2^n) [Fomin et al., WG 2004], and the Set Cover Conjecture (SeCoCo) [Cygan et al., TALG 2016] asserts that for every fixed epsilon>0, no algorithm can solve Set Cover in time 2^{(1-epsilon)n} poly(m), even if set sizes are bounded by Delta=Delta(epsilon). We show strong connections between this problem and kTree, a special case of Subgraph Isomorphism where the input is an n-node graph G and a k-node tree T, and the goal is to determine whether G has a subgraph isomorphic to T. First, we propose a weaker conjecture Log-SeCoCo, that allows input sets of size Delta=O(1/epsilon * log n), and show that an algorithm breaking Log-SeCoCo would imply a faster algorithm than the currently known 2^n poly(n)-time algorithm [Koutis and Williams, TALG 2016] for Directed nTree, which is kTree with k=n and arbitrary directions to the edges of G and T. This would also improve the running time for Directed Hamiltonicity, for which no algorithm significantly faster than 2^n poly(n) is known despite extensive research. Second, we prove that if p-Partial Cover, a parameterized version of Set Cover that requires covering at least p elements, cannot be solved significantly faster than 2^n poly(m) (an assumption even weaker than Log-SeCoCo) then kTree cannot be computed significantly faster than 2^k poly(n), the running time of the Koutis and Williams\u27 algorithm

Rainbow perfect matchings and Hamilton cycles in the random geometric graph

Given a graph on n vertices and an assignment of colours to the edges, a rainbow Hamilton cycle is a cycle of length n visiting each vertex once and with pairwise different colours on the edges. Similarly (for even n) a rainbow perfect matching is a collection of independent edges with pairwise different colours. In this note we show that if we randomly colour the edges of a random geometric graph with sufficiently many colours, then a.a.s. the graph contains a rainbow perfect matching (rainbow Hamilton cycle) if and only if the minimum degree is at least 1 (respectively, at least 2). More precisely, consider n points (i.e. vertices) chosen independently and uniformly at random from the unit d‐dimensional cube for any fixed . Form a sequence of graphs on these n vertices by adding edges one by one between each possible pair of vertices. Edges are added in increasing order of lengths (measured with respect to the norm, for any fixed ). Each time a new edge is added, it receives a random colour chosen uniformly at random and with repetition from a set of colours, where a sufficiently large fixed constant. Then, a.a.s. the first graph in the sequence with minimum degree at least 1 must contain a rainbow perfect matching (for even n), and the first graph with minimum degree at least 2 must contain a rainbow Hamilton cycle

The intersection of longest paths in a graph.

In this thesis we examine the famous conjecture that every three longest paths in a graph intersect, and add to the classes of graphs for which it is known that this conjecture holds. This conjecture arose from a question asked by Gallai in 1966, the question of whether all of the longest paths in a graph intersect (Gallai's question). In 1969, Walther found a graph in which the longest paths do not all intersect, answering Gallai's question. Since then, many other graphs in which the longest paths do not all intersect have been found. However there are also many classes of graphs for which the longest paths all intersect, such as series-parallel graphs and dually chordal graphs. Finding such classes of graphs is an active area of research and in this thesis we add to these classes of graphs. We begin by investigating Gallai's question for a speci c class of graphs. A theta graph is a graph consisting of three paths with a pair of common endpoints and no other common vertices. A generalised theta graph is a graph with at least one block that consists of at least three paths with a pair of common endpoints and no other common vertices. We show that for a subclass of generalised theta graphs, all of the longest paths intersect. Next, we consider the conjecture that every three longest paths of a graph intersect. We prove that, for every graph with n vertices and at most n + 5 edges, every three longest paths intersect. Finally, we use computational methods to investigate whether all longest paths intersect, or every three longest paths intersect, for several classes of graphs. Two graphs are homeomorphic if each can be obtained from the same graph H by a series of subdivisions. We show that, for every simple connected graph G that is homeomorphic to a simple connected graph with at most 7 vertices, all of the longest paths of G intersect. Additionally, we show that, for every simple connected graph G homeomorphic to a simple connected graph with n vertices, n + 6 edges, and minimum vertex degree 3, all of the longest paths of G intersect. We then show that for every graph with n vertices and at most n + 5 edges, every three longest paths intersect, independently verifying this result. We also present results for several additional classes of graphs with conditions on the blocks, maximum degree of the vertices, and other properties of the graph, showing that every three longest paths intersect or every six longest paths intersect for these graphs

Fast winning strategies in positional games

Abstract For the unbiased Maker-Breaker game, played on the hypergraph H, let τ M (H) be the smallest integer t such that Maker can win the game within t moves (if the game is a Breaker's win then set τ M (H) = ∞). Similarly, for the unbiased Avoider-Enforcer game played on H, let τ E (H) be the smallest integer t such that Enforcer can win the game within t moves (if the game is an Avoider's win then set τ E (H) = ∞). In this paper, we investigate τ M and τ E and determine their value for various positional games

Packings and coverings with Hamilton cycles and on-line Ramsey theory

A major theme in modern graph theory is the exploration of maximal packings and minimal covers of graphs with subgraphs in some given family. We focus on packings and coverings with Hamilton cycles, and prove the following results in the area. • Let ε > 0, and let $G$ be a large graph on n vertices with minimum degree at least (1=2+ ε)n. We give a tight lower bound on the size of a maximal packing of $G$ with edge-disjoint Hamilton cycles. • Let $T$ be a strongly k-connected tournament. We give an almost tight lower bound on the size of a maximal packing of $T$ with edge-disjoint Hamilton cycles. • Let log $^1$$^1$$^7$ $n$/$n$≤$p$≤1-$n$$^-$$^1$$^/$$^8$. We prove that $G$$_n$$_,$$_p$ may a.a.s be covered by a set of ⌈Δ($G$$_n$$_,$$_p$)/2⌉ Hamilton cycles, which is clearly best possible. In addition, we consider some problems in on-line Ramsey theory. Let r($G$,$H$) denote the on-line Ramsey number of $G$ and $H$. We conjecture the exact values of r ($P$$_k$,$P$$_ℓ$) for all $k$≤ℓ. We prove this conjecture for $k$=2, prove it to within an additive error of 10 for $k$=3, and prove an asymptotically tight lower bound for $k$=4. We also determine r($P$$_3$,$C$$_ℓ$ exactly for all ℓ

Graphs and subgraphs with bounded degree

"The topology of a network (such as a telecommunications, multiprocessor, or local area network, to name just a few) is usually modelled by a graph in which vertices represent 'nodes' (stations or processors) while undirected or directed edges stand for 'links' or other types of connections, physical or virtual. A cycle that contains every vertex of a graph is called a hamiltonian cycle and a graph which contains a hamiltonian cycle is called a hamiltonian graph. The problem of the existence of a hamiltonian cycle is closely related to the well known problem of a travelling salesman. These problems are NP-complete and NP-hard, respectively. While some necessary and sufficient conditions are known, to date, no practical characterization of hamiltonian graphs has been found. There are several ways to generalize the notion of a hamiltonian cycle. In this thesis we make original contributions in two of them, namely k-walks and r-trestles." --Abstract.Doctor of Philosoph