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Questions and Answers
If a graph is not bipartite, then it must contain an odd cycle.
If a graph is not bipartite, then it must contain an odd cycle.
True (A)
The join of two graphs always results in a bipartite graph.
The join of two graphs always results in a bipartite graph.
False (B)
The Cartesian product of two bipartite graphs is always bipartite.
The Cartesian product of two bipartite graphs is always bipartite.
True (A)
The proof assumes that the graph under consideration is connected.
The proof assumes that the graph under consideration is connected.
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If a graph has an even number of vertices, then all vertices cannot have different degrees.
If a graph has an even number of vertices, then all vertices cannot have different degrees.
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The complete graph of order 5, denoted by $K_5$, has a size of 10 edges.
The complete graph of order 5, denoted by $K_5$, has a size of 10 edges.
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The complement of a graph with 6 vertices and 8 edges has a size of 10 edges.
The complement of a graph with 6 vertices and 8 edges has a size of 10 edges.
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A path of order 7, denoted by $P_7$, is a cycle.
A path of order 7, denoted by $P_7$, is a cycle.
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If a graph G is bipartite, it cannot contain any cycles.
If a graph G is bipartite, it cannot contain any cycles.
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The empty graph of order 4, denoted by $K_4$, has 4 edges.
The empty graph of order 4, denoted by $K_4$, has 4 edges.
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A graph with 10 vertices and 15 edges is a complete graph.
A graph with 10 vertices and 15 edges is a complete graph.
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The complement of a cycle of order 4, denoted by $C_4$, is also a cycle.
The complement of a cycle of order 4, denoted by $C_4$, is also a cycle.
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A graph is bipartite if and only if it has at least one odd cycle.
A graph is bipartite if and only if it has at least one odd cycle.
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Flashcards
Bipartite Graph
Bipartite Graph
A graph whose vertices can be divided into two disjoint sets where each edge connects a vertex from one set to a vertex from the other set.
Odd Cycle
Odd Cycle
A cycle in a graph with an odd number of edges.
Join of Graphs
Join of Graphs
The graph formed by combining two graphs and adding edges between every pair of vertices from both graphs.
Cartesian Product of Graphs
Cartesian Product of Graphs
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Contradiction Proof
Contradiction Proof
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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 Graph
Size of a Graph
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Graph Complement (Ḡ)
Graph Complement (Ḡ)
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Odd Cycle Theorem
Odd Cycle Theorem
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Study Notes
Common Graph Classes
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Path (Pn): A graph where vertices can be labeled consecutively (u1, u2, ..., un) such that edges are u1u2, u2u3, ..., un-1un. The path of order n is denoted by Pn.
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Cycle (Cn): A graph of order n ≥ 3 where vertices can be labeled consecutively (u1, u2, ..., un) and edges are u1u2, u2u3, ..., un-1un, unu1. The cycle of order n is denoted by Cn.
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Complete Graph (Kn): A graph where every pair of distinct vertices is connected by an edge. A complete graph of order n is denoted by Kn. The size of a complete graph of order 'n' is given by n(n-1)/2.
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Complement of a Graph (G): The complement Gc of a graph G has the same vertices but edges uv in Gc if and only if uv is not present in G.
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Bipartite Graph: A graph whose vertices can be divided into two disjoint and independent sets A and B such that every edge connects a vertex in A to a vertex in B (i.e., no edges connect vertices within the same set). A nontrivial graph is bipartite if and only if it contains no odd cycles.
Definitions & Notation
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Join (G + H): If G and H are graphs, then G + H is the graph consisting of G ∪ H (the union of the vertices and edges of G and H), and all edges joining a vertex of G and a vertex of H.
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Cartesian Product (G × H): If G and H are graphs, the Cartesian Product G × H is a graph with vertex set V(G × H) = V(G) × V(H). Two distinct vertices (u, v) and (x, y) are joined by an edge if either u = x and v y ∈ E(H) or v = y and u x ∈ E(G)
Theorem
- Theorem 1.12: A non-trivial graph G is bipartite if and only if G does not have any odd cycles.
Additional Notes
- There is no graph of order n ≥ 2 in which all vertices have different degrees.
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Description
Test your knowledge on common graph classes including Path, Cycle, Complete Graph, and the Complement of a Graph. This quiz covers definitions and properties of these essential graph structures. Perfect for students and enthusiasts of graph theory!
