The connectivity and Hamiltonian properties of second-order circuit graphs of wheel cycle matroids

Let [Formula: see text] be the [Formula: see text]th-order circuit graph of a simple connected matroid M. The first-order circuit graph is also called a circuit graph. There are lots of results about connectivity and Hamiltonian properties of circuit graph of matroid, while there are few related results on the second-order circuit graph of a matroid. This paper mainly focuses on the connectivity and Hamiltonian properties of the second-order circuit graphs of the cycle matroid of wheels. It determines the minimum degree and connectivity of these graphs, and proves that the second-order circuit graph of the cycle matroid of a wheel is uniformly Hamiltonian.

A Note on Circuit Graphs

We give a short proof of Gao and Richter's theorem that every circuit graph contains a closed walk visiting each vertex once or twice.