65 problems

Let $P$) be the Petersen graph, and let $\phi_2(P)$ denote its $2$-dimensional flow number, the infimum of the real numbers $r$ for which $P$ has a $(r,2)$-dimensional nowhere-zero…

Tutte's conjecture. Every nonplanar snark $G$ has the Petersen graph as a minor.

Jaeger–Swart conjecture. There are no cyclically $7$-edge-connected snarks.

Fleischner's dominating circuit conjecture. Every cyclically $4$-edge-connected snark has a dominating circuit.

Snark order-to-oddness ratio conjecture. The order of $G$ satisfies

Let $G$ be a cyclically $5$-edge-connected odd $2$-factored snark, meaning that $G$ is a snark, every cycle in every $2$-factor of $G$ is odd, and no edge cut of size at most four…

The 7/5-conjecture. Every bridgeless graph has a cycle cover of length at most

A permutation snark is a cycle permutation graph that is not $3$-edge-colourable; a graph is cyclically $5$-edge-connected if removing fewer than five edges cannot separate it into…

Petersen Coloring Conjecture.

Let $G$ be a hypohamiltonian snark, namely a snark such that deleting any vertex leaves a Hamiltonian graph, and let defect denote colouring defect. Steffen's conjecture. Every hyp…

Strong-snark conjecture. If the perfect matching index of $G$ is greater than $4$ and $G$ is not the Petersen graph, then $G$ is a strong snark.

Every-cubic-graph 3-edge-coloring conjecture. Every cubic graph is 3-edge-colorable.

Strong-snark conjecture. With the sole exception of the Petersen graph, every nontrivial snark with perfect matching index at least $5$ is strong; equivalently, all its edges are s…

Let $G$ be a cyclically $5$-edge-connected odd $2$-factored snark, where a snark is a bridgeless cubic graph of chromatic index four and odd 2-factored means that every cycle in ev…

Fiol et al.'s conjecture.

Jaeger's conjecture. Snarks with cyclic connectivity or girth greater than $6$ do not exist.

Let $G$ be a generalized Blanuša snark, and let $i_{[kR]}(G)$ denote its independent $[k]$-Roman domination number and $i(G)$ its independent domination number. Independent [k]-Rom…

A snark is a cyclically $4$-edge-connected cubic graph of girth at least $5$ that is not 3-edge-colorable. Mohar's projective-plane snark conjecture. The only snark embeddable in t…

Let $G$ be a cubic graph, meaning that every vertex has degree $3$, and let its girth be the length of its shortest cycle. A cubic graph is Type 2 when its total chromatic number i…

Let $G$ be a snark. Define the resistance $r(G)$ to be the minimum number of edges that can be removed from $G$ so that the resulting graph is $3$-edge-colorable, and define…

Let $G$ be a snark, let $F$ be a $2$-factor of $G$, and let $M$ be a matching in $F$ such that there exists an $F$-matching $\mathcal{P}$ in $H=G-M$. Let $B$ be the corresponding l…

Let $G$ be a bridgeless cubic graph, let $F$ be a $2$-factor of $G$, and let $M$ be a matching in $F$. Put $H=G-M$. An $F$-matching is the matching notion defined in the paper, and…

Let $G$ be a bridgeless cubic graph. A proper $A$-coloring is a proper edge-coloring of $G$ by the non-zero elements of an abelian group $A$ such that the sum of the colors on the…

Poor-edge conjecture. If $G\not=P_{10}$, then $G$ has a normal $5$-edge-coloring with at least one poor edge. Moreover, if additionally $G\not=P_{10}^{\Delta}$, then $G$ has a norm…

Let an irreducible snark be a snark for which deleting any pair of distinct vertices produces a $3$-edge-colourable graph, and let the girth of a graph be the length of its shortes…