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B6) Show that the following graphs Hamiltonian are hot a a) d d 137 c) ol

B6) Show that the following graphs Hamiltonian are hot a a) d d 137 c) ol

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Problem 36 **Objective:** Show that the following graphs are not Hamiltonian. #### Graph Descriptions **a)** - This graph is composed of two triangles joined by a shared vertex. - Vertices are labeled as: \(a, b, c, d, e, f, g, h\). - There is a triangle formed by the vertices \(a, f, g\). - A second triangle is formed by \(b, e, f\). - Both triangles meet at vertex \(f\). - Additionally, a link exists forming a triangle with vertices \(b, c, d\). **b)** - This diagram is a complete bipartite graph known as \(K_{3,3}\). - There are six vertices: \(a, b, c, d, e, f\). - Three vertices at the top (\(a, b, c\)) are connected to three at the bottom (\(d, e, f\)). - Edges criss-cross, making connections between the non-adjacent vertices of the two groups. **c)** - This graph forms a star structure. - Vertices are labeled \(a, b, c, d, e\). - There is a central vertex \(c\) connected to four peripheral vertices \(a, b, d, e\). - Vertices \(a\) and \(b\) are connected by a line, so are \(d\) and \(e\). **d)** - This graph consists of a quadrilateral formed by a single triangle joined at one vertex to another point. - It has five total vertices in the shape, forming two separate triangles connected by one common vertex. #### Analysis **Notes:** - A Hamiltonian graph is one that contains a Hamiltonian cycle, a loop that visits each vertex exactly once and returns to the starting vertex. - All four graphs exhibit characteristics that prevent the formation of such a cycle, usually due to excessive intersections or separate components that cannot be traversed without revisiting vertices. **Educational Insight:** Understanding why certain graphs are non-Hamiltonian is crucial for students studying graph theory, as it reinforces the structural properties and constraints affecting cycle formation.
Transcribed Image Text:### Problem 36 **Objective:** Show that the following graphs are not Hamiltonian. #### Graph Descriptions **a)** - This graph is composed of two triangles joined by a shared vertex. - Vertices are labeled as: \(a, b, c, d, e, f, g, h\). - There is a triangle formed by the vertices \(a, f, g\). - A second triangle is formed by \(b, e, f\). - Both triangles meet at vertex \(f\). - Additionally, a link exists forming a triangle with vertices \(b, c, d\). **b)** - This diagram is a complete bipartite graph known as \(K_{3,3}\). - There are six vertices: \(a, b, c, d, e, f\). - Three vertices at the top (\(a, b, c\)) are connected to three at the bottom (\(d, e, f\)). - Edges criss-cross, making connections between the non-adjacent vertices of the two groups. **c)** - This graph forms a star structure. - Vertices are labeled \(a, b, c, d, e\). - There is a central vertex \(c\) connected to four peripheral vertices \(a, b, d, e\). - Vertices \(a\) and \(b\) are connected by a line, so are \(d\) and \(e\). **d)** - This graph consists of a quadrilateral formed by a single triangle joined at one vertex to another point. - It has five total vertices in the shape, forming two separate triangles connected by one common vertex. #### Analysis **Notes:** - A Hamiltonian graph is one that contains a Hamiltonian cycle, a loop that visits each vertex exactly once and returns to the starting vertex. - All four graphs exhibit characteristics that prevent the formation of such a cycle, usually due to excessive intersections or separate components that cannot be traversed without revisiting vertices. **Educational Insight:** Understanding why certain graphs are non-Hamiltonian is crucial for students studying graph theory, as it reinforces the structural properties and constraints affecting cycle formation.
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