Answer true or false and in each case either give a proof or provide a counterexample.
(a) A Hamiltonian graph contains no proper cycles.
(b) Every vertex in a Hamiltonian graph has degree 2.
(c) [BB] Every Eulerian graph is Hamiltonian.
(d) Every Hamiltonian graph is Eulerian.
The Correct Answer and Explanation is :
Let’s examine each statement one by one:
(a) A Hamiltonian graph contains no proper cycles.
False.
A Hamiltonian graph is one that contains a Hamiltonian cycle, meaning a cycle that visits every vertex exactly once and returns to the starting vertex. However, this does not preclude the presence of other proper cycles (cycles that do not include all vertices in the graph). For example, a graph could contain a Hamiltonian cycle and also have other cycles that are smaller, such as triangles or quadrilaterals.
Counterexample: Consider a graph with vertices {A, B, C, D}, where edges are AB, BC, CD, DA, and AC. This graph contains a Hamiltonian cycle (A, B, C, D, A) and also a proper cycle (A, C, A).
(b) Every vertex in a Hamiltonian graph has degree 2.
False.
In a Hamiltonian graph, the existence of a Hamiltonian cycle does not imply that every vertex has degree 2. For example, vertices can have a higher degree while still supporting a Hamiltonian cycle. The degree of vertices in Hamiltonian graphs can vary, and there is no restriction that forces them all to have degree 2.
Counterexample: A graph with vertices {A, B, C, D, E} and edges AB, BC, CD, DA, AE is Hamiltonian but the vertex A has degree 2, B has degree 2, C has degree 2, D has degree 2, and E has degree 1. Notice that one vertex (E) has degree 1, but the graph is still Hamiltonian.
(c) Every Eulerian graph is Hamiltonian.
False.
An Eulerian graph is one in which every vertex has an even degree and there exists a closed Eulerian path (a path that visits every edge exactly once). However, this does not imply that the graph contains a Hamiltonian cycle, which is a cycle that visits every vertex exactly once. The two properties are unrelated.
Counterexample: A cycle graph with 4 vertices (A-B-C-D-A) is Eulerian (each vertex has degree 2) but not Hamiltonian, as the graph does not have a Hamiltonian cycle that visits all vertices and edges in a specific manner.
(d) Every Hamiltonian graph is Eulerian.
False.
A Hamiltonian graph contains a Hamiltonian cycle, but this does not imply that every vertex has an even degree, which is necessary for the graph to be Eulerian. Therefore, a Hamiltonian graph is not guaranteed to be Eulerian.
Counterexample: Consider a triangle (graph with vertices A, B, C and edges AB, BC, CA), which is Hamiltonian because it contains a Hamiltonian cycle (A, B, C, A). However, each vertex has degree 2, and thus the graph cannot be Eulerian because Euler’s theorem requires all vertices to have an even degree.
In conclusion, each of the given statements is false, and we provided counterexamples to demonstrate why. The concepts of Hamiltonian cycles and Eulerian paths/cycles are related but do not imply each other in general.