326,645 research outputs found

    The early evolution of the H-free process

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    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 nn \to \infty, the minimum degree in G is at least cn1(vH2)/(eH1)(logn)1/(eH1)cn^{1-(v_H-2)/(e_H-1)}(\log n)^{1/(e_H-1)}. This gives new lower bounds for the Tur\'an numbers of certain bipartite graphs, such as the complete bipartite graphs Kr,rK_{r,r} with r5r \ge 5. When H is a complete graph KsK_s with s5s \ge 5 we show that for some C>0, with high probability the independence number of G is at most Cn2/(s+1)(logn)11/(eH1)Cn^{2/(s+1)}(\log n)^{1-1/(e_H-1)}. This gives new lower bounds for Ramsey numbers R(s,t) for fixed s5s \ge 5 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

    Peaceful Colourings

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    We introduce peaceful colourings, a variant of hh-conflict free colourings. We call a colouring with no monochromatic edges pp-peaceful if for each vertex vv, there are at most pp neighbours of vv coloured with a colour appearing on another neighbour of vv. An hh-conflict-free colouring of a graph is a (vertex)-colouring with no monochromatic edges so that for every vertex vv, the number of neighbours of vv which are coloured with a colour appearing on no other neighbour of vv is at least the minimum of hh and the degree of vv. If GG is Δ\Delta-regular then it has an hh-conflict free colouring precisely if it has a (Δh)(\Delta-h)-peaceful colouring. We focus on the minimum pΔp_\Delta of those pp for which every graph of maximum degree Δ\Delta has a pp-peaceful colouring with Δ+1\Delta+1 colours. We show that pΔ>(11eo(1))Δp_\Delta > (1-\frac{1}{e}-o(1))\Delta and that for graphs of bounded codegree, pΔ(11e+o(1))Δp_\Delta \leq (1-\frac{1}{e}+o(1))\Delta. We ask if the latter result can be improved by dropping the bound on the codegree. As a partial result, we show that pΔ80008001Δp_\Delta \leq \frac{8000}{8001}\Delta for sufficiently large Δ\Delta

    Eulerian subgraphs and Hamiltonicity of claw -free graphs

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    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

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    Given an infinite family G{\mathcal G} of graphs and a monotone property P{\mathcal P}, an (upper) threshold for G{\mathcal G} and P{\mathcal P} is a "fastest growing" function p:N[0,1]p: \mathbb{N} \to [0,1] such that limnPr(Gn(p(n))P)=1\lim_{n \to \infty} \Pr(G_n(p(n)) \in {\mathcal P})= 1 for any sequence (Gn)nN(G_n)_{n \in \mathbb{N}} over G{\mathcal G} with limnV(Gn)=\lim_{n \to \infty}\lvert V(G_n) \rvert = \infty, where Gn(p(n))G_n(p(n)) is the random subgraph of GnG_n such that each edge remains independently with probability p(n)p(n). In this paper we study the upper threshold for the family of HH-minor free graphs and for the graph property of being (r1)(r-1)-degenerate, which is one fundamental graph property with many applications. Even a constant factor approximation for the upper threshold for all pairs (r,H)(r,H) 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 (r1)(r-1)-degenerate for a large class of pairs (r,H)(r,H), including all graphs HH of minimum degree at least rr and all graphs HH with no vertex-cover of size at most rr, and provide lower bounds for the rest of the pairs of (r,H)(r,H). The results generalize to arbitrary proper minor-closed families and the properties of being rr-colorable, being rr-choosable, or containing an rr-regular subgraph, respectively

    Embedding problems and Ramsey-Turán variations in extremal graph theory

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    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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