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81.
82.
A hypergraph is b‐simple if no two distinct edges share more than b vertices. Let m(r, t, g) denote the minimum number of edges in an r‐uniform non‐t‐colorable hypergraph of girth at least g. Erd?s and Lovász proved that A result of Szabó improves the lower bound by a factor of r2?? for sufficiently large r. We improve the lower bound by another factor of r and extend the result to b‐simple hypergraphs. We also get a new lower bound for hypergraphs with a given girth. Our results imply that for fixed b, t, and ? > 0 and sufficiently large r, every r‐uniform b‐simple hypergraph with maximum edge‐degree at most trr1?? is t‐colorable. Some results hold for list coloring, as well. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009 相似文献
83.
Using PMR spectroscopy, we have demonstrated the existence of an imine-enamine tautomerism and optically active forms in tropanone Schiff's bases. We have studied the effect of solvents and substituents on the tautomeric equilibrium of the system. By reaction with acid chlorides, we obtain the N-acylation products of the enamine form of the Schiff's bases.Translated from Khimiya Geterotsiklicheskikh Soedinenii, Vol. 34, No. 3, pp. 381–386, March, 1994. 相似文献
84.
85.
A graph ofn vertices and
edges contains aTK
t on at most 7t
2 logt/ vertices. This answers a question of P. Erds. 相似文献
86.
LetP(k,r;n) denote the containment order generated by thek-element andr-element subsets of ann-element set, and letd(k,r;n) be its dimension. Previous research in this area has focused on the casek=1.P(1,n–1;n) is the standard example of ann-dimensional poset, and Dushnik determined the value ofd(1,r;n) exactly, whenr2
. Spencer used the Erdös-Szekeres theorem to show thatd(1, 2;n) lg lgn, and he used the concept of scrambling families of sets to show thatd(1,r;n)=(lg lgn) for fixedr. Füredi, Hajnal, Rödl and Trotter proved thatd(1, 2;n)=lg lgn+(1/2+o(1))lg lg lgn. In this paper, we concentrate on the casek2. We show thatP(2,n–2;n) is (n–1)-irreducible, and we investigated(2,r;n) whenr2
, obtaining the exact value for almost allr.The research was supported in part by NSF grant DMS 9201467.The research was supported in part by the Universities in Russia program. 相似文献
87.
Belostotskii A. M. Kostochka L. M. Skoldinov A. P. 《Chemistry of Heterocyclic Compounds》1982,18(12):1280-1284
The photochemical oxidation of 1,2,2,6,6-pentamethyl-4-piperidol by ketones occurs exclusively at the methylamino group and, in the case of photolytically stable ketones, may lead to products of dimerization of the aminoalkyl radicals and recombination of the aminoalkyl and ketyl radicals and to a product of N-demethylation of the starting amino alcohol. When ketones that are unstable with respect to irradiation are used, photooxidation competes to a considerable extent with photodecomposition of such ketones. Spatial proximity of the aryl and -methyl groups are observed for the products of reductive addition of the ketones on the basis of the PMR spectra.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 12, pp. 1657–1661, December, 1982. 相似文献
88.
Noga Alon Alexandr Kostochka Benjamin Reiniger Douglas B. West Xuding Zhu 《Israel Journal of Mathematics》2016,212(1):315-335
Using the classical analysis resolution of singularities algorithm of [G4], we generalize the theorems of [G3] on Rn sublevel set volumes and oscillatory integrals with real phase function to functions over an arbitrary local field of characteristic zero. The p-adic cases of our results provide new estimates for exponential sums as well as new bounds on how often a function f(x), such as a polynomial with integer coefficients, is divisible by various powers of a prime p when x is an integer. Unlike many papers on such exponential sums and p-adic oscillatory integrals, we do not require the Newton polyhedron of the phase to be nondegenerate, but rather as in [G3] we have conditions on the maximum order of the zeroes of certain polynomials corresponding to the compact faces of this Newton polyhedron. 相似文献
89.
A graph G is ‐colorable if can be partitioned into two sets and so that the maximum degree of is at most j and of is at most k. While the problem of verifying whether a graph is (0, 0)‐colorable is easy, the similar problem with in place of (0, 0) is NP‐complete for all nonnegative j and k with . Let denote the supremum of all x such that for some constant every graph G with girth g and for every is ‐colorable. It was proved recently that . In a companion paper, we find the exact value . In this article, we show that increasing g from 5 further on does not increase much. Our constructions show that for every g, . We also find exact values of for all g and all . 相似文献
90.
The Hadwiger number η(G) of a graph G is the largest integer n for which the complete graph K
n
on n vertices is a minor of G. Hadwiger conjectured that for every graph G, η(G) ≥ χ(G), where χ(G) is the chromatic number of G. In this paper, we study the Hadwiger number of the Cartesian product of graphs.
As the main result of this paper, we prove that for any two graphs G
1 and G
2 with η(G
1) = h and η(G
2) = l. We show that the above lower bound is asymptotically best possible when h ≥ l. This asymptotically settles a question of Z. Miller (1978).
As consequences of our main result, we show the following:
Alexandr Kostochka: Research of this author is supported in part by NSF grant DMS-0650784 and grant 06-01-00694 of the Russian
Foundation for Basic Research. 相似文献
1. | Let G be a connected graph. Let be the (unique) prime factorization of G. Then G satisfies Hadwiger’s conjecture if k ≥ 2 log log χ(G) + c′, where c′ is a constant. This improves the 2 log χ(G) + 3 bound in [2]. |
2. | Let G 1 and G 2 be two graphs such that χ(G 1) ≥ χ(G 2) ≥ c log1.5(χ(G 1)), where c is a constant. Then satisfies Hadwiger’s conjecture. |
3. | Hadwiger’s conjecture is true for G d (Cartesian product of G taken d times) for every graph G and every d ≥ 2. This settles a question by Chandran and Sivadasan [2]. (They had shown that the Hadiwger’s conjecture is true for G d if d ≥ 3). |