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?
- The graph has a minimum length path between any two vertices
- The graph has no odd cycles (correct)
- The graph is nontrivial
- The graph is connected
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?
- Because the edge vw is of odd length
- Because the sum of the lengths of the u-v path, vw edge, and w-u path is odd (correct)
- Because the u-v and w-u paths are of even length
- Because the u-v and w-u paths are of odd length
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)?
- Whether u and v are adjacent in G and x and y are adjacent in H
- Whether u = x and v = y, or u = y and v = x, or u = v and x = y
- Whether either u = x and v = y, or u = y and v = x
- Whether u and v are adjacent in G or x and y are adjacent in H (correct)
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?
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?
What is the size of the complete graph Kn?
What is the size of the complete graph Kn?
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?
What is the order of the empty graph K̄n?
What is the order of the empty graph K̄n?
Which of the following statements is TRUE for bipartite graphs?
Which of the following statements is TRUE for bipartite graphs?
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?
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?
What is a path graph?
What is a path graph?
What is the order of a cycle graph Cn?
What is the order of a cycle graph Cn?
Flashcards
Bipartite Graph
Bipartite Graph
A graph whose vertices can be divided into two disjoint sets where edges only connect vertices from different sets.
Odd Cycle
Odd Cycle
A cycle in a graph that contains an odd number of edges.
Graph Join
Graph Join
Combines two graphs into one, adding edges between every vertex of the first and the second graph.
Cartesian Product of Graphs
Cartesian Product of Graphs
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Graph Degree
Graph Degree
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Path Graph
Path Graph
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Cycle Graph
Cycle Graph
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Complete Graph
Complete Graph
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Order of a Graph
Order of a Graph
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Size of a Complete Graph
Size of a Complete Graph
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Graph Complement
Graph Complement
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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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