Odd cycles and image-cycles in hypergraphs Original Research Article

A image-cycle of a hypergraph is a cycle including an edge that contains at least three base points of the cycle. We show that if a hypergraph image has no image-cycle, and image, for every edge image, then image with equality if and only if H is obtained from a hypertree by doubling its edges. This result reminiscent of Bergeʹs and Lovászʹs similar inequalities implies that 3-uniform hypergraphs with n vertices and n edges have image-cycles, and 3-uniform simple hypergraphs with n vertices and image edges have image-cycles. Both results are sharp. Since the presence of a image-cycle implies the presence of an odd cycle, both results are sharp for odd cycles as well. However, for linear 3-uniform hypergraphs the thresholds are different for image-cycles and for odd cycles. Linear 3-uniform hypergraphs with n vertices and with minimum degree two have image-cycles when image and have odd cycles when image and these are sharp results apart from the values of the constants. Most of our proofs use the concept of edge-critical (minimally 2-connected) graphs introduced by Dirac and by Plummer. In fact, the hypergraph results—in disguise—are extremal results for bipartite graphs that have no cycles with chords