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Register a new userA degree sequence version of the Kuhn-Osthus tiling theorem
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A fundamental result of Kuhn and Osthus [The minimum degree threshold for perfect graph packings, Combinatorica, 2009] determines up to an additive constant the minimum degree threshold that forces a graph to contain a perfect H-tiling. We prove a degree sequence version of this result which allows for a significant number of vertices to have lower degree.
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We give the following extension of Baranys colorful Caratheodory theorem: Let M be an oriented matroid and N a matroid with rank function r, both defined on the same ground set V and satisfying rank(M) < rank(N). If every subset A of V with r(V - A) < rank (M) contains a positive circuit of M, then some independent set of N contains a positive circuit of M.
Komlos [Komlos: Tiling Turan Theorems, Combinatorica, 2000] determined the asymptotically optimal minimum-degree condition for covering a given proportion of vertices of a host graph by vertex-disjoint copies of a fixed graph H, thus essentially extending the Hajnal-Szemeredi theorem which deals with the case when H is a clique. We give a proof of a graphon version of Komloss theorem. To prove this graphon version, and also to deduce from it the original statement about finite graphs, we use the machinery introduced in [Hladky, Hu, Piguet: Tilings in graphons, arXiv:1606.03113]. We further prove a stability version of Komloss theorem.
In two papers, Little and Sellers introduced an exciting new combinatorial method for proving partition identities which is not directly bijective. Instead, they consider various sets of weighted tilings of a $1 times infty$ board with squares and dominoes, and for each type of tiling they construct a generating function in two different ways, which generates a $q$-series identity. Using this method, they recover quite a few classical $q$-series identities, but Eulers Pentagonal Number Theorem is not among them. In this paper, we introduce a key parameter when constructing the generating functions of various sets of tilings which allows us to recover Eulers Pentagonal Number Theorem along with an infinite family of generalizations.
For a collection $mathbf{G}={G_1,dots, G_s}$ of not necessarily distinct graphs on the same vertex set $V$, a graph $H$ with vertices in $V$ is a $mathbf{G}$-transversal if there exists a bijection $phi:E(H)rightarrow [s]$ such that $ein E(G_{phi(e)})$ for all $ein E(H)$. We prove that for $|V|=sgeq 3$ and $delta(G_i)geq s/2$ for each $iin [s]$, there exists a $mathbf{G}$-transversal that is a Hamilton cycle. This confirms a conjecture of Aharoni. We also prove an analogous result for perfect matchings.
Let G be a simple connected graph of order n with degree sequence d_1, d_2, ..., d_n in non-increasing order. The spectral radius rho(G) of G is the largest eigenvalue of its adjacency matrix. For each positive integer L at most n, we give a sharp upper bound for rho(G) by a function of d_1, d_2, ..., d_L, which generalizes a series of previous results.
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