Proof of a conjecture of Voss on bridges of longest cycles

Abstract

Bridges are a classical concept in structural graph theory and play a fundamental role in the study of cycles. A conjecture of Voss from 1991 asserts that if disjoint bridges $B_1,B_2,\dots ,B_k$ of a longest cycle $L$ in a $2$-connected graph overlap in a tree-like manner (i.e., induce a tree in the {\it overlap graph} of $L$), then the total {\it length} of these bridges is at most half the length of $L$. Voss established this for $k\le 3$ and used it as a key tool in his 1991 monograph on cycles and bridges. In this paper, we confirm the conjecture in full via a reduction to a cycle covering problem.