A randomized embedding algorithm for trees

In this paper, we propose a simple and natural randomized algorithm to embed a tree T in a given graph G. The algorithm can be viewed as a "self-avoiding tree-indexed random walk". The order of the tree T can be as large as a constant fraction of the order of the graph G, and the maximum degree of T can be close to the minimum degree of G. We show that our algorithm works in a variety of interesting settings. For example, we prove that any graph of minimum degree d without 4-cycles contains every tree of order d2 and maximum degree at most d - 2 d - 2. As there exist d-regular graphs without 4-cycles and with O(d2 ) vertices, this result is optimal up to constant factors. We prove similar nearly tight results for graphs of given girth and graphs with no complete bipartite subgraph Ks,t.

• Type: Journal
• Year: 2010
• Where: COMBINATORICA
• Authors: Benny Sudakov, Jan Vondrák
• Keywords: COMBINATORICA 2010, Machine Learning, Maximum Degree, Minimum Degree, Natural Randomized Algorithm