Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Only b-d

## Problem 36: Show that the Following Graphs are not Hamiltonian **Graph (a):** - Vertices: \(a, b, c, d, e, f, g, h\) - Structure: The graph is composed of two triangles linked together by a common vertex \(f\). Triangle one includes vertices \(a, f, g, h\), and the second triangle includes vertices \(b, f, e\). The vertex \(b\) is connected to vertex \(c\) which connects to vertex \(d\). - Explanation: Involves assessing each edge and vertex path to determine the impossibility of a Hamiltonian cycle, where each vertex is visited once without repetition. **Graph (b):** - Vertices: \(a, b, c, d, e\) - Structure: A complete bipartite graph represented in a quadrilateral layout. Includes diagonals intersecting between nonadjacent vertices. - Explanation: Analyze edge connections and intersections to illustrate that retracing steps is unavoidable, which precludes a Hamiltonian cycle. **Graph (c):** - Vertices: \(a, b, c, d, e\) - Structure: The graph is a bow-tie shape or hourglass structure with overlapping edges at vertex \(c\). The upper part connects vertices \(a\) and \(b\) to \(c\), and the lower part connects vertices \(d\) and \(e\) to \(c\). - Explanation: Checking possible routes, there’s necessary repetition of the intersection point resulting in no Hamiltonian cycle. **Graph (d):** - Vertices: Not labeled - Structure: Two disconnected figures composed of triangles. The left shape has three points linked, one isolated from the right triangle shape. - Explanation: As the graph is disconnected, it inherently cannot have a Hamiltonian cycle, as all vertices cannot be visited in a continuous path. Each graph demonstrates the necessary characteristics to be non-Hamiltonian by either containing disconnections, intersections voiding unique paths, or vertex repetitions—characteristics precluding a Hamiltonian cycle.
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Transcribed Image Text:## Problem 36: Show that the Following Graphs are not Hamiltonian **Graph (a):** - Vertices: \(a, b, c, d, e, f, g, h\) - Structure: The graph is composed of two triangles linked together by a common vertex \(f\). Triangle one includes vertices \(a, f, g, h\), and the second triangle includes vertices \(b, f, e\). The vertex \(b\) is connected to vertex \(c\) which connects to vertex \(d\). - Explanation: Involves assessing each edge and vertex path to determine the impossibility of a Hamiltonian cycle, where each vertex is visited once without repetition. **Graph (b):** - Vertices: \(a, b, c, d, e\) - Structure: A complete bipartite graph represented in a quadrilateral layout. Includes diagonals intersecting between nonadjacent vertices. - Explanation: Analyze edge connections and intersections to illustrate that retracing steps is unavoidable, which precludes a Hamiltonian cycle. **Graph (c):** - Vertices: \(a, b, c, d, e\) - Structure: The graph is a bow-tie shape or hourglass structure with overlapping edges at vertex \(c\). The upper part connects vertices \(a\) and \(b\) to \(c\), and the lower part connects vertices \(d\) and \(e\) to \(c\). - Explanation: Checking possible routes, there’s necessary repetition of the intersection point resulting in no Hamiltonian cycle. **Graph (d):** - Vertices: Not labeled - Structure: Two disconnected figures composed of triangles. The left shape has three points linked, one isolated from the right triangle shape. - Explanation: As the graph is disconnected, it inherently cannot have a Hamiltonian cycle, as all vertices cannot be visited in a continuous path. Each graph demonstrates the necessary characteristics to be non-Hamiltonian by either containing disconnections, intersections voiding unique paths, or vertex repetitions—characteristics precluding a Hamiltonian cycle.
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