99 problems

Every finite graph $G=(V,E)$ that is $2$-tough is fractionally Hamiltonian. Here, $G$ is $2$-tough if, for every vertex set $S\subseteq V$ with $c(G-S)>1$, one has $|S|\ge 2c(G-S)$…

For every sufficiently large integer $d$ and every sufficiently large $n$, every $n$-vertex $d$-regular sublinear expander contains a Hamiltonian cycle.

Lovász's conjecture. Every connected vertex-transitive graph has a Hamilton path.

Seymour's conjecture. For positive integers $n$ and $k$ with $n\geq k+1$ and $k\geq2$, if

Let $G=\langle S\rangle$ be a two-ended group, and suppose that its commutator subgroup satisfies … The Hamiltonian circle conjecture. The Cayley graph … has a hamiltonian circle.…

Krivelevich–Sudakov conjecture. There exists $C>0$ such that if

Let a weakly $(n,\beta)$-graph be a graph in the sense defined in the source, and let $G$ be Hamiltonian when it contains a Hamilton cycle. Hamiltonicity conjecture for weakly…

Hamilton-circle conjecture. The graph $G$ has a Hamilton circle if every ball of radius $2$ in $G$ is 2-connected and

Kühn–Osthus conjecture. The digraph $G$ contains a Hamilton cycle. The source introduces this as the digraph analogue of the $t=2$ case of the Bollobás–Häggkvist conjecture; no res…

Let $G$ be a $3$-regular, $3$-connected planar graph. Tait's conjecture. Every such graph is Hamiltonian. The conjecture was disproved by W. T. Tutte, who constructed a counterexam…

Let $G$ be a graph. It is claw-free if it has no induced subgraph isomorphic to a claw, and $G$ is 4-connected if deleting fewer than four vertices does not disconnect it. Matthews…

Let $G$) be a 3-connected non-Hamiltonian graph. Let $K_{3,4}$ be the complete bipartite graph, let $\mathfrak{Q}^+$ be obtained from the cube by adding a new vertex adjacent to th…

Hoàng's conjecture. Let $n\ge 3$ and $t\ge 1$ be integers. If $G$ is $t$-tough and, for every $i<\frac n2$, $d_i\le i$ implies $d_{n-i+t}\ge n-i$, then $G$ is Hamiltonian.

Let $G$ be a Dirac graph on $n$ vertices, meaning a graph with minimum degree at least $n/2$, and let its edge-colouring be proper if every pair of incident edges receives differen…

Let $k$ be an integer with $k\ge 4$, and let $G$ be a graph. Here $P_n$ denotes the path on $n$ vertices, $P_2\cup kP_1$ is the disjoint union of a two-vertex path and $k$ isolated…

Let $G$ be the graph under consideration, with Laplacian eigenvalues $\mu_2$ and $\mu_n$, and say that $G$ is Hamiltonian when it contains a Hamilton cycle. Gu's conjecture. There…

Let $G=\langle S\rangle$ be a one-ended generalized quasi-dihedral group, and let $\operatorname{Cay}(G,S)$ denote its Cayley graph. A Hamiltonian double ray is a spanning double r…

Let $G$ be a $3$-regular, $3$-connected bipartite graph. Tutte's conjecture. Every such graph is Hamiltonian. The supplied text introduces this conjecture after noting that the kno…

Hendry's conjecture. Every Hamiltonian chordal graph is fully cycle extendible.

Let $K_n$ be the complete graph on $n$ vertices. An edge-colouring is $k$-bounded if no colour appears on more than $k$ edges. A subgraph is rainbow if no two of its edges have the…

Let $G$ be a graph, and call the connected graph with degree sequence $3,3,3,1,1,1$ a net, with its vertices of degree $1$ called endvertices. Call $G$ claw-free if it has no induc…

Nash-Williams's conjecture. Suppose that $G$ is a graph on $n$ vertices with minimum degree

Sheehan's conjecture. There is no finite $r$-regular graph with a unique Hamilton cycle for any $r>2$.

Let $G$ be a hamiltonian loopless graph with $n$ vertices, and let $P(G,t)$ denote its chromatic polynomial. Thomassen's conjecture. … The conjecture extends the known zero-free in…

Bedert–Drăganić–Müyesser–Pavez-Signé conjecture. There is an absolute constant $C$ such that, for every connected Cayley graph $G$ of order $n$ and degree $d$, the condition