Research Summary and Plans Connectivity of Cages

My research revolves around structural and extremal aspects of Graph Theory, particularly problems involving girth and distance, trees, cycles in graphs, and some variations of Ramsey theory. I have also done some work in generalized graph colorings and graph labellings. My other interests include graph decomposi-tions and packings, perfect graphs, matching theory, hypergraphs and coding theory. Preprints of my papers are available at The girth of a graph G is the length of a shortest cycle in G. The girth is a fundamental parameter of a graph. Most work involving the girth of graphs focuses on two types of problems. In one type of problems, we study structures in graphs whose girth is large. In the other type of problems, we study the relationships between the girth and other parameters of a graph. We consider problems of both types in this section. The distance between two vertices u; v in a graph G is the length of a shortest path connecting u; v in G. Girth and distance are closely related graph parameters. Arguments involving the girth often boil down to the analysis of the distances between pairs in a properly chosen subgraph. Trees in graphs with large girth One elegant simple theorem in graph theory states that every graph with minimum degree at least k contains every tree with k edges. This is best possible, since the complete graph on k vertices has minimum degree k ? 1, and does not contain any tree with k edges as a subgraph. For graphs with large girth, however, the minimum degree requirement can be relaxed. As proved by Brandt and Dobson BD] in 1996, every graph G with girth at least 5 and minimum degree at least k=2 contains every tree with k edges whose maximum degree does not exceed the maximum degree of G. In particular, this result implies the well-known Erd} os-SS os conjecture for graphs with girth at least 5. The Erd} os-SS os conjecture states that every graph of order n with more than (1=2)n(k ?1) edges contains every tree with k edges. When the girth is larger, the minimum degree condition can be further relaxed. Based on their above result, Dobson BD,D] made the general conjecture that every graph G with girth at least 2t + 1 and minimum degree at least k=t contains every tree T with k edges whose maximum degree does not exceed …