Stanley Selkow

Worcester Polytechnic Institute | WPI · Department of Computer Science

The typical problem in (generalized) Ramsey theory is to find the order of the largest monochromatic member of a family \({\mathcal{F}}\) (for example matchings, paths, cycles, connected subgraphs) that must be present in any edge coloring of a complete graph K n with t colors. Another area is to find the minimum number of monochromatic members of...

In a landmark paper, Erdős et al. (1991) [3] proved that if G is a complete graph whose edges are colored with r colors then the vertex set of G can be partitioned into at most cr2logr monochromatic, vertex disjoint cycles for some constant c. Sárközy extended this result to non-complete graphs, and Sárközy and Selkow extended it to k-regular subgr...

A Gallai-coloring of a complete graph is an edge coloring such that no triangle is colored with three distinct colors. Gallai-colorings occur in various contexts such as the theory of partially ordered sets (in Gallai's original paper) or information theory. Gallai-colorings extend 2-colorings of the edges of complete graphs. They actually turn out...

A graph G on n vertices is called a Dirac graph if it has minimum degree at least n=2. The distance dist G (u; v) is de ned as the number of edges in a shortest subpath of G joining u and v. In this paper we show that in a Dirac graph G, for every small enough subset A of the vertices, we can distribute the vertices of A along a Hamiltonian cycle C...

The First-Fit (or Grundy) chromatic number of G, written as ÇFF(G), is defined as the maximum number of classes in an ordered partition of V(G) into independent sets so that each vertex has a neighbor in each set earlier than its own. The well-known Nordhaus--Gaddum inequality states that the sum of the ordinary chromatic numbers of an n-vertex gra...

Given a graph L , in this article we investigate the anti‐Ramsey number χ S (n,e,L), defined to be the minimum number of colors needed to edge‐color some graph G ( n , e ) with n vertices and e edges so that in every copy of L in G all edges have different colors. We call such a copy of L totally multicolored (TMC). In 7 among many other interestin...

We let G(r)(n,m) denote the set of r-uniform hypergraphs with n vertices and m edges, and f(r)(n,p,s) is the smallest m such that every member of G(r)(n,m) contains a member of G(r)(p,s). In this paper we are interested in the growth of f(r)(n,p,s) for fixed values r,p and s. Brown, Erdos and Sós [Some external problems on r-graphs, in: New Directi...

We give some lower bounds on the certificate complexity of some problems concerning stable marriage, answering a question of Gusfield and Irving.

We let G ( r)( n,m) denote the set of r-uniform hypergraphs with n vertices and m edges, and f ( r)( n,p,s) is the smallest m such that every member of G ( r)( n,m) contains amember of G ( r)( p,s). In this paper we are interested in fixed values r,p and s for which f ( r)( n,p,s) grows quadratically with n. A probabilistic construction of Brown, E...

Given graphs G and H, an edge coloring of G is called an (H,q)-coloring if the edges of every copy of H ⊂ G together receive at least q colors. Let r(G,H,q) denote the minimum number of colors in a (H,q)-coloring of G. In 9 Erdős and Gyárfás studied r(Kn,Kp,q) if p and q are fixed and n tends to infinity. They determined for every fixed p the small...

Given graphs G and H, an edge coloring of G is an (H,q)-coloring if the edges of every copy of H⊂G together receive at least q colors. Let r(G,H,q) denote the minimum number of colors in a (H,q)-coloring of G. The authors study the behaviour of r(K n,n ,K p,p ,q), namely for those values of q for which r(K n,n ,K p,p ,q) is between the linear and t...

In response to a question of Bondy, bounds are established on the minimum number of Hamiltonian cycles in all graphs of order n and minimum degree at least n/2.

We let G (n, m) denote the set of r-uniform hypergraphs with n vertices and m edges, and f(r)(n,p, s) is the smallest m such that every member of G (n, m) contains a member of G (p, s). In this paper we are interested in the growth of f(r)(n,p,s) for fixed values r,p and s. Brown, Erd6s and T. S5s ([2]) proved that for r > k _> 2 and s _> 3we have...

We give a simple quantitative proof that for every natural number p _> 3 and real number 5 > 0, there is a natural number No = No(p, 5) such that for N _> No, every set of at least 5N 2 points of [N] 2 contains a set ofp points that determine at least p - [log 2 p] isosceles right-angle triangles; i.e. triples in the form ((a, b), (a +a,b),(a,b+ a)...

We let G (n, m) denote the set of r-uniform hypergraphs with n vertices andrn edges, and f(O(n,p, s) is the smallest m such that every member of G(0 (n, m)contains a member of G (p, s). In this paper we are interested in fixed values r,pand s for which f(r) (n, p, s) grows quadratically with n. A probabilistic constructionof Brown, ErdSs and T. S6s...

For fixed integers p and q, an edge coloring of K n is called a (p; q)-coloring if the edges of K n in every subset of p vertices are colored with at least q distinct colors. Let f(n; p; q) be the smallest number of colors needed for a (p; q)-coloring of K n . In [3] Erdos and Gy'arf'as studied this function if p and q are fixed and n tends to infi...

