5 problems

Let $G$ be an essentially $4$-connected planar graph on $n$ vertices, and let $\operatorname{circ}(G)$ denote the length of a longest cycle in $G$. Circumference conjecture. Every…

Let $M$ be a connected matroid with rank function $r$. Its circumference is the maximum size of a circuit of $M$, when $M$ has at least one circuit, and its pathwidth is denoted by…

A graph's circumference is the maximum number of vertices in a cycle of the graph. A graph is vertex-transitive if, for every pair of vertices $u,v$, there is an automorphism mappi…

Sharp circumference bounds. The following inequalities hold:

Let $G$ be a graph of order $n$ with minimum degree … Write $n=3t+r$ with $r\in\{0,1,2\}$, and let … be a $2$-edge colouring. The exact monochromatic circumference conjecture. The…