Ramsey Numbers for Trees of Small Maximum Degree

For a tree T we write and , , for the sizes of the vertex classes of T as a bipartite graph. It is shown that for T with maximum degree , the obvious lower bound for the Ramsey number R(T,T) of is asymptotically the correct value for R(T,T). Received December 15, 1999 RID=" " ID=" " The first and third authors were partially supported by NSERC. The second author was partially supported by KBN grant 2 P03A 021 17.     

Steiner Numbers in Graphs

Abstract

The Steiner distance d(S) of a set S of vertices in a connected graph G is the minimum size of a connected subgraph of G that contains S. The Steiner number s(G) of a connected graph G of order p is the smallest positive integer m for which there exists a set S of m vertices of G such that d(S) = p—1. A smallest set S of vertices of a connected graph G of order p for which d(S) = p—1 is called a Steiner spanning set of G. It is shown that every connected graph has a unique Steiner spanning set. If G is a connected graph of order p and k is an integer with 0 ≤ k ≤ p—1, then the kth Steiner number sk(G) of G is the smallest positive integer m for which there exists a set S of m vertices of G such that d(S) = k. The sequence s_o(G),s₁ (G),…,8_p-1(G) is called the Steiner sequence of G. Steiner sequences for trees are characterized.     

Trees in tournaments

Letf(n) be the smallest integer such that every tournament of orderf(n) contains every oriented tree of ordern. Sumner has just conjectures thatf(n)=2n–2, and F. K. Chung has shown thatf(n)(1+o(1))nlog₂ n. Here we show thatf(n)12n andf(n)(4+o(1))n.     

Colorful Isomorphic Spanning Trees in Complete Graphs

Can a complete graph on an even number n (> 4) of vertices be properly edge-colored with n − 1 colors in such a way that the edges can be partitioned into edge-disjoint colorful isomorphic spanning trees? A spanning treee is colorful if all n − 1 colors occur among its edges. It is proved that this is possible to accomplish whenever n is a power of two.Received July 24, 2001     

On Graphs With Small Ramsey Numbers,II

There exists a constant C such that for every d-degenerate graphs G ₁ and G ₂ on n vertices, Ramsey number R(G ₁,G ₂) is at most Cn, where is the minimum of the maximum degrees of G ₁ and G ₂.* The work of this author was supported by the grants 99-01-00581 and 00-01-00916 of the Russian Foundation for Fundamental Research and the Dutch-Russian Grant NWO-047-008-006. The work of this author was supported by the NSF grant DMS-9704114.     

Trees with extremal numbers of maximal independent sets including the set of leaves

A subset S of vertices of a graph G is independent if no two vertices in S are adjacent. In this paper we study maximal (with respect to set inclusion) independent sets in trees including the set of leaves. In particular the smallest and the largest number of these sets among n-vertex trees are determined characterizing corresponding trees. We define a local augmentation of trees that preserves the number of maximal independent sets including the set of leaves.     

Induced Ramsey Numbers

   We investigate the induced Ramsey number of pairs of graphs (G, H). This number is defined to be the smallest possible order of a graph Γ with the property that, whenever its edges are coloured red and blue, either a red induced copy of G arises or else a blue induced copy of H arises. We show that, for any G and H with , we have

Sharp Bounds For Some Multicolor Ramsey Numbers

Let H ₁,H ₂, . . .,H _k+1 be a sequence of k+1 finite, undirected, simple graphs. The (multicolored) Ramsey number r(H ₁,H ₂,...,H _k+1) is the minimum integer r such that in every edge-coloring of the complete graph on r vertices by k+1 colors, there is a monochromatic copy of H _i in color i for some 1ik+1. We describe a general technique that supplies tight lower bounds for several numbers r(H ₁,H ₂,...,H _k+1) when k2, and the last graph H _k+1 is the complete graph K _m on m vertices. This technique enables us to determine the asymptotic behaviour of these numbers, up to a polylogarithmic factor, in various cases. In particular we show that r(K ₃,K ₃,K _m ) = (m ³ poly logm), thus solving (in a strong form) a conjecture of Erdos and Sós raised in 1979. Another special case of our result implies that r(C ₄,C ₄,K _m ) = (m ² poly logm) and that r(C ₄,C ₄,C ₄,K _m ) = (m ²/log² m). The proofs combine combinatorial and probabilistic arguments with spectral techniques and certain estimates of character sums.* Research supported in part by a State of New Jersey grant, by a USA Israeli BSF grant and by a grant from the Israel Science Foundation. Research supported by NSF grant DMS 9704114.     

Homogeneous trees are bilipschitz equivalent

We prove that any two locally finite homogeneous trees with valency greater than 3 are bilipschitz equivalent. This implies that the quotienth ¹(G)/h ^k (G), whereh ^k (G) is thekthL ₂-Betti number ofG, is not a quasi-isometry invariant.     

Trees and continuous mappings into the real line

Trees of height ω₁ are characterized in terms of continuous mappings to the real line. In particular Souslin trees are characterized as those uncountable trees with no uncountable continuous image in the real line. Trees which can not be continuously embeded in the real line are also characterized.     

Spanning Trees with Vertices Having Large Degrees

Let G be a connected simple graph, and let f be a mapping from to the set of integers. This paper is concerned with the existence of a spanning tree in which each vertex v has degree at least . We show that if for any nonempty subset , then a connected graph G has a spanning tree such that for all , where is the set of neighbors v of vertices in S with , , and is the degree of x in T. This is an improvement of several results, and the condition is best possible.     

