A graph has 21 edges, two vertices of degree 5, four vertices of degree 3, and...
Discrete Mathematics 6: A: Draw a graph with 5 vertices and the requisite number of edges to show that if four of the vertices have degree 2, it would be impossible for the 5 vertex to have degree 1. Repetition of edges is not permitted. (There may not be two different bridges connecting the same pair of vertices.) B: Draw a graph with 4 vertices and determine the largest number of edges the graph can have, assuming repetition of 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...
Choose the true statement. There exists a graph with 7 vertices of degree 1, 2, 2, 3, 4, 4 and 5, respectively. the four other possible answers are false There exists a bipartite graph with 14 vertices and 13 edges. There exists a planar and connected graph with 5 vertices, 6 edges and 4 faces. There exists a graph with 5 vertices of degree 2, 3, 4, 5 and 6, respectively.
Question 16. A maximal plane graph is a plane graph G = (V, E) with n ≥ 3 vertices such that if we join any two non-adjacent vertices in G, we obtain a non-plane graph. (a) Draw a maximal plane graphs on six vertices. (b) Show that a maximal plane graph on n points has 3n − 6 edges and 2n − 4 faces. (c) A triangulation of an n-gon is a plane graph whose infinite face boundary is a...
A graph has 4 vertices of degrees 3, 3, 4, 4. (a) How many edges such a graph have? (b) Draw two non isomorphic such graphs. (c) Explain why there is no such simple graph
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.
Suppose that T is a tree with four vertices of degree 3, six vertices of degree 4, one vertex of degree 5, and 8 vertices of degree 6. No other vertices of T have degree 3 or more. How many leaf vertices does T have?
2. If possible, draw a simple graph with 11 edges and all vertices are of degree 3. If no such graph exists, explain why.
6. (4pts) Draw a simple graph with nine edges and all vertices of degree 3. (this is possible)
A graph has 320 vertices and 9650 edges. It is also known that this graph has 10 vertices of degree 100. Show that it has at least one vertex of degree less than 60.