7 problems

A signed graph is coverable if it admits a signed circuit cover. A signed circuit $k$-cover is a family of signed circuits in which every edge is covered exactly $k$ times. Fan's c…

Fan's weighted refinement. If there is a $\mathbb{Z}_2\times\mathbb{Z}_2$-flow $f$ such that

Let $G$ be a $2$-edge-connected graph, and let $scc(G)$ denote the minimum length of a circuit cover of $G$, where the length of a circuit cover is the sum of the lengths of its ci…

6-cover conjecture. Every flow-admissible signed graph has a family of signed circuits such that every edge belongs to exactly $6$ members of the family.

Let $G$ be a 2-edge-connected graph, and let a circuit double cover be a family of circuits in which every edge of $G$ is covered twice. Circuit Double Cover Conjecture. Every 2-ed…

Let $G$ be a 2-edge-connected cubic graph. A shortest circuit cover of $G$ is a circuit cover having minimum total length, denoted by $\operatorname{scc}(G)$. Alon and Tarsi's conj…

Let $G$ be a bridgeless graph, and let a circuit cover be a collection of circuits in which every edge of $G$ belongs to at least one circuit. Its length is the sum of the lengths…