A Note on Alon–Tarsi Shortest Cycle Cover Conjecture

Abstract

The shortest cycle cover conjecture (SCC conjecture), proposed by Alon and Tarsi, asserts that every bridgeless cubic graph $G$ has a cycle cover with a total length at most $\frac{21}{15}\mid E\left(G\right)\mid$. Tarsi further proposed a related conjecture, the ${\mathrm{SCC}}_3$ conjecture, which states that every bridgeless cubic graph $G$ has a 3-cycle cover with a total length at most $\frac{22}{15}\mid E\left(G\right)\mid$. In this paper, we prove that every cyclically odd-$\left(4k+1\right)$-edge-connected cubic graph $G$ has a 3-cycle cover with a total length at most $\frac{20}{15}\mid E\left(G\right)\mid +\frac{1}{2k+1}\mid E\left(G\right)\mid$. Consequently, the SCC conjecture and the ${\mathrm{SCC}}_3$ conjecture are verified in this paper for cubic graphs with cyclic odd-edge-connectivity at least 29 and 17, respectively. Additionally, for cubic graphs that satisfies Kaiser–Raspaud conjecture (i.e., every bridgeless cubic graph has two perfect matchings $M_1$ and $M_2$ and a parity subgraph $J$, such that $M_1\cap M_2\cap J=\varnothing$), the SCC conjecture and the ${\mathrm{SCC}}_3$ conjecture are verified for graphs with girth at least 20 and 10, respectively.