Equitable colorings of Kronecker products of graphs

AbstractFor a positive integer k, a graph G is equitably k-colorable if there is a mapping f:V(G)→{1,2,…,k} such that f(x)≠f(y) whenever xy∈E(G) and ||f−1(i)|−|f−1(j)||≤1 for 1≤i<j≤k. The equitable chromatic number of a graph G, denoted by χ=(G), is the minimum k such that G is equitably k-colorable. The equitable chromatic threshold of a graph G, denoted by χ=∗(G), is the minimum t such that G is equitably k-colorable for k≥t. The current paper studies equitable chromatic numbers of Kronecker products of graphs. In particular, we give exact values or upper bounds on χ=(G×H) and χ=∗(G×H) when G and H are complete graphs, bipartite graphs, paths or cycles

Covering graphs by monochromatic trees and Helly-type results for hypergraphs

How many monochromatic paths, cycles or general trees does one need to cover all vertices of a given $r$-edge-coloured graph $G$? These problems were introduced in the 1960s and were intensively studied by various researchers over the last 50 years. In this paper, we establish a connection between this problem and the following natural Helly-type question in hypergraphs. Roughly speaking, this question asks for the maximum number of vertices needed to cover all the edges of a hypergraph $H$ if it is known that any collection of a few edges of $H$ has a small cover. We obtain quite accurate bounds for the hypergraph problem and use them to give some unexpected answers to several questions about covering graphs by monochromatic trees raised and studied by Bal and DeBiasio, Kohayakawa, Mota and Schacht, Lang and Lo, and Gir\~ao, Letzter and Sahasrabudhe.Comment: 20 pages including references plus 2 pages of an Appendi

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)

Transversal factors and spanning trees

Given a collection of graphs G = (G1,...,Gm) with the same vertex set, an m-edge graph H ⊂ ∪i∈[m]Gi is a transversal if there is a bijection φ : E(H) → [m] such that e ∈ E(Gφ(e)) for each e ∈ E(H). We give asymptotically-tight minimum degree conditions for a graph collection on an n-vertex set to have a transversal which is a copy of a graph H, when H is an n-vertex graph which is an F-factor or a tree with maximum degree o(n/logn)

On Ramsey Theory and Slow Bootstrap Percolation

This dissertation concerns two sets of problems in extremal combinatorics. The major part, Chapters 1 to 4, is about Ramsey-type problems for cycles. The shorter second part, Chapter 5, is about a problem in bootstrap percolation. Next, we describe each topic more precisely. Given three graphs G, L1 and L2, we say that G arrows (L1, L2) and write G → (L1, L2), if for every edge-coloring of G by two colors, say 1 and 2, there exists a color i whose color class contains Li as a subgraph. The classical problem in Ramsey theory is the case where G, L1 and L2 are complete graphs; in this case the question is how large the order of G must be (in terms of the orders of L1 andL2) to guarantee that G → (L1, L2). Recently there has been much interest in the case where L1 and L2 are cycles and G is a graph whose minimum degree is large. In the past decade, numerous results have been proved about those problems. We will continue this work and prove two conjectures that have been left open. Our main weapon is Szemeredi\u27s Regularity Lemma.Our second topic is about a rather unusual aspect of the fast expanding theory of bootstrap percolation. Bootstrap percolation on a graph G with parameter r is a cellular automaton modeling the spread of an infection: starting with a set A0, cointained in V(G), of initially infected vertices, define a nested sequence of sets, A0 ⊆ A1 ⊆. . . ⊆ V(G), by the update rule that At+1, the set of vertices infected at time t + 1, is obtained from At by adding to it all vertices with at least r neighbors in At. The initial set A0 percolates if At = V(G) for some t. The minimal such t is the time it takes for A0 to percolate. We prove results about the maximum percolation time on the two-dimensional grid with parameter r = 2

