What is a 1-factor in graph theory?
In graph theory, a factor of a graph G is a spanning subgraph, i.e., a subgraph that has the same vertex set as G. In particular, a 1-factor is a perfect matching, and a 1-factorization of a k-regular graph is an edge coloring with k colors. A 2-factor is a collection of cycles that spans all vertices of the graph.
What is a bipartite graph 1 point?
Explanation: A graph G1(V, E) is called bipartite if its vertex set V(G) can be decomposed into two non-empty disjoint subsets V1(G1) and V2(G1) in such a way that each edge e ∈ E(G) has its one end joint in V1(G1) and other endpoint in V2(G1).
Is there a bipartite graph that is 1 colorable?
Theorem 2.7 (Bipartite Colorings) If G is a bipartite graph with a positive num- ber of edges, then G is 2-colorable. If G is bipartite with no edges, it is 1-colorable.
What does it mean when a graph is bipartite?
A bipartite graph, also called a bigraph, is a set of graph vertices decomposed into two disjoint sets such that no two graph vertices within the same set are adjacent.
What is the factor of 1?
The number 1 has only one factor which is 1 and 1 is a common factor to all numbers.
Does every regular graph have a 1-factor?
Every 1-regular graph has exc max ( G ) = 0 , as it consists entirely of a 1-factor. is a 2-regular graph with exc max ( G ) = 0 if and only if is an even cycle or disjoint pair of even cycles.
What is a bipartite graph give an example?
A graph G=(V, E) is called a bipartite graph if its vertices V can be partitioned into two subsets V1 and V2 such that each edge of G connects a vertex of V1 to a vertex V2. It is denoted by Kmn, where m and n are the numbers of vertices in V1 and V2 respectively. Example: Draw the bipartite graphs K2, 4and K3 ,4.
What is bipartite graph in Mcq?
Explanation: A graph is said to be bipartite if it can be divided into two independent sets A and B such that each edge connects a vertex from A to B. Check this: Data Structure MCQ | Programming MCQs.
What is colorable in graph theory?
A graph is said to be k-colorable if it can be properly colored using k colors. For example, a bipartite graph is 2-colorable.
Is every 2-colorable graph bipartite?
Bipartite graphs may be characterized in several different ways: A graph is bipartite if and only if it does not contain an odd cycle. A graph is bipartite if and only if it is 2-colorable, (i.e. its chromatic number is less than or equal to 2).
How do you represent a bipartite graph?
We represent a complete bipartite graph by Km,n where m is the size of the first set and n is the size of the second set. So a K3,2 complete bipartite graph has three vertices in the first set, two vertices in the second, and every vertex in the second set is connected to the vertices in the first set.