Generalizing a result of Erdős, Gyárfás and Pyber we show that there exists a constant c such that for any integers r,k⩾2 and for any coloring of the edges of a complete graph with r colors, its vertices can be partitioned into at most rc(rlogr+k) connected monochromatic k-regular subgraphs and vertices. We also show that the same result holds for...

A Hamiltonian graph G of order n is k-ordered, 2 ≤ k ≤ n, if for every sequence v1, v2, . . . , vk of k distinct vertices of G, there exists a Hamiltonian cycle that encounters v1, v2, . . . , vk in this order. Define f (k, n) as the smallest integer m for which any graph on n vertices with minimum degree at least m is a k-ordered Hamiltonian graph...

Gagliardi et al. (1996, unpublished manuscript) defined an irregular multigraph to be a loopless multigraph with degree sequence n, n − 1,…, 1, and they posed the problem of determining the number of different irregular multigraphs fn on n vertices. In Gagliardi et al. (1996) they showed that if n ≡ 0 or 3 (mod 4) then fn > n − 1. In this note our...

A Hamiltonian graph G of order n is k-ordered, 2 ≤ k ≤ n, if for every sequence v1, v2, …, vk of k distinct vertices of G, there exists a Hamiltonian cycle that encounters v1, v2, …, vk in this order. Define f(k, n) as the smallest integer m for which any graph on n vertices with minimum degree at least m is a k-ordered Hamiltonian graph. In this a...

A generating,function is developed to express the number,of labeled graphs,with a fixed number,of points and cutpoints in terms of the generating,function of the number,of blocks. An asymptotic,bound,is derived for the number,of connected graphs,with any number,of cutpoints. ACKNOWLEDGEMENT: The referees’ comments,improved,the exposition of

: A generating function is developed to express the number of labeled graphs with a fixed number of points and cutpoints in terms of the generating function of the number of blocks. An asymptotic bound is derived for the number of connected graphs with any number of cutpoints. ACKNOWLEDGEMENT: The referees' comments improved the exposition of this...

The abstract logical data structure for the BANG file directory is a multiway tree structure with one node for each bucket in the file. Under assumptions of “perfect hashing” or “growth on data principle”, we model the growth of the tree. The average cost for search and insertion is found to be logarithmic in the file size. The order constant is sm...

A Hamiltonian graph $G$ of order $n$ is $k$-ordered, $2\leq k \leq n$, if for every sequence $v_1, v_2, \ldots ,v_k$ of $k$ distinct vertices of $G$, there exists a Hamiltonian cycle that encounters $v_1, v_2, \ldots , v_k$ in this order. In this paper, answering a question of Ng and Schultz, we give a sharp bound for the minimum degree guaranteein...

The n-component graph of a graph G is the intersection graph having a point corresponding to each n-component of G and a line joining two points whenever the corresponding n-components of G share at least one point. We give a characterization of graphs which are n-component graphs of some graph, thus generalizing a theorem of Harary (1963).

Caro (1979) and Wei (1981) established a bound on the size of an independent set of a graph as a function of its degrees. In case the degrees of each vertex's neighbors are also known, we establish a lower bound which is tighter for most graphs.

Murphy reviewed lower bounds on the independence number of graphs in terms of degrees, and then he presented a new bound which was at least as strong as the others. We present a new lower bound, as well as a strengthening of Murphy's bound.

Bentley, Weide and Yao (1980) have shown that, subject to certain assumptions, nearest neighbors in k-dimensional space can be computed in O(3k) expected time. This paper presents a new method for finding nearest neighbors in any bounded region. The algorithm employs the Voronoi tessellation to allow nearest neighbor retrievals to be performed in e...

In this paper, we examine the efficiency of the k-d tree for retrieving from a file of fixed-length binary key records the best match to a given input word. We provide guidelines for determining if the search of the tree will provide any savings when compared with an exhaustive search.

A straightforward linear time canonical labeling algorithm is shown to apply to almost all graphs (i.e. all but $o(2^{( \begin{subarray}{l} n \\ 2 \end{subarray} )} )$) of the $2^{( \begin{subarray}{l} n \\ 2 \end{subarray} )} $ graphs on n vertices). Hence, for almost all graphs X, any graph Y can be easily tested for isomorphism to X by an extrem...

A Grundy n-coloring of a finite graph is a coloring of the points of the graph with the non-negative integers smaller than n such that each point is adjacent to some point of each smaller color but to none of the same color. The Grundy number of a graph is the maximum n for which it has a Grundy n-coloring. Characterizations are given of the famili...

Algorithms are introduced which recursively merge neighboring vertices of a graph, conditional upon a test performed on the labels of the vertices and the are between them. A characterization is given of the class of algorithms which produce a unique result, independent of the order in which the graph is scanned. Examples and motivation are drawn f...

The computation of a number of picture properties which involve connectivity and component counting is considered. The computational model consists of a one-dimensional array of finite state automata which scans a digital picture, one row at a time, in one-pass. The inherent complexity of a picture property is reflected in the memory requirements o...

A modeling technique, the Capitalist model, is outlined for analyzing multi-dimensional file structures. It is particularly appropriate where the data keys may have non-uniform data distributions. The basis of the model is to assume that the current distribution of data in an existing file is a good predictor of the underlying "true" distribution....