HIST-Critical Graphs and Malkevitch's Conjecture

In a given graph, a HIST is a spanning tree without $2$-valent vertices. Motivated by developing a better understanding of HIST-free graphs, i.e. graphs containing no HIST, in this article's first part we study HIST-critical graphs, i.e. HIST-free graphs in which every vertex-deleted subgraph does contain a HIST (e.g. a triangle). We give an almost complete characterisation of the orders for which these graphs exist and present an infinite family of planar examples which are $3$-connected and in which nearly all vertices are $4$-valent. This leads naturally to the second part in which we investigate planar $4$-regular graphs with and without HISTs, motivated by a conjecture of Malkevitch, which we computationally verify up to order $22$. First we enumerate HISTs in antiprisms, whereafter we present planar $4$-regular graphs with and without HISTs, obtained via line graphs. Finally, we confirm Malkevitch's conjecture for the family of line graphs of cyclically $4$-edge connected cubic graphs.