326,645 research outputs found
The early evolution of the H-free process
The H-free process, for some fixed graph H, is the random graph process
defined by starting with an empty graph on n vertices and then adding edges one
at a time, chosen uniformly at random subject to the constraint that no H
subgraph is formed. Let G be the random maximal H-free graph obtained at the
end of the process. When H is strictly 2-balanced, we show that for some c>0,
with high probability as , the minimum degree in G is at least
. This gives new lower bounds for
the Tur\'an numbers of certain bipartite graphs, such as the complete bipartite
graphs with . When H is a complete graph with we show that for some C>0, with high probability the independence number of
G is at most . This gives new lower bounds
for Ramsey numbers R(s,t) for fixed and t large. We also obtain new
bounds for the independence number of G for other graphs H, including the case
when H is a cycle. Our proofs use the differential equations method for random
graph processes to analyse the evolution of the process, and give further
information about the structure of the graphs obtained, including asymptotic
formulae for a broad class of subgraph extension variables.Comment: 36 page
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The Chromatic Structure of Dense Graphs
This thesis focusses on extremal graph theory, the study of how local constraints on a graph affect its macroscopic structure. We primarily consider the chromatic structure: whether a graph has or is close to having some (low) chromatic number.
Chapter 2 is the slight exception. We consider an induced version of the classical Turán problem. Introduced by Loh, Tait, Timmons, and Zhou, the induced Turán number ex(n, {H, F-ind}) is the greatest number of edges in an n-vertex graph with no copy of H and no induced copy of F. We asymptotically determine ex(n, {H, F-ind}) for H not bipartite and F neither an independent set nor a complete bipartite graph. We also improve the upper bound for ex(n, {H, K_{2, t}-ind}) as well as the lower bound for the clique number of graphs that have some fixed edge density and no induced K_{2, t}.
The next three chapters form the heart of the thesis. Chapters 3 and 4 consider the Erdős-Simonovits question for locally r-colourable graphs: what are the structure and chromatic number of graphs with large minimum degree and where every neighbourhood is r-colourable? Chapter 3 deals with the locally bipartite case and Chapter 4 with the general case.
While the subject of Chapters 3 and 4 is a natural local to global colouring question, it is also essential for determining the minimum degree stability of H-free graphs, the focus of Chapter 5. Given a graph H of chromatic number r + 1, this asks for the minimum degree that guarantees that an H-free graph is close to r-partite. This is analogous to the classical edge stability of Erdős and Simonovits. We also consider the question for the family of graphs to which H is not homomorphic, showing that it has the same answer.
Chapter 6 considers sparse analogues of the results of Chapters 3 to 5 obtaining the thresholds at which the sparse problem degenerates away from the dense one.
Finally, Chapter 7 considers a chromatic Ramsey problem first posed by Erdős: what is the greatest chromatic number of a triangle-free graph on vertices or with m edges? We improve the best known bounds and obtain tight (up to a constant factor) bounds for the list chromatic number, answering a question of Cames van Batenburg, de Joannis de Verclos, Kang, and Pirot
Peaceful Colourings
We introduce peaceful colourings, a variant of -conflict free colourings.
We call a colouring with no monochromatic edges -peaceful if for each vertex
, there are at most neighbours of coloured with a colour appearing
on another neighbour of . An -conflict-free colouring of a graph is a
(vertex)-colouring with no monochromatic edges so that for every vertex ,
the number of neighbours of which are coloured with a colour appearing on
no other neighbour of is at least the minimum of and the degree of .
If is -regular then it has an -conflict free colouring precisely
if it has a -peaceful colouring. We focus on the minimum
of those for which every graph of maximum degree has a
-peaceful colouring with colours. We show that and that for graphs of bounded codegree, . We ask if the latter result can be improved
by dropping the bound on the codegree. As a partial result, we show that
for sufficiently large
Eulerian subgraphs and Hamiltonicity of claw -free graphs
Let C(l, k) denote the class of 2-edge-connected graphs of order n such that a graph G ∈ C(l, k) if and only if for every edge cut S ⊆ E(G) with |S| ≤ 3, each component of G - S has order at least n-kl . We prove that If G ∈ C(6, 0), then G is supereulerian if and only if G cannot be contracted to K2,3, K 2,5 or K2,3(e), where e ∈ E(K2,3) and K2,3(e) stands for a graph obtained from K2,3 by replacing e by a path of length 2. Previous results by Catlin and Li, and by Broersma and Xiong are extended.;We also investigate the supereulerian graph problems within planar graphs, and we prove that if a 2-edge-connected planar graph G is at most three edges short of having two edge-disjoint spanning trees, then G is supereulerian except a few classes of graphs. This is applied to show the existence of spanning Eulerian subgraphs in planar graphs with small edge cut conditions. We determine several extremal bounds for planar graphs to be supereulerian.;Kuipers and Veldman conjectured that any 3-connected claw-free graph with order n and minimum degree delta ≥ n+610 is Hamiltonian for n sufficiently large. We prove that if H is a 3-connected claw-free graph with sufficiently large order n, and if delta(H) ≥ n+510 , then either H is hamiltonian, or delta( H) = n+510 and the Ryjac˘ek\u27s closure cl( H) of H is the line graph of a graph obtained from the Petersen graph P10 by adding n-1510 pendant edges at each vertex of P10
Phase Transition of Degeneracy in Minor-Closed Families
Given an infinite family of graphs and a monotone property
, an (upper) threshold for and is a
"fastest growing" function such that for any sequence over with , where is the random subgraph of such that each
edge remains independently with probability .
