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Register a new user$H$-Decomposition of $r$-graphs when $H$ is an $r$-graph with exactly $k$ independent edges
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Let $phi_H^r(n)$ be the smallest integer such that, for all $r$-graphs $G$ on $n$ vertices, the edge set $E(G)$ can be partitioned into at most $phi_H^r(n)$ parts, of which every part either is a single edge or forms an $r$-graph isomorphic to $H$. The function $phi^2_H(n)$ has been well studied in literature, but for the case $rge 3$, the problem that determining the value of $phi_H^r(n)$ is widely open. Sousa (2010) gave an asymptotic value of $phi_H^r(n)$ when $H$ is an $r$-graph with exactly 2 edges, and determined the exact value of $phi_H^r(n)$ in some special cases. In this paper, we first give the exact value of $phi_H^r(n)$ when $H$ is an $r$-graph with exactly 2 edges, which improves Sousas result. Second we determine the exact value of $phi_H^r(n)$ when $H$ is an $r$-graph consisting of exactly $k$ independent edges.
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It is known that for $Omega subset mathbb{R}^{2}$ an unbounded convex domain and $H>0$, there exists a graph $Gsubset mathbb{R}^{3}$ of constant mean curvature $H$ over $Omega $ with $partial G=$ $partial Omega $ if and only if $Omega $ is included in a strip of width $1/H$. In this paper we obtain results in $mathbb{H}^{2}times mathbb{R}$ in the same direction: given $Hin left( 0,1/2right) $, if $Omega $ is included in a region of $mathbb{ H}^{2}times left{ 0right} $ bounded by two equidistant hypercycles $ell(H)$ apart, we show that, if the geodesic curvature of $partial Omega $ is bounded from below by $-1,$ then there is an $H$-graph $G$ over $Omega $ with $partial G=partial Omega$. We also present more refined existence results involving the curvature of $partialOmega,$ which can also be less than $-1.$
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The ErdH{o}s-Simonovits stability theorem states that for all epsilon >0 there exists alpha >0 such that if G is a K_{r+1}-free graph on n vertices with e(G) > ex(n,K_{r+1}) - alpha n^2, then one can remove epsilon n^2 edges from G to obtain an r-partite graph. Furedi gave a short proof that one can choose alpha=epsilon. We give a bound for the relationship of alpha and varepsilon which is asymptotically sharp as epsilon to 0.
Let $A(G)$ be the adjacency matrix of a graph $G$ with $lambda_{1}(G)$, $lambda_{2}(G)$, ..., $lambda_{n}(G)$ being its eigenvalues in non-increasing order. Call the number $S_k(G):=sum_{i=1}^{n}lambda_{i}^k(G) (k=0,1,...,n-1)$ the $k$th spectral moment of $G$. Let $S(G)=(S_0(G),S_1(G),...,S_{n-1}(G))$ be the sequence of spectral moments of $G$. For two graphs $G_1$ and $G_2$, we have $G_1prec_sG_2$ if $S_i(G_1)=S_i(G_2) (i=0,1,...,k-1)$ and $S_k(G_1)<S_k(G_2)$ for some $kin {1,2,...,n-1}$. Denote by $mathscr{G}_n^k$ the set of connected $n$-vertex graphs with $k$ cut edges. In this paper, we determine the first, the second, the last and the second last graphs, in an $S$-order, among $mathscr{G}_n^k$, respectively.
By using the Szemeredi Regularity Lemma, Alon and Sudakov recently extended the classical Andrasfai-Erd~os-Sos theorem to cover general graphs. We prove, without using the Regularity Lemma, that the following stronger statement is true. Given any (r-1)-partite graph H whose smallest part has t vertices, and any fixed c>0, there exists a constant C such that whenever G is an n-vertex graph with minimum degree at least ((3r-4)/(3r-1)+c)n, either G contains H, or we can delete at most Cn^(2-1/t) edges from G to yield an r-partite graph.
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