Graphs without $K_4$-minors

Abstract

In [Discrete Math.75(1989)69-99], Bondy conjectured that if $G$ is a 2-edge-connected simple graph with $n$ vertices, then $G$ admits a double cycle cover with at most $n–1$ cycles. In this note, we prove this conjecture for graphs without subdivision of $K_4$ and characterize all the extremal graphs.