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1.
What is the largest number of edges in a graph of order n and girth g? For d-regular graphs, essentially the best known answer is provided by the Moore bound. This result is extended here to cover irregular
graphs as well, yielding an affirmative answer to an old open problem ([4] p. 163, problem 10).
Received: June 27, 2000 Final version received: July 3, 2001 相似文献
2.
We prove that for every fixed k and ? ≥ 5 and for sufficiently large n, every edge coloring of the hypercube Qn with k colors contains a monochromatic cycle of length 2 ?. This answers an open question of Chung. Our techniques provide also a characterization of all subgraphs H of the hypercube which are Ramsey, that is, have the property that for every k, any k‐edge coloring of a sufficiently large Qn contains a monochromatic copy of H. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 196–208, 2006 相似文献
3.
Peter G. Doyle Colin Mallows Alon Orlitsky Larry Shepp 《Israel Journal of Mathematics》1989,67(1):34-42
Letn cells be arranged in a ring, or alternatively, in a row. Initially, all cells are unmarked. Sequentially, one of the unmarked
cells is chosen at random and marked until, aftern steps, each cell is marked. After thekth cell has been marked the configuration of marked cells defines some number of islands: maximal sets of adjacent marked
cells. Let ξ
k
denote the random number of islands afterk cells have been marked.
We give explicit expressions for moments of products of ξ
k
’s and for moments of products of 1/ξ
k
’s.
These are used in a companion paper to prove that if a random graph on the natural number is made by drawing an edge betweeni≧1 andj>i with probabilityλ/j, then the graph is almost surely connected ifλ>1/4 and almost surely disconnected ifλ≦1/4. 相似文献
4.
Noga Alon Oded Goldreich Johan Hstad Ren Peralta 《Random Structures and Algorithms》1992,3(3):289-304
We present three alternative simple constructions of small probability spaces on n bits for which any k bits are almost independent. The number of bits used to specify a point in the sample space is (2 + o(1)) (log log n + k/2 + log k + log 1/?), where ? is the statistical difference between the distribution induced on any k bit locations and the uniform distribution. This is asymptotically comparable to the construction recently presented by Naor and Naor (our size bound is better as long as ? < 1/(k log n)). An additional advantage of our constructions is their simplicity. 相似文献
5.
Letm 3 andk 1 be two given integers. Asub-k-coloring of [n] = {1, 2,...,n} is an assignment of colors to the numbers of [n] in which each color is used at mostk times. Call an
arainbow set if no two of its elements have the same color. Thesub-k-Ramsey number sr(m, k) is defined as the minimumn such that every sub-k-coloring of [n] contains a rainbow arithmetic progression ofm terms. We prove that((k – 1)m
2/logmk) sr(m, k) O((k – 1)m
2 logmk) asm , and apply the same method to improve a previously known upper bound for a problem concerning mappings from [n] to [n] without fixed points.Research supported in part by Allon Fellowship and by a Bat Sheva de-Rothschild grant.Research supported in part by the AKA Research Fund of the Hungarian Academy of Sciences, grant No. 1-3-86-264. 相似文献
6.
As is well known, Lovász Local Lemma implies that everyd-uniformd-regular hypergraph is 2-colorable, providedd 9. We present a different proof of a slightly stronger result; everyd-uniformd-regular hypergraph is 2-colorable, providedd 8.Research supported in part by Allon Fellowship and by a grant from the United States Israel Binational Science Foundation. 相似文献
7.
Forr2 letp(n, r) denote the maximum cardinality of a subsetA ofN={1, 2,...,n} such that there are noBA and an integery with
b=y
r. It is shown that for any>0 andn>n(), (1+o(1))21/(r+1)
n
(r–1)/(r+1)p(n, r)n
+2/3 for allr5, and that for every fixedr6,p(n, r)=(1+o(1))·21/(r+1)
n
(r–1)/(r+1) asn. Letf(n, m) denote the maximum cardinality of a subsetA ofN such that there is noBA the sum of whose elements ism. It is proved that for 3n
6/3+mn
2/20 log2
n andn>n(), f(n, m)=[n/s]+s–2, wheres is the smallest integer that does not dividem. A special case of this result establishes a conjecture of Erds and Graham.Research supported in part by Allon Fellowship, by a Bat-Sheva de Rothschild Grant and by the Fund for Basic Research administered by the Israel Academy of Sciences. 相似文献
8.
A regressive function (also called a regression or contractive mapping) on a partial order P is a function mapping P to itself such that (x)x. A monotone k-chain for is a k-chain on which is order-preserving; i.e., a chain x
1<...ksuch that (x
1)...(xk). Let P
nbe the poset of integer intervals {i, i+1, ..., m} contained in {1, 2, ..., n}, ordered by inclusion. Let f(k) be the least value of n such that every regression on P
nhas a monotone k+1-chain, let t(x,j) be defined by t(x, 0)=1 and t(x,j)=x
t(x,j–1). Then f(k) exists for all k (originally proved by D. White), and t(2,k) < f(K) <t( + k, k) , where k 0 as k. Alternatively, the largest k such that every regression on P
nis guaranteed to have a monotone k-chain lies between lg*(n) and lg*(n)–2, inclusive, where lg*(n) is the number of appliations of logarithm base 2 required to reduce n to a negative number. Analogous results hold for choice functions, which are regressions in which every element is mapped to a minimal element. 相似文献
9.
Vinyl acetate–vinyl alcohol copolymers were fractionated in order to obtain their degree of hydrolysis distributions. In the method employed for fractionation, the differences in molecular weight of copolymer did not affect the fractional separation. Degree of hydrolysis distributions was found to be broad, with a pronounced maximum at a low degree of hydrolysis. Viscosity measurements were performed both for the precipitated fractions and unfractionated polymer. The Huggins constant was found to increase with a decrease in the degree of hydrolysis of polyvinyl acetate. These results were interpreted in terms of a polymer molecular association. From values of Huggins constants, comparative information about copolymer “blockiness” is also established. 相似文献
10.
Harary [8] calls a finite, simple graphG asum graph if one can assign to eachv
i
V(G) a labelx
i
so that{v
i
,v
j
}E(G) iffx
i
+x
j
=x
k
for somek. We generalize this notion by replacing x
i
+x
j
with an arbitrary symmetric polynomialf(x
i
,x
j
). We show that for eachf, not all graphs are f-graphs. Furthermore, we prove that for everyf and every graphG we can transformG into anf-graph via the addition of |E(G)| isolated vertices. This result is nearly best possible in the sense that for allf and for
, there is a graphG withn vertices andm edges which, even after the addition ofm-O(n logn) isolated vetices, is not andf-graph.Research supported in part by a U.S.A.-Israel Binational Science Foundation and by a Bergmann Memorial Grant.Research supported in part by the Office of Naval Research contract number N00014-85-K0622. 相似文献