Elsevier

Discrete Applied Mathematics

Volume 284, 30 September 2020, Pages 622-625
Discrete Applied Mathematics

Note
Non-hamiltonian 1-tough triangulations with disjoint separating triangles

https://doi.org/10.1016/j.dam.2020.03.053Get rights and content

Abstract

In this note, we consider triangulations of the plane. Ozeki and the second author asked whether there are non-hamiltonian 1-tough triangulations in which every two separating triangles are disjoint. We answer this question in the affirmative and strengthen a result of Nishizeki by proving that there are infinitely many non-hamiltonian 1-tough triangulations with pairwise disjoint separating triangles.

Introduction

In this note, a triangulation shall be a plane 3-connected graph in which every face is a triangle. (Triangulations are also known as maximal planar graphs, since the addition of any edge renders the graph non-planar.) For a possibly disconnected graph G, denote by c(G) the number of connected components of G. In a triangulation G, a triangle T is said to be separating if c(GT)>1. For triangles T and T in G the distance between T and T shall be the number of edges of a shortest path in G between vV(T) and vV(T) for all possible combinations of v and v.

Answering a question of Böhme, Harant, and Tkáč [2], Böhme and Harant [1] showed that for any non-negative integer d there exists a non-hamiltonian triangulation with seven separating triangles every two of which lie at distance at least d. Ozeki and the second author [8] proved that the result holds even if we replace ‘seven’ by ‘six’. We note that no non-hamiltonian triangulation with fewer than six separating triangles is known, while Jackson and Yu [6] showed that every triangulation with at most three separating triangles is hamiltonian. (It was recently proven that this result’s generalisation to polyhedral graphs—where 3-vertex-cuts replace separating triangles—is valid, as well [3].)

Chvátal [4] introduced the toughness of a non-complete graph G as t(G)=min|X|c(GX):XV(G),c(GX)>1.The toughness of a complete graph is convened to be . A graph G is t-tough whenever tt(G). Chvátal observed that every hamiltonian graph is 1-tough [4]. In 1979 he raised the question whether l-toughness is a sufficient condition for a triangulation to be hamiltonian, and Nishizeki settled this by proving that there is a non-hamiltonian 1-tough triangulation [7]. (Dillencourt [5] showed that there exists a smaller such triangulation, namely one of order 15, and Tkáč [10] proved that there exists such a triangulation of order 13, and no smaller one. Tkáč’s triangulation contains seven separating triangles.)

Recently, Ozeki and the second author asked whether there are non-hamiltonian 1-tough triangulations in which every two separating triangles are disjoint, see [8, Remark (a)]. We now answer this question in the affirmative and strengthen Nishizeki’s result.

Section snippets

Result

Theorem

There exist infinitely many non-hamiltonian 1-tough triangulations with pairwise disjoint separating triangles.

For the proof of this theorem we will use the following lemma.

Lemma

Nishizeki [7]

Let G be a graph and SV(G). If for a vertex v in G, the graph Gv is 1-tough, and if c(GS)>|S|, then v does not belong to S but all of its neighbours do.

Proof of the Theorem

In the first part of the proof, we construct a triangulation G with the desired properties, and in the second part we present an infinite family. Consider the circular

Acknowledgements

Fujisawa’s work is supported by the Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B) 16H03952 and Grant-in-Aid for Scientific Research (C) 17K05349. Zamfirescu’s research is supported by a Postdoctoral Fellowship of the Research Foundation Flanders (FWO) .

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