Dirac's Condition for Spanning Halin Subgraphs

TL;DR

This paper extends Dirac's classical Hamiltonian graph theorem to show that sufficiently large graphs with high minimum degree contain spanning Halin subgraphs, which are also pancyclic, highlighting rich Hamiltonian properties.

Contribution

It proves that for large enough graphs with minimum degree at least (n+1)/2, a spanning Halin subgraph exists, generalizing Dirac's theorem.

Findings

Graphs with minimum degree ≥ (n+1)/2 contain spanning Halin subgraphs.

Such spanning Halin subgraphs are also pancyclic.

The result applies to sufficiently large graphs (n ≥ n0).

Abstract

Let $G$ be an $n$-vertex graph with $n\ge 3$. A classic result of Dirac from 1952 asserts that $G$ is hamiltonian if $\delta(G)\ge n/2$. Dirac's theorem is one of the most influential results in the study of hamiltonicity and by now there are many related known results\,(see, e.g., J. A. Bondy, Basic Graph Theory: Paths and Circuits, Chapter 1 in: {\it Handbook of Combinatorics Vol.1}). A {\it Halin graph} is a planar graph consisting of two edge-disjoint subgraphs: a spanning tree of at least 4 vertices and with no vertex of degree 2, and a cycle induced on the set of the leaves of the spanning tree. Halin graphs possess rich hamiltonicity properties such as being hamiltonian, hamiltonian connected, and almost pancyclic. As a continuous "generalization" of Dirac's theorem, in this paper, we show that there exists a positive integer $n_0$ such that any graph $G$ with $n\ge n_0$ vertices and $\delta(G)\ge (n+1)/2$ contains a spanning Halin subgraph. In particular, it contains a spanning Halin subgraph which is also pancyclic.