Note on bipartite graph tilings

Let s<t be two fixed positive integers. We study what are the minimum degree conditions for a bipartite graph G, with both color classes of size n=k(s+t), which ensure that G has a K_{s,t}-factor. Exact result for large n is given. Our result extends the work of Zhao, who determined the minimum degree threshold which guarantees that a bipartite graph has a K_{s,s}-factor.Comment: 6 pages, no figures; statement of the main theorem corrected (thanks to Andrzej Czygrinow and Louis DeBiasio); to appear in SIAM Journal on Discrete Mathematic

Tilings in randomly perturbed graphs: Bridging the gap between Hajnal‐Szemerédi and Johansson‐Kahn‐Vu

A perfect Kr-tiling in a graph G is a collection of vertex-disjoint copies of Kr that together cover all the vertices in G. In this paper we consider perfect Kr-tilings in the setting of randomly perturbed graphs; a model introduced by Bohman, Frieze, and Martin [7] where one starts with a dense graph and then adds m random edges to it. Specifically, given any fixed 0 < < 1 − 1∕r we determine how many random edges one must add to an n-vertex graph G of minimum degree (G) ≥ n to ensure that, asymptotically almost surely, the resulting graph contains a perfect Kr-tiling. As one increases we demonstrate that the number of random edges required “jumps” at regular intervals, and within these intervals our result is best-possible. This work therefore closes the gap between the seminal work of Johansson, Kahn and Vu [25] (which resolves the purely random case, that is, = 0) and that of Hajnal and Szemerédi [18] (which demonstrates that for ≥ 1 − 1∕r the initial graph already houses the desired perfect Kr-tiling)

Covering and tiling hypergraphs with tight cycles

Given $3 \leq k \leq s$, we say that a $k$-uniform hypergraph $C^k_s$ is a tight cycle on $s$ vertices if there is a cyclic ordering of the vertices of $C^k_s$ such that every $k$ consecutive vertices under this ordering form an edge. We prove that if $k \ge 3$ and $s \ge 2k^2$, then every $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $(1/2 + o(1))n$ has the property that every vertex is covered by a copy of $C^k_s$. Our result is asymptotically best possible for infinitely many pairs of $s$ and $k$, e.g. when $s$ and $k$ are coprime. A perfect $C^k_s$-tiling is a spanning collection of vertex-disjoint copies of $C^k_s$. When $s$ is divisible by $k$, the problem of determining the minimum codegree that guarantees a perfect $C^k_s$-tiling was solved by a result of Mycroft. We prove that if $k \ge 3$ and $s \ge 5k^2$ is not divisible by $k$ and $s$ divides $n$, then every $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $(1/2 + 1/(2s) + o(1))n$ has a perfect $C^k_s$-tiling. Again our result is asymptotically best possible for infinitely many pairs of $s$ and $k$, e.g. when $s$ and $k$ are coprime with $k$ even.Comment: Revised version, accepted for publication in Combin. Probab. Compu

Rooted structures in graphs: a project on Hadwiger's conjecture, rooted minors, and Tutte cycles

Hadwigers Vermutung ist eine der anspruchsvollsten Vermutungen für Graphentheoretiker und bietet eine weitreichende Verallgemeinerung des Vierfarbensatzes. Ausgehend von dieser offenen Frage der strukturellen Graphentheorie werden gewurzelte Strukturen in Graphen diskutiert. Eine Transversale einer Partition ist definiert als eine Menge, welche genau ein Element aus jeder Menge der Partition enthält und sonst nichts. Für einen Graphen G und eine Teilmenge T seiner Knotenmenge ist ein gewurzelter Minor von G ein Minor, der T als Transversale seiner Taschen enthält. Sei T eine Transversale einer Färbung eines Graphen, sodass es ein System von kanten-disjunkten Wegen zwischen allen Knoten aus T gibt; dann stellt sich die Frage, ob es möglich ist, die Existenz eines vollständigen, in T gewurzelten Minors zu gewährleisten. Diese Frage ist eng mit Hadwigers Vermutung verwoben: Eine positive Antwort würde Hadwigers Vermutung für eindeutig färbbare Graphen bestätigen. In dieser Arbeit wird ebendiese Fragestellung untersucht sowie weitere Konzepte vorgestellt, welche bekannte Ideen der strukturellen Graphentheorie um eine Verwurzelung erweitern. Beispielsweise wird diskutiert, inwiefern hoch zusammenhängende Teilmengen der Knotenmenge einen hoch zusammenhängenden, gewurzelten Minor erzwingen. Zudem werden verschiedene Ideen von Hamiltonizität in planaren und nicht-planaren Graphen behandelt.Hadwiger's Conjecture is one of the most tantalising conjectures for graph theorists and offers a far-reaching generalisation of the Four-Colour-Theorem. Based on this major issue in structural graph theory, this thesis explores rooted structures in graphs. A transversal of a partition is a set which contains exactly one element from each member of the partition and nothing else. Given a graph G and a subset T of its vertex set, a rooted minor of G is a minor such that T is a transversal of its branch set. Assume that a graph has a transversal T of one of its colourings such that there is a system of edge-disjoint paths between all vertices from T; it comes natural to ask whether such graphs contain a minor rooted at T. This question of containment is strongly related to Hadwiger's Conjecture; indeed, a positive answer would prove Hadwiger's Conjecture for uniquely colourable graphs. This thesis studies the aforementioned question and besides, presents several other concepts of attaching rooted relatedness to ideas in structural graph theory. For instance, whether a highly connected subset of the vertex set forces a highly connected rooted minor. Moreover, several ideas of Hamiltonicity in planar and non-planar graphs are discussed