In this paper we study the upper threshold for the family of -minor free
graphs and for the graph property of being -degenerate, which is one
fundamental graph property with many applications. Even a constant factor
approximation for the upper threshold for all pairs is expected to be
very difficult by its close connection to a major open question in extremal
graph theory. We determine asymptotically the thresholds (up to a constant
factor) for being -degenerate for a large class of pairs ,
including all graphs of minimum degree at least and all graphs with
no vertex-cover of size at most , and provide lower bounds for the rest of
the pairs of . The results generalize to arbitrary proper minor-closed
families and the properties of being -colorable, being -choosable, or
containing an -regular subgraph, respectively
Embedding problems and Ramsey-Turán variations in extremal graph theory
In this dissertation, we will focus on a few problems in extremal graph theory. The first chapter consists of some basic terms and tools.
In Chapter 2, we study a conjecture of Mader on embedding subdivisions of cliques. Improving a bound by Mader, Bollobás and Thomason, and independently Komlós and Szemerédi proved that every graph with average degree d contains a subdivision of K_[Ω(√d)]. The disjoint union of complete bipartite graph K_(r,r) shows that their result is best possible. In particular, this graph does not contain a subdivision of a clique of order w(r). However, one can ask whether their bound can be improved if we forbid such structures. There are various results in this direction, for example Kühn and Osthus proved that their bound can be improved if we forbid a complete bipartite graph of fixed size. Mader proved that that there exists a function g(r) such that every graph G with ẟ(G) ≥ r and girth at least g(r) contains a TK_(r+1). He also asked about the minimum value of g(r). Furthermore, he conjectured that C_4-freeness is enough to guarantee a clique subdivision of order linear in average degree. Some major steps towards these two questions were made by Kühn and Osthus, such as g(r) ≤ 27 and g(r) ≤ 15 for large enough r. In an earlier result, they proved that for C_4-free graphs one can find a subdivision of a clique of order almost linear in minimum degree. Together with József Balogh and Hong Liu, we proved that every C_(2k)-free graph, for k ≥ 3, has such a subdivision of a large clique. We also proved the dense case of Mader's conjecture in a stronger sense.
In Chapter 3, we study a graph-tiling problem. Let H be a fixed graph on h vertices and G be a graph on N vertices such that h|n. An H-factor is a collection of n/h vertex-disjoint copies of H in G. The problem of finding sufficient conditions for a graph G to have an H-factor has been extensively studied; most notable is the celebrated Hajnal-Szemerédi Theorem which states that every n-vertex graph G with ẟ(G) ≥ (1-1/r)n has a K_r-factor. The case r=3 was proved earlier by Corrádi and Hajnal. Another type of problems that have been studied over the past few decades are the so-called Ramsey-Turán problems. Erdős and Sós, in 1970, began studying a variation on Turán's theorem: What is the maximum number of edges in an n-vertex, K_r-free graph G if we add extra conditions to avoid the very strict structure of Turán graph. In particular, what if besides being K_r-free, we also require α(G) = o(n) . Since the extremal example for the Hajnal-Szemerédi theorem is very similar to the Turán graph, one can similarly ask how stable is this extremal example. With József Balogh and Theodore Molla, we proved that for an n-vertex graph G with α(G) = o(n), if ẟ(G) ≥ (1/2+o(1))n then G has a triangle factor. This minimum degree condition is asymptotically best possible. We also consider a fractional variant of the Corrádi-Hajnal Theorem, settling the triangle case of a conjecture of Balogh, Kemkes, Lee, and Young.
In Chapter 4, we first consider a Ramsey-Turán variant of a theorem of Erd\Ho s. In 1962, he proved that for any r > l ≥ 2, among all K_r-free graphs, the (r-1)-partite Turán graph has the maximum number of copies of K_l. We consider a Ramsey-Turán-type variation of Erdős's result. In particular, we define RT(F,H,f(n)) to be the maximum number of copies of F in an H-free graph with n-vertices and independence number at most f(n). We study this function for different graphs F and H. Recently, Balogh, Hu and Simonovits proved that the Ramsey-Turán function for even cliques experiences a jump. We show that the function RT(K_3,H,f(n)) has a similar behavior when H is an even clique. We also study the sparse analogue of a theorem of Bollobás and Gy\Ho ri about the maximum number of triangles that a C_5-free graph can have. Finally, we consider a Ramsey-Turán variant of a function studied by Erdős and Rothschild about the maximum number of edge-colorings that an n-vertex graph can have without a monochromatic copy of a given graph
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