Choose the true statement. There exists a graph with 7 vertices of degree 1, 2, 2,...
(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.
Consider the following propositions P and Q. P: For all graph G, if G has 6 vertices of degree 3, 3, 3, 3, 3 and 3, respectively, then G is planar. Q: For all graph G, if G has 6 vertices of degree 3, 3, 3, 3, 3 and 3, respectively, then G is not planar. Choose the true statement. P is true and Q is true P is false and Q is false P is false and Q is...
(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...
Recall the definition of the degree of a vertex in a graph. a) Suppose a graph has 7 vertices, each of degree 2 or 3. Is the graph necessarily connected ? b) Now the graph has 7 vertices, each degree 3 or 4. Is it necessarily connected? My professor gave an example in class. He said triangle and a square are graph which are not connected yet each vertex has degree 2. (Paul Zeitz, The Art and Craft of Problem...
solve with steps 1. (20 points) True or false. Justify. Every planar graph is 4-colorable /2 The number of edges in a simple graph G is bounded by n(n 1) where n is the number of vertices. The number of edges of a simple connected graph G is at least n-1 where n is the number of vertices. Two graphs are isomorphic if they have the same number of vertices and 1) the same mumber of edges 1. (20 points)...
A graph has 21 edges, two vertices of degree 5, four vertices of degree 3, and all remaining vertices have degree 2. How many vertices does the graph have? 12 10 16 O 14
2. If possible, draw a simple graph with 11 edges and all vertices are of degree 3. If no such graph exists, explain why.
Answer the following true or false questions with a brief justification. A) There exists an undirected graph on 6 vertices whose degrees are 4, 5, 8, 9, 3, 6. B) Every undirected graph with n vertices and n − 1 edges is a tree. C) Let G be an undirected graph. Suppose u and v are the only vertices of odd degree in G. Then G contains a u-v path.
4. (10 points) (a) An undirected graph has 6 vertices and 13 edges. It is known three vertices have degree 3, one has degree 4, and another one has degree 7. Find the degree of the remaining vertex. (b) For each of the following graphs, determine if it is bipartite, complete, and/or a tree. Give a brief written or graphical justification for your answers (you may address multiple graphs at the same time). iii.
Choose the true statement. If a graph G admits an Eulerian path, then G is connected. If a graph G admits an Eulerian path, then G admits a Hamiltonian path. If a graph G admits a Hamiltonian path, then G admits an Eulerian path. the four other possible answers are false If a graph G is connected, then G admits an Eulerian path.