A Note on Gr\"{u}nbaum's Conjecture about Longest Cycles and Paths

Let $c(G)$ denote the circumference of a graph $G$, i.e., the number of vertices in its longest cycle. For positive integers $n$ and $k$ with $n>k$, let $\varGamma(n;k)$ be the class of graphs of order $n$ with $c(G) = n-k$ such that every induced subgraph of order $n-k$ is Hamiltonian. When $k=$, the class $\varGamma(n; 1)$ coincides with the family of hypohamiltonian graphs-non-Hamiltonian graphs in which the deletion of any single vertex yields a Hamiltonian graph.Replacing Hamiltonian with traceable and $c(G)$ with $p(G)$, the order of a longest path, defines the analogous class $\varPi(n;k)$.Gr\"{u}nbaum (1974) conjectured that both $\varGamma(n; k)$ and $\varPi(n; k)$ are empty for all $n>k \ge 2$. In this note, we first establish upper bounds on the maximum degree of graphs in the classes $\varGamma(n; k)$ and $\varPi(n; k)$. Using these bounds, we show that $\varGamma(n; k)$ is empty when $n<k^2+2k+3$, and that $\varPi(n; k)$ is empty when $n<k^2+2k+2$. These results provide further evidence supporting Gr\"{u}nbaum's conjecture.