Sparse Kneser graphs are Hamiltonian

For integers $k\geq 1$ and $n\geq 2k+1$, the Kneser graph $K(n,k)$ is the graph whose vertices are the $k$-element subsets of $\{1,\ldots,n\}$ and whose edges connect pairs of subsets that are disjoint. The Kneser graphs of the form $K(2k+1,k)$ are also known as the odd graphs. We settle an old problem due to Meredith, Lloyd, and Biggs from the 1970s, proving that for every $k\geq 3$, the odd graph $K(2k+1,k)$ has a Hamilton cycle. This and a known conditional result due to Johnson imply that all Kneser graphs of the form $K(2k+2^a,k)$ with $k\geq 3$ and $a\geq 0$ have a Hamilton cycle. We also prove that $K(2k+1,k)$ has at least $2^{2^{k-6}}$ distinct Hamilton cycles for $k\geq 6$. Our proofs are based on a reduction of the Hamiltonicity problem in the odd graph to the problem of finding a spanning tree in a suitably defined hypergraph on Dyck words

Determining The Word Processing Software to Meet the Needs of The Business User

Computerized word processing with computer software packages is common throughout industry today. However, selection of the appropriate software is a complicated procedure and requires a careful examination and comparison of the complexity of the software (features available) with the ease of learning to use the software. With the wide variety of products that run on IBM PC\u27s and the IBM compatibles, it is possible that the business manager can find one that adequately fills both needs. When selecting an appropriate word processing software package, one must first determine how the product will be used so that the purchaser can weigh the advantages and disadvantages of easy versus powerful packages and come up with an adequate compromise. Some of the easier-to-use programs can make great learning tools, especially for computer novices. They also work nicely if the major word processing application is the production of short letters and memos. However, if the user is expected to do a great deal of tabular typing or produce hundreds of form letters or many long reports, the business might be better off sacrificing some of those user-friendly features in favor of powerful capabilities that will make these tasks easier. On long reports, for example, production goes faster with software that automatically writes tables of contents and indexes, and numbers, chapters and lists. The decisions involved in selecting an appropriate word processing software package often revolve around how long a learning curve the business can_ afford and which sophisticated features are needed. Some of the more powerful packages require extensive training and practice before users can use then competently and with efficiency. The friendlier packages save learning time, but in the long run might cost time because they do not automate many production tasks. Another ease-of-use feature is the quality and organization of the software instruction manuals. Some manuals explain in detail how to use the program; others just list the features of the program. Most of the documentation included with software packages could use improvement. For many buyers, availability of features ranks higher than ease-of-use. In this study the word processing g software packages were evaluated, and a comparison was done to determine which features were available from a list of more than three hundred features, reporting and foreign language ability. including scientific No word processing software package does everything, and those. that do many things often have a variety of weaknesses. Since packages with more features are more complicated to use, the purchaser is wise to buy software which contains only the features the users will needs. In addition, it is always wise to insist on a demonstration of the needed features before purchasing. Balancing ease-of-use and power of word processing software packages is a continuing process. This study\u27 s goal is to make the selection of computer programs for individuals and business firms more efficient because they can now see an evaluation and comparison. Of the different programs available