Cyclic base ordering of certain degenerate graphs

Abstract

Let $G$ be a connected graph. A cyclic base ordering of $G$ is a cyclic ordering of elements in $E\left(G\right)$ such that every cyclically consecutive $\left|V\left(G\right)\right|-1$ edges form a spanning tree of $G$. The density of $G$ is defined to be $d\left(G\right)=\left|E\left(G\right)\right|/(\left|V\left(G\right)\right|-1)$; and $G$ is uniformly dense if $d\left(H\right)\le d\left(G\right)$ for every connected subgraph $H$ of $G$. It was conjectured by Kajitani, Ueno and Miyano that $G$ has a cyclic base ordering if and only if $G$ is uniformly dense. We show that the conjecture holds for maximal 2-degenerate graphs and graphs with uniform ear decompositions. As applications, book graphs, broken fan and broken wheel graphs have cyclic base ordering. We also study cyclic base ordering of double wheel graphs and the square of cycles.