878 research outputs found

    Decomposing highly edge-connected graphs into homomorphic copies of a fixed tree

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    The Tree Decomposition Conjecture by Bar\'at and Thomassen states that for every tree TT there exists a natural number k(T)k(T) such that the following holds: If GG is a k(T)k(T)-edge-connected simple graph with size divisible by the size of TT, then GG can be edge-decomposed into subgraphs isomorphic to TT. So far this conjecture has only been verified for paths, stars, and a family of bistars. We prove a weaker version of the Tree Decomposition Conjecture, where we require the subgraphs in the decomposition to be isomorphic to graphs that can be obtained from TT by vertex-identifications. We call such a subgraph a homomorphic copy of TT. This implies the Tree Decomposition Conjecture under the additional constraint that the girth of GG is greater than the diameter of TT. As an application, we verify the Tree Decomposition Conjecture for all trees of diameter at most 4.Comment: 18 page

    Spanning embeddings of arrangeable graphs with sublinear bandwidth

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    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

    Partitioning random graphs into monochromatic components

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    Erd\H{o}s, Gy\'arf\'as, and Pyber (1991) conjectured that every rr-colored complete graph can be partitioned into at most r1r-1 monochromatic components; this is a strengthening of a conjecture of Lov\'asz (1975) in which the components are only required to form a cover. An important partial result of Haxell and Kohayakawa (1995) shows that a partition into rr monochromatic components is possible for sufficiently large rr-colored complete graphs. We start by extending Haxell and Kohayakawa's result to graphs with large minimum degree, then we provide some partial analogs of their result for random graphs. In particular, we show that if p(27lognn)1/3p\ge \left(\frac{27\log n}{n}\right)^{1/3}, then a.a.s. in every 22-coloring of G(n,p)G(n,p) there exists a partition into two monochromatic components, and for r2r\geq 2 if p(rlognn)1/rp\ll \left(\frac{r\log n}{n}\right)^{1/r}, then a.a.s. there exists an rr-coloring of G(n,p)G(n,p) such that there does not exist a cover with a bounded number of components. Finally, we consider a random graph version of a classic result of Gy\'arf\'as (1977) about large monochromatic components in rr-colored complete graphs. We show that if p=ω(1)np=\frac{\omega(1)}{n}, then a.a.s. in every rr-coloring of G(n,p)G(n,p) there exists a monochromatic component of order at least (1o(1))nr1(1-o(1))\frac{n}{r-1}.Comment: 27 pages, 2 figures. Appears in Electronic Journal of Combinatorics Volume 24, Issue 1 (2017) Paper #P1.1

    Square Property, Equitable Partitions, and Product-like Graphs

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    Equivalence relations on the edge set of a graph GG that satisfy restrictive conditions on chordless squares play a crucial role in the theory of Cartesian graph products and graph bundles. We show here that such relations in a natural way induce equitable partitions on the vertex set of GG, which in turn give rise to quotient graphs that can have a rich product structure even if GG itself is prime.Comment: 20 pages, 6 figure

    Parabolic theory of the discrete p-Laplace operator

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    We study the discrete version of the pp-Laplacian. Based on its variational properties we discuss some features of the associated parabolic problem. Our approach allows us in turn to obtain interesting information about positivity and comparison principles as well as compatibility with the symmetries of the graph. We conclude briefly discussing the variational properties of a handful of nonlinear generalized Laplacians appearing in different parabolic equations.Comment: 35 pages several corrections and enhancements in comparison to the v
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