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Xing-Zhi Zhan

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Preprint Aug 2026

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

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 $d\ge 3$...

Xing-Zhi Zhan · 0 citations

Preprint Aug 2026

A proof of Bickle's conjecture on collapsible graphs

A graph $G$ is said to be $k$-collapsible if $G$ has minimum degree $k$ and every non-null proper induced subgraph of $G$ has minimum degree less than $k.$ In 2018, Bickle conjectured that the minimum number of vertices of degree $k$ in a $k$-collapsible graph of order $n$ with $k\ge 3$ is ${\rm max}\{\lceil 2n/(2k-1)\...

Xing-Zhi Zhan · 0 citations