The Number of Subtrees of Trees with Given Degree Sequence

This article investigates some properties of the number of subtrees of a tree with given degree sequence. These results are used to characterize trees with the given degree sequence that have the largest number of subtrees, which generalize the recent results of Kirk and Wang (SIAM J Discrete Math 22 (2008), 985–995). These trees coincide with those which were proven by Wang and independently Zhang et al. (2008) to minimize the Wiener index. We also provide a partial ordering of the extremal trees with different degree sequences, some extremal results follow as corollaries.     

A Class of M-matrices whose Graphs are Trees

In this article, we characterize generalized ultrametric matrices whose inverses are tree-diagonal. This generalizes the results of McDonald, Nabben, Neumann, Schneider and Tsatsomeros for tri-diagonal matrices.     

On Bipartite Graphs with Linear Ramsey Numbers

Dedicated to the memory of Paul Erdős We provide an elementary proof of the fact that the ramsey number of every bipartite graph H with maximum degree at most is less than . This improves an old upper bound on the ramsey number of the n-cube due to Beck, and brings us closer toward the bound conjectured by Burr and Erdős. Applying the probabilistic method we also show that for all and there exists a bipartite graph with n vertices and maximum degree at most whose ramsey number is greater than for some absolute constant c>1. Received December 1, 1999 RID="*" ID="*" Supported by NSF grant DMS-9704114 RID="**" ID="**" Supported by KBN grant 2 P03A 032 16     

Total Variation Distance for Poisson Subset Numbers

Let n be an integer and A₀,..., A_k random subsets of {1,..., n} of fixed sizes a₀,..., a_k, respectively chosen independently and uniformly. We provide an explicit and easily computable total variation bound between the distance from the random variable , the size of the intersection of the random sets, to a Poisson random variable Z with intensity λ = EW. In particular, the bound tends to zero when λ converges and for all j = 0,..., k, showing that W has an asymptotic Poisson distribution in this regime. Received February 24, 2005     

All graphs with maximum degree three whose complements have 4-cycle decompositions

Let G be the set that contains precisely the graphs on n vertices with maximum degree 3 for which there exists a 4-cycle system of their complement in K_n. In this paper G is completely characterized.     

First Observations on Prefab Posets’ Whitney Numbers

We introduce a natural partial order ≤ in structurally natural finite subsets of the cobweb prefabs sets recently constructed by the present author. Whitney numbers of the second kind of the corresponding subposet which constitute Stirling-like numbers’ triangular array — are then calculated and the explicit formula for them is provided. Next — in the second construction — we endow the set sums of prefabiants with such an another partial order that their Bell-like numbers include Fibonacci triad sequences introduced recently by the present author in order to extend famous relation between binomial Newton coefficients and Fibonacci numbers onto the infinity of their relatives among whom there are also the Fibonacci triad sequences and binomial-like coefficients (incidence coefficients included). The first partial order is F-sequence independent while the second partial order is F-sequence dependent where F is the so-called admissible sequence determining cobweb poset by construction. An F-determined cobweb poset’s Hasse diagram becomes Fibonacci tree sheathed with specific cobweb if the sequence F is chosen to be just the Fibonacci sequence. From the stand-point of linear algebra of formal series these are generating functions which stay for the so-called extended coherent states of quantum physics. This information is delivered in the last section. Presentation (November 2006) at the Gian-Carlo Rota Polish Seminar .     

Trees and Markov Convexity

We show that an infinite weighted tree admits a bi-Lipschitz embedding into Hilbert space if and only if it does not contain arbitrarily large complete binary trees with uniformly bounded distortion. We also introduce a new metric invariant called Markov convexity, and show how it can be used to compute the Euclidean distortion of any metric tree up to universal factors. The research of the first author was conducted while he was at U. C. Berkeley and the Institute for Advanced Study.     

Random Minimum Length Spanning Trees in Regular Graphs

r -regular n-vertex graph G with random independent edge lengths, each uniformly distributed on (0, 1). Let mst(G) be the expected length of a minimum spanning tree. We show that mst(G) can be estimated quite accurately under two distinct circumstances. Firstly, if r is large and G has a modest edge expansion property then , where . Secondly, if G has large girth then there exists an explicitly defined constant such that . We find in particular that . Received: Februray 9, 1998     

Classes of graphs that exclude a tree and a clique and are not vertex ramsey

A class of graphs is vertex Ramsey if for allH there existsG such that for all partitions of the vertices ofG into two parts, one of the parts contains an induced copy ofH. Forb (T,K) is the class of graphs that induce neitherT norK. LetT(k, r) be the tree with radiusr such that each nonleaf is adjacent tok vertices farther from the root than itself. Gyárfás conjectured that for all treesT and cliquesK, there exists an integerb such that for allG in Forb(T,K), the chromatic number ofG is at mostb. Gyárfás' conjecture implies a weaker conjecture of Sauer that for all treesT and cliquesK, Forb(T,K) is not vertex Ramsey. We use techniques developed for attacking Gyárfás' conjecture to prove that for allq, r and sufficiently largek, Forb(T(k,r),K _q ) is not vertex Ramsey.Research partially supported by Office of Naval Research grant N00014-90-J-1206.