11 problems

Let $G$ be an $r$-connected graph with $r\ge 2$. A longest cycle is a cycle in $G$ of maximum length. Smith's conjecture. Every pair of longest cycles in $G$ intersects in at least…

Let $G$ be a $k$-connected graph, let $\delta(G)$ be its minimum degree, and let $r$ be an integer satisfying $\delta(G)\geq r\geq 2k$. For an integer $\ell\geq 1$, a cycle $C$ in…

Let $G$ be a $k$-connected graph, let $C$ be a longest cycle of $G$, and let $\delta$ denote the minimum degree of $G$. Suppose that $G-V(C)$ contains a path with at least $k-1$ ve…

Let $n$ and $k$ be positive integers with $n>k$, and let $\Gamma(n;k)$ be the class of graphs of order $n$ whose circumference is $n-k$ and every induced subgraph of order $n-k$ is…

Scott Smith's conjecture. The cycles $C_1$ and $C_2$ intersect in at least $k$ vertices:

Voss's conjecture.

Linear-forest chord conjecture. Every longest cycle of $G$ passing through $F$ has a chord.

Gu–Jia–Wu conjecture. Every longest cycle containing $e$ has a chord.

A triangle-free graph is a graph containing no cycle of length three. Let $n$ and $g$ be integers with $n,g\geq 4$. Zamfirescu's question. Do there exist 3-regular, 2-connected, tr…

Let $G$ be a $k$-connected graph with $k\geq 2$. A longest cycle is a cycle of maximum length in $G$. Smith's conjecture. Every pair of longest cycles in $G$ intersect in at least…

Erdős's conjecture. If $c>1$, then w.h.p. [the statement is incomplete in the supplied excerpt].