22 problems

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.

Bondy's small cycle double cover conjecture. Every simple $2$-edge-connected graph on $n$ vertices has a cycle double cover consisting of at most $n-1$ cycles.

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

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

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…

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…

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…

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 a Hist-snark be a snark with a Hist, where a Hist is a spanning tree having only vertices of degree three and one, and the outer cycles are the vertex-disjoint cycles induced b…

Amiable-coloring conjecture. If $R$ is a $3$-row graph such that $R_C$ is eulerian, then $R$ admits an amiable coloring.

Kotzig-frame existence conjecture. Every cyclically $4$-edge connected cubic graph has a Kotzig-frame.

Connected 3-edge-colorable spanning-minor conjecture. If a cubic bridgeless graph contains a connected 3-edge-colorable cubic graph as a spanning minor, then $G$ has a $6$-even-sub…

Semi-Kotzig-frame conjecture. Every bridgeless cubic graph with a semi-Kotzig frame has a $6$-even-subgraph double cover.

Häggkvist–Markström conjecture. Every $3$-connected cubic graph contains a Kotzig graph as a spanning minor.