Graphs such that every two edges are contained in a shortest cycle

Abstract

A graph G is said to have depth δ if every path of length δ + 1 is contained in a shortest cycle. First we answer by the negative a problem of Neumaier [2], by constructing for every δ, a graph of depth δ which is neither a cycle nor a uniform subdivision of another graph.

Then we characterize the graphs G such that every two edges are contained in a shortest cycle and we show that G is a uniform subdivision of a regular graph of girth 2D, a semi-regular graph of girth 2D or a multigraph on two vertices.