Hamiltonian cycles in maximal planar graphs and planar triangulations

In this thesis we study planar graphs, in particular, maximal planar graphs and general planar triangulations. In Chapter 1 we present the terminology and notations that will be used throughout the thesis and review some elementary results on graphs that we shall need. In Chapter 2 we study the fundamentals of planarity, since it is the cornerstone of this thesis. We begin with the famous Euler's Formula which will be used in many of our results. Then we discuss another famous theorem in graph theory, the Four Colour Theorem. Lastly, we discuss Kuratowski's Theorem, which gives a characterization of planar graphs. In Chapter 3 we discuss general properties of a maximal planar graph, G particularly concerning connectivity. First we discuss maximal planar graphs with minimum degree i, for i = 3; 4; 5, and the subgraph induced by the vertices of G with the same degree. Finally we discuss the connectivity of G, a maximal planar graph with minimum degree i. Chapter 4 will be devoted to Hamiltonian cycles in maximal planar graphs. We discuss the existence of Hamiltonian cycles in maximal planar graphs. Whitney proved that any maximal planar graph without a separating triangle is Hamiltonian, where a separating triangle is a triangle such that its removal disconnects the graph. Chen then extended Whitney's results and allowed for one separating triangle and showed that the graph is still Hamiltonian. Helden also extended Chen's result and allowed for two separating triangles and showed that the graph is still Hamiltonian. G. Helden and O. Vieten went further and allowed for three separating triangles and showed that the graph is still Hamiltonian. In the second section we discuss the question by Hakimi and Schmeichel: what is the number of cycles of length p that a maximal planar graph on n vertices could have in terms of n? Then in the last section we discuss the question by Hakimi, Schmeichel and Thomassen: what is the minimum number of Hamiltonian cycles that a maximal planar graph on n vertices could have, in terms of n? In Chapter 5, we look at general planar triangulations. Note that every maximal planar graph on n ≥ 3 vertices is a planar triangulation. In the first section we discuss general properties of planar triangulations and then end with Hamiltonian cycles in planar triangulations

A semi-exact degree condition for Hamilton cycles in digraphs

The paper is concerned with directed versions of Posa's theorem and Chvatal's theorem on Hamilton cycles in graphs. We show that for each a>0, every digraph G of sufficiently large order n whose outdegree and indegree sequences d_1^+ \leq ... \leq d_n^+ and d_1^- \leq >... \leq d_n^- satisfy d_i^+, d_i^- \geq min{i + a n, n/2} is Hamiltonian. In fact, we can weaken these assumptions to (i) d_i^+ \geq min{i + a n, n/2} or d^-_{n - i - a n} \geq n-i; (ii) d_i^- \geq min{i + a n, n/2} or d^+_{n - i - a n} \geq n-i; and still deduce that G is Hamiltonian. This provides an approximate version of a conjecture of Nash-Williams from 1975 and improves a previous result of K\"uhn, Osthus and Treglown

Enumeration of Hamiltonian cycles in certain generalized Petersen graphs

AbstractThe generalized Petersen graph P(n, k) has vertex set V={u0, u1, …, un−1, v0, v1, …, vn−1} and edge set E={uiui+1, uivi, vivi+k∥ for 0≤i≤n−1 with indices taken modulo n}. The classification of the Hamiltonicity of generalized Petersen graphs was begun by Watkins, continued by Bondy and Bannai, and completed by Alspach. We now determine the precise number of Hamiltonian cycles present in each of the graphs P(n, 2). This more detailed information allows us to identify an infinite family of counterexamples to a conjecture of Greenwell and Kronk who had suggested a relation between uniquely 3-edge-colorable cubic graphs and the number of Hamiltonian cycles present

Parity balance of the ii-th dimension edges in Hamiltonian cycles of the hypercube

Let $n\geq 2$ be an integer, and let $i\in\{0,...,n-1\}$. An $i$-th dimension edge in the $n$-dimensional hypercube $Q_n$ is an edge ${v_1}{v_2}$ such that $v_1,v_2$ differ just at their $i$-th entries. The parity of an $i$-th dimension edge \edg{v_1}{v_2} is the number of 1's modulus 2 of any of its vertex ignoring the $i$-th entry. We prove that the number of $i$-th dimension edges appearing in a given Hamiltonian cycle of $Q_n$ with parity zero coincides with the number of edges with parity one. As an application of this result it is introduced and explored the conjecture of the inscribed squares in Hamiltonian cycles of the hypercube: Any Hamiltonian cycle in $Q_n$ contains two opposite edges in a 4-cycle. We prove this conjecture for $n \le 7$, and for any Hamiltonian cycle containing more than $2^{n-2}$ edges in the same dimension. This bound is finally improved considering the equi-independence number of $Q_{n-1}$, which is a concept introduced in this paper for bipartite graphs

