full explanation... **Problem A5:**For each graph \( G \) and \( H \) below, determine whether it has a Hamiltonian cycle. Justify your answer by showing the Hamiltonian cycle if it exists, or proving that it does not exist.**Graph Descriptions:**1. **Graph \( G \):** - Graph \( G \) is structured as a 4x4 grid consisting of nodes arranged in a square lattice. - Nodes are labeled from 11 to 44, inclusive, such that each row and each column contains four nodes. - Nodes \(11, 12, 13, 14\) form the first row, nodes \(21, 22, 23, 24\) form the second row, nodes \(31, 32, 33, 34\) form the third row, and nodes \(41, 42, 43, 44\) form the fourth row. - Each node is connected to its adjacent nodes vertically, horizontally, but not diagonally. 2. **Graph \( H \):** - Graph \( H \) is a hexagonal-like structure with nodes labeled from 12 to 34. - The top row consists of nodes \(12, 13, 14\). - The middle row consists of nodes \(21, 22, 23, 24, 25\). - The bottom row consists of nodes \(32, 33, 34\). - Each node is connected in a hexagonal pattern, where nodes are connected horizontally with their neighbors and diagonally connecting to the nodes above and below as appropriate.**Detailed Instructions:**For each graph:1. Determine if there is a Hamiltonian cycle present.2. To justify: - If a Hamiltonian cycle exists, depict it by highlighting the cycle in the graph. - If no Hamiltonian cycle exists, provide reasoning or a proof to demonstrate why it is not possible to have a Hamiltonian cycle.A Hamiltonian cycle is defined as a cycle that visits every vertex exactly once and returns to the starting vertex.**Notes:**- Graph \( G \) consists of a more regular rectangular grid pattern.- Graph \( H \) has a more complex hexagonal pattern, presenting a different challenge in determining the Hamiltonian cycle. These graphs visually represent nodes connected in two distinct patterns which require careful analysis to identify potential Hamiltonian cycles

Hamiltonian cycles are named after William Rowan Hamilton, an Irish mathematician who studied paths…

Problem A5: For each graph G and H below, determine whether it has a hamiltonian cycle. Justify your answer, by showing the hamiltonian cycle if it exists, or proving that it does not exist. (11) (12) (13) (14) (21) (31) (41) (22) (42) 23 (33) (43) (24) 34 (44) G (22) (32) (13) (23 (33) (14) (24) (34) (25) H

Problem A5: For each graph G and H below, determine whether it has a hamiltonian cycle. Justify your answer, by showing the hamiltonian cycle if it exists, or proving that it does not exist. (11) (12) (13) (14) (21) (31) (41) (22) (42) 23 (33) (43) (24) 34 (44) G (22) (32) (13) (23 (33) (14) (24) (34) (25) H

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

full explanation...

**Problem A5:**
For each graph \( G \) and \( H \) below, determine whether it has a Hamiltonian cycle. Justify your answer by showing the Hamiltonian cycle if it exists, or proving that it does not exist.

**Graph Descriptions:**

1. **Graph \( G \):**
- Graph \( G \) is structured as a 4x4 grid consisting of nodes arranged in a square lattice.
- Nodes are labeled from 11 to 44, inclusive, such that each row and each column contains four nodes.
- Nodes \(11, 12, 13, 14\) form the first row, nodes \(21, 22, 23, 24\) form the second row, nodes \(31, 32, 33, 34\) form the third row, and nodes \(41, 42, 43, 44\) form the fourth row.
- Each node is connected to its adjacent nodes vertically, horizontally, but not diagonally.

2. **Graph \( H \):**
- Graph \( H \) is a hexagonal-like structure with nodes labeled from 12 to 34.
- The top row consists of nodes \(12, 13, 14\).
- The middle row consists of nodes \(21, 22, 23, 24, 25\).
- The bottom row consists of nodes \(32, 33, 34\).
- Each node is connected in a hexagonal pattern, where nodes are connected horizontally with their neighbors and diagonally connecting to the nodes above and below as appropriate.

**Detailed Instructions:**

For each graph:
1. Determine if there is a Hamiltonian cycle present.
2. To justify:
- If a Hamiltonian cycle exists, depict it by highlighting the cycle in the graph.
- If no Hamiltonian cycle exists, provide reasoning or a proof to demonstrate why it is not possible to have a Hamiltonian cycle.

A Hamiltonian cycle is defined as a cycle that visits every vertex exactly once and returns to the starting vertex.

**Notes:**
- Graph \( G \) consists of a more regular rectangular grid pattern.
- Graph \( H \) has a more complex hexagonal pattern, presenting a different challenge in determining the Hamiltonian cycle.

These graphs visually represent nodes connected in two distinct patterns which require careful analysis to identify potential Hamiltonian cycles

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