53 research outputs found

    Improper colourings inspired by Hadwiger’s conjecture

    Get PDF
    Hadwiger’s Conjecture asserts that every Kt-minor-free graph has a proper (t − 1)-colouring. We relax the conclusion in Hadwiger’s Conjecture via improper colourings. We prove that every Kt-minor-free graph is (2t − 2)-colourable with monochromatic components of order at most 1/2 (t − 2). This result has no more colours and much smaller monochromatic components than all previous results in this direction. We then prove that every Kt-minor-free graph is (t − 1)-colourable with monochromatic degree at most t − 2. This is the best known degree bound for such a result. Both these theorems are based on a decomposition method of independent interest. We give analogous results for Ks,t-minorfree graphs, which lead to improved bounds on generalised colouring numbers for these classes. Finally, we prove that graphs containing no Kt-immersion are 2-colourable with bounded monochromatic degree

    Spanning embeddings of arrangeable graphs with sublinear bandwidth

    Full text link
    The Bandwidth Theorem of B\"ottcher, Schacht and Taraz [Mathematische Annalen 343 (1), 175-205] gives minimum degree conditions for the containment of spanning graphs H with small bandwidth and bounded maximum degree. We generalise this result to a-arrangeable graphs H with \Delta(H)<sqrt(n)/log(n), where n is the number of vertices of H. Our result implies that sufficiently large n-vertex graphs G with minimum degree at least (3/4+\gamma)n contain almost all planar graphs on n vertices as subgraphs. Using techniques developed by Allen, Brightwell and Skokan [Combinatorica, to appear] we can also apply our methods to show that almost all planar graphs H have Ramsey number at most 12|H|. We obtain corresponding results for graphs embeddable on different orientable surfaces.Comment: 20 page

    Local resilience of spanning subgraphs in sparse random graphs

    Get PDF
    For each real Îł>0Îł>0 and integers Δ≄2Δ≄2 and k≄1k≄1, we prove that there exist constants ÎČ>0ÎČ>0 and C>0C>0 such that for all p≄C(log⁥n/n)1/Δp≄C(log⁥n/n)1/Δ the random graph G(n,p)G(n,p) asymptotically almost surely contains – even after an adversary deletes an arbitrary (1/k−γ1/k−γ)-fraction of the edges at every vertex – a copy of every n-vertex graph with maximum degree at most Δ, bandwidth at most ÎČn and at least Cmax⁥{p−2,p−1log⁥n}Cmax⁥{p−2,p−1log⁥n} vertices not in triangles

    Embedding large subgraphs into dense graphs

    Full text link
    What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F-packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect F-packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect F-packing. The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy and Szemeredi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F-packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved

    On sufficient conditions for Hamiltonicity in dense graphs

    Full text link
    We study structural conditions in dense graphs that guarantee the existence of vertex-spanning substructures such as Hamilton cycles. It is easy to see that every Hamiltonian graph is connected, has a perfect fractional matching and, excluding the bipartite case, contains an odd cycle. Our main result in turn states that any large enough graph that robustly satisfies these properties must already be Hamiltonian. Moreover, the same holds for embedding powers of cycles and graphs of sublinear bandwidth subject to natural generalisations of connectivity, matchings and odd cycles. This solves the embedding problem that underlies multiple lines of research on sufficient conditions for Hamiltonicity in dense graphs. As applications, we recover and establish Bandwidth Theorems in a variety of settings including Ore-type degree conditions, P\'osa-type degree conditions, deficiency-type conditions, locally dense and inseparable graphs, multipartite graphs as well as robust expanders

    Forcing spanning subgraphs via Ore type conditions

    Get PDF
    Abstract We determine an Ore type condition that allows the embedding of 3-colourable bounded degree graphs of sublinear bandwidth: For all ∆, Îł &gt; 0 there are ÎČ, n 0 &gt; 0 such that for all n ≄ n 0 the following holds. Let G = (V, E) and H be n-vertex graphs such that H is 3-colourable, has maximum degree ∆(H) ≀ ∆ and bandwidth bw(H) ≀ ÎČn, and G satisfies deg(u) + deg(v) ≄

    The bandwidth theorem in sparse graphs

    Get PDF
    The bandwidth theorem [Mathematische Annalen, 343(1):175–205, 2009] states that any n-vertex graph G with minimum degree [Formula Presented] contains all n-vertex k-colourable graphs H with bounded maximum degree and bandwidth o(n). We provide sparse analogues of this statement in random graphs as well as pseudorandom graphs. More precisely, we show that for p ≫[Formula Presented] asymptotically almost surely each spanning subgraph G of G(n, p) with minimum degree [Formula Presented] pn contains all n-vertex k-colourable graphs H with maximum degree ∆, bandwidth o(n), and at least Cp−2 vertices not contained in any triangle. A similar result is shown for sufficiently bijumbled graphs, which, to the best of our knowledge, is the first resilience result in pseudorandom graphs for a rich class of spanning subgraphs. Finally, we provide improved results for H with small degeneracy, which in particular imply a resilience result in G(n, p) with respect to the containment of spanning bounded degree trees for p ≫[Formula Presented]

    Embedding into bipartite graphs

    Full text link
    The conjecture of Bollob\'as and Koml\'os, recently proved by B\"ottcher, Schacht, and Taraz [Math. Ann. 343(1), 175--205, 2009], implies that for any Îł>0\gamma>0, every balanced bipartite graph on 2n2n vertices with bounded degree and sublinear bandwidth appears as a subgraph of any 2n2n-vertex graph GG with minimum degree (1+Îł)n(1+\gamma)n, provided that nn is sufficiently large. We show that this threshold can be cut in half to an essentially best-possible minimum degree of (12+Îł)n(\frac12+\gamma)n when we have the additional structural information of the host graph GG being balanced bipartite. This complements results of Zhao [to appear in SIAM J. Discrete Math.], as well as Hladk\'y and Schacht [to appear in SIAM J. Discrete Math.], who determined a corresponding minimum degree threshold for Kr,sK_{r,s}-factors, with rr and ss fixed. Moreover, it implies that the set of Hamilton cycles of GG is a generating system for its cycle space.Comment: 16 pages, 2 figure
    • 

    corecore