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Database System Concepts
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
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The class I'm taking is computer science discrete structures. 

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**Educational Content: Euler Paths and Circuits** **Problem Statement:** Which of the following graphs contain an Euler path? Which contain an Euler circuit? (a) \( K_4 \) (b) \( K_5 \) (c) \( K_{5,7} \) (d) \( K_{2,7} \) (e) \( C_7 \) (f) \( P_7 \) **Definitions:** - **Euler Path:** A trail in a graph that visits every edge exactly once. An Euler path exists if a graph has exactly zero or two vertices of odd degree. - **Euler Circuit:** A circuit in a graph that visits every edge exactly once and returns to the starting vertex. An Euler circuit exists if all vertices in the graph have even degrees. **Graph Notations:** - **\( K_n \):** A complete graph on \( n \) vertices, where there is an edge between every pair of vertices. - **\( K_{m,n} \):** A complete bipartite graph where vertices are divided into two disjoint sets of \( m \) and \( n \) vertices, and every vertex from one set is connected to every vertex in the other set. - **\( C_n \):** A cycle graph on \( n \) vertices, forming a single loop. - **\( P_n \):** A path graph on \( n \) vertices, forming a single straight line without any loops.
Transcribed Image Text:**Educational Content: Euler Paths and Circuits** **Problem Statement:** Which of the following graphs contain an Euler path? Which contain an Euler circuit? (a) \( K_4 \) (b) \( K_5 \) (c) \( K_{5,7} \) (d) \( K_{2,7} \) (e) \( C_7 \) (f) \( P_7 \) **Definitions:** - **Euler Path:** A trail in a graph that visits every edge exactly once. An Euler path exists if a graph has exactly zero or two vertices of odd degree. - **Euler Circuit:** A circuit in a graph that visits every edge exactly once and returns to the starting vertex. An Euler circuit exists if all vertices in the graph have even degrees. **Graph Notations:** - **\( K_n \):** A complete graph on \( n \) vertices, where there is an edge between every pair of vertices. - **\( K_{m,n} \):** A complete bipartite graph where vertices are divided into two disjoint sets of \( m \) and \( n \) vertices, and every vertex from one set is connected to every vertex in the other set. - **\( C_n \):** A cycle graph on \( n \) vertices, forming a single loop. - **\( P_n \):** A path graph on \( n \) vertices, forming a single straight line without any loops.
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