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Báo cáo toán học: "On the number of independent sets in a tree"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: On the number of independent sets in a tree. | On the number of independent sets in a tree Hiu-Fai Law Mathematical Institute Oxford University OX1 3LB U.K. lawh@maths.ox.ac.uk Submitted Feb 9 2010 Accepted Mar 15 2010 Published Mar 22 2010 Mathematics Subject Classifications 05C69 05C05 Abstract We show in a simple way that for any k m E N there exists a tree T such that the number of independent sets of T is congruent to k modulo m. This resolves a conjecture of Wagner Almost all trees have an even number of independent sets Electron. J. Combin. 16 2009 R93 . 1 The number of independent sets in a tree A set of vertices in a graph G is called independent if the set induces no edges. We write i G for the number of independent sets in G i G is often known as the Fibonacci number or in mathematical chemistry as the Merrifield-Simmons index or the a-index. The study was initiated by Prodinger and Tichy in 4 . In particular they showed that among trees of the same order the maximum and minimum Fibonacci numbers are attained by the star and the path respectively. The name stems from the fact that the Fibonacci numbers of paths are the usual Fibonacci numbers. Indeed as the empty set is independent i P0 1 i P1 2 and i Pn i Pn-1 i Pn-2 for n 2. The inverse question asks for a positive integer k whether there exists a graph G such that i G k. Clearly there does as i Kk-1 k note that the empty set is independent . The question becomes more interesting if we restrict ourselves to certain classes of graphs. For the class of bipartite graphs Linek 3 answered the question affirmatively. Here we are interested in the class of trees. For k E N we say that k is constructible if there exists a tree T such that i T k. For example 1 2 3 are constructible from the paths P0 P1 P2 respectively but 4 is not. In 3 Linek raised the following conjecture see also 2 . Conjecture 1 3 . There are only finitely many positive integers that are not con-structible. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 N18 1 An interesting paper of .