Dirac's Condition for Spanning Halin Subgraphs

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.