🤖 AI Summary
This work addresses Malkevitch’s conjecture that every planar 4-regular graph contains a homeomorphically irreducible spanning tree (HIST), i.e., a spanning tree with no vertices of degree two.
Method: We formally define and characterize HIST-critical graphs—graphs that are HIST-free but become HIST-containing upon deletion of any vertex—and systematically construct infinite families of planar 3-connected 4-regular HIST-critical graphs. We further employ computer-assisted enumeration and structural analysis to verify the conjecture exhaustively for all planar 4-regular graphs on at most 22 vertices.
Contribution/Results: Our construction refutes Malkevitch’s conjecture in general, while the exhaustive verification confirms its validity for small-order instances. This resolves two long-standing gaps: the first systematic characterization of critical structures for HIST existence, and the first complete computational validation for graphs up to order 22. The results establish a new framework for studying structural properties of spanning trees in combinatorial graph theory.
📝 Abstract
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.