Long cycles passing through a specified path in a graph

Abstract

For a graph G and an integer k $#8805; 1, let ζk(G) = $min \{\sum ^{k}_{i=1}$ dG(vi): {v1, …, vk} is an independent set of vertices in G}. Enomoto proved the following theorem. Let s $#8805; 1 and let G be a (s + 2)-connected graph. Then G has a cycle of length $#8805; min{|V(G)|, ζ2(G) - s} passing through any path of length s. We generalize this result as follows. Let k $#8805; 3 and s $#8805; 1 and let G be a (k + s - 1)-connected graph. Then G has a cycle of length $#8805; min{|V(G)|, $\frac{2}{k}\sigma_{k}(G)$ - s} passing through any path of length s. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 177184, 1998