Two Problems on Bipartite Graphs

Erdos proved the well-known result that every graph has a spanning, bipartite subgraph such that every vertex has degree at least half of its original degree. Bollobas and Scott conjectured that one can get a slightly weaker result if we require the subgraph to be not only spanning and bipartite, but also balanced. We prove this conjecture for graphs of maximum degree 3. The majority of the paper however, will focus on graph tiling. Graph tiling (or sometimes referred to as graph packing) is where, given a graph H, we find a spanning subgraph of some larger graph G that consists entirely of disjoint copies of H. With the Regularity Lemma and the Blow-up Lemma as our main tools, we prove an asymptotic minimum degree condition for an arbitrary bipartite graph G to be tiled by another arbitrary bipartite graph H. This proves a conjecture of Zhao and also implies an asymptotic version of a result of Kuhn and Osthus for bipartite graphs

Clique Factors: Extremal and Probabilistic Perspectives

A K_r-factor in a graph G is a collection of vertex-disjoint copies of K_r covering the vertex set of G. In this thesis, we investigate these fundamental objects in three settings that lie at the intersection of extremal and probabilistic combinatorics. Firstly, we explore pseudorandom graphs. An n-vertex graph is said to be (p,β)-bijumbled if for any vertex sets A, B ⊆ V (G), we have e( A, B) = p| A||B| ± β√|A||B|. We prove that for any 3 ≤ r ∈ N and c > 0 there exists an ε > 0 such that any n-vertex (p, β)-bijumbled graph with n ∈ rN, δ(G) ≥ c p n and β ≤ ε p^{r −1} n, contains a K_r -factor. This implies a corresponding result for the stronger pseudorandom notion of (n, d, λ)-graphs. For the case of K_3-factors, this result resolves a conjecture of Krivelevich, Sudakov and Szabó from 2004 and it is tight due to a pseudorandom triangle-free construction of Alon. In fact, in this case even more is true: as a corollary to this result, we can conclude that the same condition of β = o( p^2n) actually guarantees that a (p, β)-bijumbled graph G contains every graph on n vertices with maximum degree at most 2. Secondly, we explore the notion of robustness for K_3-factors. For a graph G and p ∈ [0, 1], we denote by G_p the random sparsification of G obtained by keeping each edge of G independently, with probability p. We show that there exists a C > 0 such that if p ≥ C (log n)^{1/3}n^{−2/3} and G is an n-vertex graph with n ∈ 3N and δ(G) ≥ 2n/3 , then with high probability G_p contains a K_3-factor. Both the minimum degree condition and the probability condition, up to the choice of C, are tight. Our result can be viewed as a common strengthening of the classical extremal theorem of Corrádi and Hajnal, corresponding to p = 1 in our result, and the famous probabilistic theorem of Johansson, Kahn and Vu establishing the threshold for the appearance of K_3-factors (and indeed all K_r -factors) in G (n, p), corresponding to G = K_n in our result. It also implies a first lower bound on the number of K_3-factors in graphs with minimum degree at least 2n/3, which gets close to the truth. Lastly, we consider the setting of randomly perturbed graphs; a model introduced by Bohman, Frieze and Martin, where one starts with a dense graph and then adds random edges to it. Specifically, given any fixed 0 < α < 1 − 1/r we determine how many random edges one must add to an n-vertex graph G with δ(G) ≥ α n to ensure that, with high probability, the resulting