The maximum number of paths of a given length in a nonhamiltonian graph

Abstract

In 1980, Paul Erdős posed the following problem: For every positive integer $n,$ determine a nonhamiltonian graph of order $n$ having the maximum number of Hamilton paths. We solve the more general problem of determining the nonhamiltonian graphs of order $n$ having the maximum number of paths of length $k$ for given integers $n$ and $k$ with $1\le k\le n-1.$ The case $k=n-1$ gives a solution to Erdős's problem and the case $k=1$ corresponds to a theorem due to Ore and Bondy.