34 problems

Bondy's cubic small cycle double cover conjecture. Every simple $2$-connected cubic graph on $n$ vertices other than $K_4$ has a cycle double cover consisting of at most $n/2$ cycl…

Let $G$ be a bridgeless graph, and let a cycle double cover be a family $\mathcal{F}$ of cycles of $G$ such that each edge of $G$ is contained in exactly two members of…

Let $\operatorname{\mathcal{G}}=(V,E)$ be a bridgeless cubic graph. A cycle double cover of $\operatorname{\mathcal{G}}$ is a set of cycles in which every arc of…

Archdeacon–Jaeger's oriented 5-cycle double cover conjecture. Every bridgeless graph $G$ has an oriented 5-cycle double cover.

Equivalent flow formulation. Every bridgeless cubic graph has a nowhere-zero $\mathbb{Z}_2^3$-flow $f$ such that every component of $(V,E\setminus F)$ contains an even number of ve…

Five-cycle double cover conjecture. Every bridgeless cubic graph has a $5$-CyDC.

Hušek–Šámal's conjecture. The graph $G$ has at least

Szekeres–Seymour conjecture. Every bridgeless cubic graph has a CyDC, equivalently a CiDC.

Graphic matroid 5-cycle double cover conjecture. Every graphic matroid without coloops has a $5$-cycle double cover.

Orientable 5-cycle double cover conjecture. Every bridgeless graph admits an orientable $5$-cycle double cover.

5-cycle double cover conjecture. Every bridgeless graph has a $5$-cycle double cover.

Strong embedding conjecture. Every $2$-connected graph has a strong embedding on a surface.

Let $G$ be a cubic $2$-connected graph on $n$ vertices, with $G\neq K_4$. Let $c(G)$ denote the minimum size of a cycle double cover of $G$, with $c(G)=\infty$ if no cycle double c…

Let $G$ be a $2$-connected graph on $n$ vertices with minimum degree at least $3$. A cycle double cover (CDC) is a collection of cycles containing every edge of $G$ exactly twice.…

Let $G$ be a bridgeless cubic graph on $n$ vertices. A cycle double cover (CDC) is a collection of cycles containing every edge of $G$ exactly twice. Hušek and Šámal's CDC-count co…

Let $G$ be a simple $2$-connected cubic graph on $n$ vertices, with $G\neq K_4$. A cycle double cover (CDC) is a collection of cycles containing every edge of $G$ exactly twice. Bo…

Let $G$ be a bridgeless connected graph and let $\Gamma$ be a signed ribbon diagram of $G$. For $n$ a positive integer, write $T(\Gamma,n,t)$ for the total face color polynomial. T…

A cycle double cover of a graph $G$ is a collection of cycles such that each edge of $G$ is contained in exactly two cycles. It is famously conjectured that every biconnected graph…

Let $G$ be a bridgeless cubic graph. A $5$-cycle double cover is a multiset of five cycles in $G$ such that every edge $e\in E(G)$ belongs to exactly two members of the multiset. 5…

Let $G$ be a bridgeless cubic graph. A cycle double cover is a collection of cycles covering every edge exactly twice, and a $2$-factor is a spanning $2$-regular subgraph. Let…

A bridgeless graph is a graph with no bridge, that is, no edge whose deletion disconnects the graph. A cycle double cover is a list of cycles in which every edge appears exactly tw…

Let $G$ be a cyclically $4$-edge connected cubic graph. For a cycle $C$ of $G$, let $G/E(C)$ denote the graph obtained by contracting the edges of $C$, and let a nowhere-zero $4$-f…

Let $G$ be a $2$-edge connected cubic graph with a decomposition into edge-disjoint subgraphs covering $E(G)$, consisting of a tree $T$ and a cycle $C$. Tree–cycle decomposition cy…

Let $G$ be a $2$-edge connected cubic graph. A cycle $C$ of $G$ is non-separating if $G-E(C)$ is connected. Non-separating cycle double cover conjecture. Every non-separating cycle…

Let $G$ be a bridgeless graph and let $C$ be a circuit of $G$. A strong cycle double cover of $G$ containing $C$ is a cycle double cover ${\cal S}$ with $C\in{\cal S}$. Strong cycl…