Draw a planar graph(with no loops or multiple edges) for each of the following properties, if...
Draw a planar graph(with no loops or multiple edges) for each of the following properties, if possible. If not possible, explain briefly why not.
b) 8 vertices, all of degree 3 ( how many edges and regions must there be)
c) has exactly 7 vertices, has an euler cycle and 3 is minimum vertex coloring number
Also please draw the graph.
Homework Answers
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Draw a planar graph(with no loops or multiple edges) for each of
the following properties, if...
8. For each of the following, either draw a undirected graph satisfying the given criteria or explain why it cannot be done. Your graphs should be simple, i.e. not having any multiple edges (more tha...
8. For each of the following, either draw a undirected graph satisfying the given criteria or explain why it cannot be done. Your graphs should be simple, i.e. not having any multiple edges (more than one edge between the same pair of vertices) or self-loops (edges with both ends at the same vertex). [10 points] a. A graph with 3 connected components, 11 vertices, and 10 edges. b. A graph with 4 connected components, 10 vertices, and 30 edges. c....
Consider this map of the New Zealand North Island showing each of the regions in a different colour. (a) (4 marks) Draw...
Consider this map of the New Zealand North Island showing each of the regions in a different colour. (a) (4 marks) Draw a planar graph representing this map such that each region corresponds to a vertex and two vertices are connected by an edge if the two regions touch each other on the map (b) (2 marks) How many vertices in your graph have an even degree and how many vertices have an uneven degree? (c) (4 marks) What is...
8. For each of the following, either draw a undirected graph satisfying the given criteria or...
8. For each of the following, either draw a undirected graph satisfying the given criteria or explain why it cannot be done. Your graphs should be simple, i.e. not having any multiple edges (more than one edge between the same pair of vertices) or self-loops (edges with both ends at the same vertex). [10 points] a. A graph with 3 connected components, 11 vertices, and 10 edges. b. A graph with 4 connected components, 10 vertices, and 30 edges. c....
12. For each of the following collection of properties, draw one graph G that satisfies them...
12. For each of the following collection of properties, draw one graph G that satisfies them all (a) G is Bipartite and contains a vertex of degree 3 (b) G is a non-planar graph with A(G) < 3 (c) G is a tree with 5 vertices and A(G) = 4
12. For each of the following collection of properties, draw one graph G that satisfies them all (a) G is Bipartite and contains a vertex of degree 3 (b) G...
(a) Suppose that a connected planar graph has six vertices, each of degree three. Into how...
(a) Suppose that a connected planar graph has six vertices, each of degree three. Into how many regions is the plane divided by a planar embedding of this graph? 1. (b) Suppose that a connected bipartite planar simple graph has e edges and v vertices. Show that є 20-4 if v > 3.
(2) Recall the following fact: In any planar graph, there exists a vertex whose degree is s 5 Use...
(2) Recall the following fact: In any planar graph, there exists a vertex whose degree is s 5 Use this fact to prove the six-color theorem: for any planar graph there exists a coloring with six colors, i.e. an assignment of six given colors (e.g. red, orange, yellow, green, blue, purple) to the vertices such that any two vertices connected by an edge have different colors. (Hint: use induction, and in the inductive step remove some verter and all edges...
A graph without multiple edges between the same pair of vertices and without loops is called...
A graph without multiple edges between the same pair of vertices and without loops is called a _______. salesman graph degree graph simple graph complex graph
2. If possible, draw a simple graph with 11 edges and all vertices are of degree...
2. If possible, draw a simple graph with 11 edges and all vertices are of degree 3. If no such graph exists, explain why.
Write down true (T) or false (F) for each statement. Statements are shown below If a...
Write down true (T) or false (F) for each statement. Statements are shown below If a graph with n vertices is connected, then it must have at least n − 1 edges. If a graph with n vertices has at least n − 1 edges, then it must be connected. If a simple undirected graph with n vertices has at least n edges, then it must contain a cycle. If a graph with n vertices contain a cycle, then it...
8. For each of the following, either draw a undirected graph satisfying the given criteria or explain why it cannot be done. Your graphs should be simple, i.e. not having any multiple edges (more than one edge between the same pair of vertices) or self-loops (edges with both ends at the same vertex). [10 points] a. A graph with 3 connected components, 11 vertices, and 10 edges. b. A graph with 4 connected components, 10 vertices, and 30 edges. c....
Consider this map of the New Zealand North Island showing each of the regions in a different colour. (a) (4 marks) Draw a planar graph representing this map such that each region corresponds to a vertex and two vertices are connected by an edge if the two regions touch each other on the map (b) (2 marks) How many vertices in your graph have an even degree and how many vertices have an uneven degree? (c) (4 marks) What is...
8. For each of the following, either draw a undirected graph satisfying the given criteria or explain why it cannot be done. Your graphs should be simple, i.e. not having any multiple edges (more than one edge between the same pair of vertices) or self-loops (edges with both ends at the same vertex). [10 points] a. A graph with 3 connected components, 11 vertices, and 10 edges. b. A graph with 4 connected components, 10 vertices, and 30 edges. c....
12. For each of the following collection of properties, draw one graph G that satisfies them all (a) G is Bipartite and contains a vertex of degree 3 (b) G is a non-planar graph with A(G) < 3 (c) G is a tree with 5 vertices and A(G) = 4
12. For each of the following collection of properties, draw one graph G that satisfies them all (a) G is Bipartite and contains a vertex of degree 3 (b) G...
(a) Suppose that a connected planar graph has six vertices, each of degree three. Into how many regions is the plane divided by a planar embedding of this graph? 1. (b) Suppose that a connected bipartite planar simple graph has e edges and v vertices. Show that є 20-4 if v > 3.
(2) Recall the following fact: In any planar graph, there exists a vertex whose degree is s 5 Use this fact to prove the six-color theorem: for any planar graph there exists a coloring with six colors, i.e. an assignment of six given colors (e.g. red, orange, yellow, green, blue, purple) to the vertices such that any two vertices connected by an edge have different colors. (Hint: use induction, and in the inductive step remove some verter and all edges...
2. If possible, draw a simple graph with 11 edges and all vertices are of degree 3. If no such graph exists, explain why.
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