Hamiltonian graphs with prescribed minimum degree and no near-spanning cycles

Abstract

In 1984, Roland H\"{a}ggkvist posed the problem of constructing Hamiltonian graphs of order $n$ with large minimum degree and no $(n-2)$-cycle. He remarked that he did not know of such a graph with minimum degree at least three. We solve this problem by proving the following two results. (1) For every integer $d\ge 3$ and every integer $n\ge 15d-14,$ there exists a Hamiltonian graph of order $n$ and minimum degree $d$ that contains no $(n-2)$-cycle. (2) For every integer $d\ge 3,$ every positive integer $k,$ and every integer $n\ge (k+1)[(d-1)(k+3)+1],$ there exists a Hamiltonian graph of order $n$ and minimum degree $d$ that contains no $(n-s)$-cycle for any $s\in\{1,2,\dots,k\}.$ The proofs are constructive. We also pose several open problems.