Edge-Disjoint Hamiltonian Cycles in De Bruijn Graphs

The purpose of this thesis is to examine the number of edge-disjoint Hamiltonian cycles in de Bruijn graphs using ideas from finite field theory, particularly linear recurring sequences. It is known that the de Bruijn graph B(d,n) admits d-1 disjoint Hamiltonian cycles when d is a power of 2, and it is conjectured that all de Bruijn graphs B(d,n) admit d-1 disjoint Hamiltonian cycles. The conjecture also states that for every de Bruijn graph there exists a Hamiltonian cycle to which a particular function, defined in chapter 4, can be applied to obtain d-2 additional Hamiltonian cycles. I have shown for several specific de Bruijn graphs that this method does not work on Hamiltonian cycles obtained using linear recurring sequences.Keywords: de Bruijn graphs, Hamiltonian cycles, interconnection networks, linear recurring sequence

The hamiltonicity and path t-coloring of Sierpiński-like graphs

AbstractA mapping ϕ from V(G) to {1,2,…,t} is called a path t-coloring of a graph G if each G[ϕ−1(i)], for 1≤i≤t, is a linear forest. The vertex linear arboricity of a graph G, denoted by vla(G), is the minimum t for which G has a path t-coloring. Graphs S[n,k] are obtained from the Sierpiński graphs S(n,k) by contracting all edges that lie in no induced Kk. In this paper, the hamiltonicity and path t-coloring of Sierpiński-like graphs S(n,k), S+(n,k), S++(n,k) and graphs S[n,k] are studied. In particular, it is obtained that vla(S(n,k))=vla(S[n,k])=⌈k/2⌉ for k≥2. Moreover, the numbers of edge disjoint Hamiltonian paths and Hamiltonian cycles in S(n,k), S+(n,k) and S++(n,k) are completely determined, respectively

Twin-constrained Hamiltonian paths on threshold graphs: an approach to the minimum score separation problem

The Minimum Score Separation Problem (MSSP) is a combinatorial problem that has been introduced in JORS 55 as an open problem in the paper industry arising in conjunction with the cutting-stock problem. During the process of producing boxes, áat papers are prepared for folding by being scored with knives. The problem is to determine if and how a given production pattern of boxes can be arranged such that a certain minimum distance between the knives can be kept. While it was originally suggested to analyse the MSSP as a specific variant of a Generalized Travelling Salesman Problem, the thesis introduces the concept of twin-constrained Hamiltonian cycles and models the MSSP as the problem of finding a twin-constrained Hamiltonian path on a threshold graph (threshold graphs are a specific type of interval graphs). For a given undirected graph G(N,E) with an even node set N and edge set E, and a bijective function b on N that assigns to every node i in N a "twin node" b(i)6=i, we define a new graph G'(N,E') by adding the edges {i,b(i)} to E. The graph G is said to have a twin-constrained Hamiltonian path with respect to b if there exists a Hamiltonian path on G' in which every node has its twin node as its predecessor (or successor). We start with presenting some general Öndings for the construction of matchings, alternating paths, Hamiltonian paths and alternating cycles on threshold graphs. On this basis it is possible to develop criteria that allow for the construction of twin-constrained Hamiltonian paths on threshold graphs and lead to a heuristic that can quickly solve a large percentage of instances of the MSSP. The insights gained in this way can be generalized and lead to an (exact) polynomial time algorithm for the MSSP. Computational experiments for both the heuristic and the polynomial-time algorithm demonstrate the efficiency of our approach to the MSSP. Finally, possible extensions of the approach are presented

Finding long cycles in graphs

We analyze the problem of discovering long cycles inside a graph. We propose and test two algorithms for this task. The first one is based on recent advances in statistical mechanics and relies on a message passing procedure. The second follows a more standard Monte Carlo Markov Chain strategy. Special attention is devoted to Hamiltonian cycles of (non-regular) random graphs of minimal connectivity equal to three