Cyclic base orderings in some classes of graphs

A cyclic base ordering of a connected graph G is a cyclic ordering of E(G) such that every |V (G)−1| cyclically consecutive edges form a spanning tree ofG. LetG be a graph with E(G) ̸= ∅ and ω(G) denote the number of components in G. The invariants d(G) and γ(G) are respectively defined as d(G) = |E(G)| |V (G)|−ω(G) and γ(G) = max{d(H)}, where H runs over all subgraphs of G with E(H) ̸= ∅. A graph G is uniformly dense if d(G) = γ(G). Kajitani et al. [8] conjectured in 1988 that a connected graph G has a cyclic base ordering if and only if G is uniformly dense. In this paper, we show that this conjecture holds for some classes of uniformly dense graphs.