This article gives a brief introduction to the On-Line Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs to generate the sequence, references, links to relevant web pages, and other

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A digraph D is called a quasi-transitive digraph (QTD) if for any triple x; y; z of distinct vertices of D such that (x; y) and (y; z) are arcs of D there is at least one arc from x to z or from z to x. Solving a conjecture by J. Bang-Jensen and J. Huang (J. Graph Theory, to appear), G. Gutin (Australas. J. Combin., to appear) described polynomial algorithms for finding a Hamiltonian cycle and a Hamiltonian path (if it exists) in a QTD. The approach taken in that paper cannot be used to find a longest path or cycle in polynomial time. We present a principally new approach that leads to polynomial algorithms for finding vertex heaviest paths and cycles in QTD's with non-negative weights on the vertices. This, in particular, provides an answer to a question by N. Alon on longest paths and cycles in QTD's. 1 Introduction A digraph D is called quasi-transitive if for any triple x; y; z of distinct vertices of D such that (x; y) and (y; z) are arcs of D there is at least one arc from x to ...

Zemlyachenko's linear time algorithm for free tree isomorphism is unique in that it also partitions the set of rooted subtrees of a given rooted tree into isomorphism equivalence classes. Unfortunately, his algorithm is very hard to follow. In this note, we use modern data structures to explain and implement Zemlyachenko's scheme. We give a full description of a free rendition of his method using some of his ideas and adding some new ones; in particular, the data structures are new. email: fdititz, itai, rodehg@cs.technion.ac.il 1 1 Introduction Free tree isomorphism is a well known problem which has many applications. A linear time algorithm to solve the problem has been developed by Hopcroft and Tarjan [2], who used it to devise a linear planarity test. Several other linear algorithms were suggested [1, pp.196-199], [3], [4]. The common approach is as follows. First, free tree isomorphism is reduced to the rooted case. The root is set canonically at the central vertex or at the ...

Given a digraph (an undirected graph, resp.) D and two positive integers f(x); g(x) for every x 2 V (D), a subgraph H of D is called a (g; f)-factor if g(x) d + H (x) = d \Gamma H (x) f(x)(g(x) dH (x) f(x), resp.) for every x 2 V (D). If f(x) = g(x) = 1 for every x, then a connected (g; f)-factor is a hamiltonian cycle. The previous research related to the topic has been carried out either for (g; f)-factors (in general, disconnected) or for hamiltonian cycles separately, even though numerous similarities between them have been recently detected. Here we consider connected (g; f)-factors in digraphs and show that several results on hamiltonian digraphs, which are generalizations of tournaments, can be extended to connected (g; f)-factors. Applications of these results to supereulerian digraphs are also obtained. 1 Introduction and terminology Given a digraph (an undirected graph, resp.) D and two positive integers f(x); g(x) for every x 2 V (D), a subgraph H of D is called ...