10,500 research outputs found

    Sharp lower bounds on the fractional matching number

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    A fractional matching of a graph G is a function f from E(G) to the interval [0,1] such that \sum_{e\in\Gamma(v)}f(e) \le 1 for each v\in V(G), where \Gamma(v) is the set of edges incident to v. The fractional matching number of G, written \alpha'_*(G), is the maximum of \sum_{e\in E(G)}f(e) over all fractional matchings f. For G with n vertices, m edges, positive minimum degree d, and maximum degree D, we prove \alpha'_*(G) \ge \max\{m/D, n-m/d, d n/(D+d)\}. For the first two bounds, equality holds if and only if each component of G is r-regular or is bipartite with all vertices in one part having degree r, where r=D for the first bound and r=d for the second. Equality holds in the third bound if and only if G is regular or is (d,D)-biregular

    A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on bounded domains

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    We investigate quantitative properties of the nonnegative solutions u(t,x)0u(t,x)\ge 0 to the nonlinear fractional diffusion equation, tu+L(um)=0\partial_t u + {\mathcal L} (u^m)=0, posed in a bounded domain, xΩRNx\in\Omega\subset {\mathbb R}^N with m>1m>1 for t>0t>0. As L{\mathcal L} we use one of the most common definitions of the fractional Laplacian (Δ)s(-\Delta)^s, 0<s<10<s<1, in a bounded domain with zero Dirichlet boundary conditions. We consider a general class of very weak solutions of the equation, and obtain a priori estimates in the form of smoothing effects, absolute upper bounds, lower bounds, and Harnack inequalities. We also investigate the boundary behaviour and we obtain sharp estimates from above and below. The standard Laplacian case s=1s=1 or the linear case m=1m=1 are recovered as limits. The method is quite general, suitable to be applied to a number of similar problems

    Bipartite induced density in triangle-free graphs

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    We prove that any triangle-free graph on nn vertices with minimum degree at least dd contains a bipartite induced subgraph of minimum degree at least d2/(2n)d^2/(2n). This is sharp up to a logarithmic factor in nn. Relatedly, we show that the fractional chromatic number of any such triangle-free graph is at most the minimum of n/dn/d and (2+o(1))n/logn(2+o(1))\sqrt{n/\log n} as nn\to\infty. This is sharp up to constant factors. Similarly, we show that the list chromatic number of any such triangle-free graph is at most O(min{n,(nlogn)/d})O(\min\{\sqrt{n},(n\log n)/d\}) as nn\to\infty. Relatedly, we also make two conjectures. First, any triangle-free graph on nn vertices has fractional chromatic number at most (2+o(1))n/logn(\sqrt{2}+o(1))\sqrt{n/\log n} as nn\to\infty. Second, any triangle-free graph on nn vertices has list chromatic number at most O(n/logn)O(\sqrt{n/\log n}) as nn\to\infty.Comment: 20 pages; in v2 added note of concurrent work and one reference; in v3 added more notes of ensuing work and a result towards one of the conjectures (for list colouring

    Multipartite hypergraphs achieving equality in Ryser's conjecture

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    A famous conjecture of Ryser is that in an rr-partite hypergraph the covering number is at most r1r-1 times the matching number. If true, this is known to be sharp for rr for which there exists a projective plane of order r1r-1. We show that the conjecture, if true, is also sharp for the smallest previously open value, namely r=7r=7. For r{6,7}r\in\{6,7\}, we find the minimal number f(r)f(r) of edges in an intersecting rr-partite hypergraph that has covering number at least r1r-1. We find that f(r)f(r) is achieved only by linear hypergraphs for r5r\le5, but that this is not the case for r{6,7}r\in\{6,7\}. We also improve the general lower bound on f(r)f(r), showing that f(r)3.052r+O(1)f(r)\ge 3.052r+O(1). We show that a stronger form of Ryser's conjecture that was used to prove the r=3r=3 case fails for all r>3r>3. We also prove a fractional version of the following stronger form of Ryser's conjecture: in an rr-partite hypergraph there exists a set SS of size at most r1r-1, contained either in one side of the hypergraph or in an edge, whose removal reduces the matching number by 1.Comment: Minor revisions after referee feedbac
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