Subdivisions of maximal 3-degenerate graphs of order $d+1$ in graphs of minimum degree $d$

Abstract

We prove that every graph of minimum degree at least $d\ge 1$ contains a subdivision of some maximal 3-degenerate graph of order $d+1$. This generalizes the classic results of Dirac ($d=3$) and Pelikán ($d=4$). We conjecture that for any planar maximal 3-degenerate graph $H$ of order $d+1$ and any graph $G$ of minimum degree at least $d$, $G$ contains a subdivision of $H$. We verify this in the case $H$ is $P_6^3$ and $P_7^3$