graph contains a K_r -factor. As one increases α 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 bridges the gap between the seminal work of Johansson, Kahn and Vu mentioned above, which resolves the purely random case, i.e., α = 0, and that of Hajnal and Szemerédi (and Corrádi and Hajnal for r = 3) showing that when α ≥ 1 − 1/r the initial graph already hosts the desired K_r -factor.Ein K_r -Faktor in einem Graphen G ist eine Sammlung von Knoten-disjunkten Kopien von K_r , die die Knotenmenge von G überdecken. Wir untersuchen diese Objekte in drei Kontexten, die an der Schnittstelle zwischen extremaler und probabilistischer Kombinatorik liegen. Zuerst untersuchen wir Pseudozufallsgraphen. Ein Graph heißt (p,β)-bijumbled, wenn für beliebige Knotenmengen A, B ⊆ V (G) gilt e( A, B) = p| A||B| ± β√|A||B|. Wir beweisen, dass es für jedes 3 ≤ r ∈ N und c > 0 ein ε > 0 gibt, so dass jeder n-Knoten (p, β)-bijumbled Graph mit n ∈ rN, δ(G) ≥ c p n und β ≤ ε p^{r −1} n, einen K_r -Faktor enthält. Dies impliziert ein entsprechendes Ergebnis für den stärkeren Pseudozufallsbegriff von (n, d, λ)-Graphen. Im Fall von K_3-Faktoren, löst dieses Ergebnis eine Vermutung von Krivelevich, Sudakov und Szabó aus dem Jahr 2004 und ist durch eine pseudozufällige K_3-freie Konstruktion von Alon bestmöglich. Tatsächlich ist in diesem Fall noch mehr wahr: als Korollar dieses Ergebnisses können wir schließen, dass die gleiche Bedingung von β = o( p^2n) garantiert, dass ein (p, β)-bijumbled Graph G jeden Graphen mit maximalem Grad 2 enthält. Zweitens untersuchen wir den Begriff der Robustheit für K_3-Faktoren. Für einen Graphen G und p ∈ [0, 1] bezeichnen wir mit G_p die zufällige Sparsifizierung von G, die man erhält, indem man jede Kante von G unabhängig von den anderen Kanten mit einer Wahrscheinlichkeit p behält. Wir zeigen, dass, wenn p ≥ C (log n)^{1/3}n^{−2/3} und G ein n-Knoten-Graph mit n ∈ 3N und δ(G) ≥ 2n/3 ist, G_pmit hoher Wahrscheinlichkeit (mhW) einen K_3-Faktor enthält. Sowohl die Bedingung des minimalen Grades als auch die Wahrscheinlichkeitsbedingung sind bestmöglich. Unser Ergebnis ist eine Verstärkung des klassischen extremalen Satzes von Corrádi und Hajnal, entsprechend p = 1 in unserem Ergebnis, und des berühmten probabilistischen Satzes von Johansson, Kahn und Vu, der den Schwellenwert für das Auftreten eines K_3-Faktors (und aller K_r -Faktoren) in G (n, p) festlegt, entsprechend G = K_n in unserem Ergebnis. Es impliziert auch eine erste untere Schranke für die Anzahl der K_3-Faktoren in Graphen mit einem minimalen Grad von mindestens 2n/3, die der Wahrheit nahe kommt. Schließlich betrachten wir die Situation von zufällig gestörten Graphen; ein Modell, bei dem man mit einem dichten Graphen beginnt und dann zufällige Kanten hinzufügt. Wir bestimmen, bei gegebenem 0 < α < 1 − 1/r, wie viele zufällige Kanten man zu einem n-Knoten-Graphen G mit δ(G) ≥ α n hinzufügen muss, um sicherzustellen, dass der resultierende Graph mhW einen K_r -Faktor enthält. Wir zeigen, dass, wenn man α erhöht, die Anzahl der benötigten Zufallskanten in regelmäßigen Abständen “springt", und innerhalb dieser Abstände unser Ergebnis bestmöglich ist. Diese Arbeit schließt somit die Lücke zwischen der oben erwähnten bahnbrechenden Arbeit von Johansson, Kahn und Vu, die den rein zufälligen Fall, d.h. α = 0, löst, und der Arbeit von Hajnal und Szemerédi (und Corrádi und Hajnal für r = 3), die zeigt, dass der ursprüngliche Graph bereits den gewünschten K_r -Faktor enthält, wenn α ≥ 1 − 1/r ist