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In this lecture, we discuss the notions of Hamiltonian cycles and paths in the context of both undirected and directed graphs. Hamiltonian Cycles and Paths. Let G be a graph. A cycle in G is a closed trail that only repeats the rst and last vertices.
We shall now study some questions on Hamiltonian paths in digraphs, proving (in particular) Rédei’s theorem on Hamiltonian paths in tournaments. We let denote the set {0, 1, 2, . . .} of all nonnegative integers. We recall some basic notions from graph theory: Definition 1.1.1.
Discrete Mathematics 01/04/2019 Lecture 16: Eulerian Tours and Hamiltonian Cycles Instructor: Goutam Paul Scribe: Souhardya Sengupta 1 Introduction In this lecture, we will take up the problem of traversing through a graph through trails and cycles. Its a natural question to ask whether we can travel through a graph following its
Hamilton Paths and Circuits •Next, we shall give the proof of the “if case” of Bondy-Chvátal’s Theorem •Proof (“if case”): Suppose on the contrary that (i) G does not have a Hamilton circuit, but (ii) G’s Hamilton closure has a Hamilton circuit. Then, consider the sequence of graphs obtained
Hamiltonian graphs and the Bondy-Chvátal Theorem This lecture introduces the notion of a Hamiltonian graph and proves a lovely the-orem due to J. Adrian Bondy and Vašek Chvátal that says—in essence—that if a graph has lots of edges, then it must be Hamiltonian. Reading: The material in today’s lecture comes from Section 1.4 of
Hamilton cycle. Take a Hamilton path v1;:::;vn, where v1vn62E. Let N (v1) = fvj 1: vjv1 2Eg. Since jN(vn)j;jN (v1)j n 2 and N(vn);N (v1) Vnfvng, there is a vertex vi2N(vn) \N (v1) Hamilton cycle in G: vnvivi 1:::v1vi+1:::vn. 2 Remark.Dirac’s Theorem is best possible. Non-Hamiltonian graphs with minimum degree dn 2 e 1: - K d(n+1)=2eand K
Definition 4.2.1: A graph with a spanning path is called traceable and this path is called a Hamiltonian path. A graph with a spanning cycle is called Hamiltonian and this cycle is known as a Hamiltonian cycle. It is clear that Hamiltonian graphs are connected; Cn and Kn are Hamiltonian but tree is not Hamil-tonian.
