Podcast
Questions and Answers
What is the key assumption made to prove that a graph without odd cycles is bipartite?
What is the key assumption made to prove that a graph without odd cycles is bipartite?
Why is the cycle formed by combining a u-v path, edge vw, and a w-u path of minimum length an odd cycle in the proof?
Why is the cycle formed by combining a u-v path, edge vw, and a w-u path of minimum length an odd cycle in the proof?
In the Cartesian product G□H, what determines whether an edge connects vertices (u, x) and (v, y)?
In the Cartesian product G□H, what determines whether an edge connects vertices (u, x) and (v, y)?
What is the key idea behind proving that there is no graph of order n ≥ 2 whose vertices have all different degrees?
What is the key idea behind proving that there is no graph of order n ≥ 2 whose vertices have all different degrees?
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Which of the following best describes the concept of the join of two graphs G and H, denoted G + H?
Which of the following best describes the concept of the join of two graphs G and H, denoted G + H?
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What is the size of the complete graph Kn?
What is the size of the complete graph Kn?
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What is the relationship between the size of a graph G and its complement Ḡ?
What is the relationship between the size of a graph G and its complement Ḡ?
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What is the order of the empty graph K̄n?
What is the order of the empty graph K̄n?
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Which of the following statements is TRUE for bipartite graphs?
Which of the following statements is TRUE for bipartite graphs?
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What is the relationship between the degree of a vertex in a graph G and its degree in the complement Ḡ?
What is the relationship between the degree of a vertex in a graph G and its degree in the complement Ḡ?
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If a graph has an odd number of vertices, is it possible for it to be bipartite?
If a graph has an odd number of vertices, is it possible for it to be bipartite?
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What is a path graph?
What is a path graph?
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What is the order of a cycle graph Cn?
What is the order of a cycle graph Cn?
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Study Notes
Common Graph Classes
- Paths: A graph is a path if its vertices can be relabeled as u₁, u₂, ..., uₙ so that the edges are u₁u₂, u₂u₃, ..., uₙ₋₁uₙ. A path of order n is denoted by Pₙ.
- Cycles: A graph of order n ≥ 3 is a cycle if its vertices can be relabeled as u₁, u₂, ..., uₙ so that the edges are u₁u₂, u₂u₃, ..., uₙ₋₁uₙ, and uₙu₁. A cycle of order n is denoted by Cₙ.
- Complete Graphs: A graph is complete if every pair of distinct vertices is adjacent. A complete graph of order n is denoted by Kₙ. The size (m) of a complete graph of order n is given by m = n(n-1)/2.
Complement of a Graph
- The complement of a graph G, denoted as Ḡ, is the graph with the same vertices as G. An edge exists between two vertices in Ḡ if and only if there is no edge between those vertices in G.
Bipartite Graphs
- A graph is bipartite if its vertices can be partitioned into two sets (A and B) such that no edge connects two vertices within the same set.
Theorem 1.12
- A nontrivial graph is bipartite if and only if it contains no odd cycles.
Definitions and Notation
- Join (G + H): If G and H are graphs, their join (G + H) is the graph that combines G and H, along with all edges connecting a vertex in G to a vertex in H.
- Cartesian Product (G × H): If G and H are graphs, their Cartesian product (G × H) is a graph where vertices are pairs of vertices from G and H, and two vertices are connected if either their first components are the same and their second components are connected in H or their second components are the same and their first components are connected in G.
Problem
- The provided image shows examples of finding the Cartesian product and the join of two graphs (P₃ and K₃).
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Description
This quiz covers various classes of graphs, including paths, cycles, complete graphs, and bipartite graphs. It also discusses the complement of a graph and relevant theorems associated with these concepts. Test your understanding of graph theory and its fundamental concepts through this engaging quiz.
