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

TL;DR

This paper investigates Gr"{u}nbaum's conjecture about the non-existence of certain graphs with long cycles or paths, providing bounds that support the conjecture's validity.

Contribution

The authors establish upper bounds on the maximum degree of graphs in specific classes and prove the classes are empty under certain size conditions, supporting Gr"{u}nbaum's conjecture.

Findings

Graphs in classes mma(n;k) and ta(n;k) are empty below certain size thresholds.

Upper bounds on maximum degree are derived for these graph classes.

Results support the conjecture that such graphs do not exist for large enough n.

Abstract

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.