16 Use the fact that every planar graph with fewer than 12 vertices has a vertex of degree <4 (Exercise 19 in Section 1.4) to prove that every planar graph with less than 12 vertices can be 4-colored.

Q: Is hypercube graph Q4 planar? If not can you provide any bounds on it crossing number?

A: No, hypercube graph Q4 is not a planner graph.

Q: I have a graph with 2021 vertices. It is not a tree but there exists some vertex whose removal makes…

A: We know that maximum number of possible edges is 3(n-2)

Q: How many nonisomoprhic connected planar graphs (V, E) are there with v = 6 and e = 10 assuming that…

A:

Q: Calculate the number of paths of length 10 starting at the vertex 1. How many of these paths end at…

A: We will use the fact that if A is the adjacency matrix of a graph .Then the number of paths of…

Q: Prove that a cycle with even number of vertexes is bipartite graph.

A: A bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets U…

Q: 2. A k-dimensional hypercube on 2^k vertices is defined recursively. The base case, a 1-dimensional…

A: The given diagram:               Here the vertices can be denoted as V and edges can be denoted as…

Q: Show that in every simple graph there is a path from every vertex of odd degree to some other vertex…

A: we have to prove that In every simple graph there is a path from every vertex of odd degree to some…

Q: Show that if a simple graph G has n vertices and at least n edges, then G must contain a cycle.

A: We have to Show that if a simple graph G has n vertices and at least n edges, then G must contain a…

Q: "If G has a walk that uses all the edges, then G is connected." I know that this statement is false…

A: If G has a walk that uses all the edges, then G is connected Clearly this statement is false For…

Q: (7) Verify whether it is possible to have a connected Eulerian graph with 3 vertices and 6 edges,…

A:

Q: find the number of the isomorphism classes of graphs with 5 vertices and 6 edges which have (i)…

A: Given the a graphs with 5 vertices and 6 edges. We have to find the number of the isomorphism…

Q: Draw an Eulerian graph which has 5 vertices having one vertex with degree 4 and other vertices with…

A: Given: To draw the Eulerian graph which has 5 vertices having one vertex with a degree 4 and the…

Q: Prove that If a connected planar simple graph has e edges and v vertices with v ≥ 3 and no circuits…

A: A bipartite graph is a simple graph whose vertices can be partitioned into two sets V1​ and V2​ such…

Q: Find the number of spanning trees that can be made of this graph using Kirchhoff's Theorem

A: The total number of spanning trees are given by using Kirchhoff's Theorem

Q: Compute the betweenness centrality of all four vertices

A: Betweenness centrality measures the extent to which a vertex lies on paths between other vertices.…

Q: Show that any connected simple planar graph on n vertices must have at most n/2 vertices of degree…

A: Consider the equation

Q: An undirected graph G with 6 vertices with the degrees 7,3,12,4,4,6 respectively, what is the length…

A: In an undirected graph, an Eulerian path is a path that traverses each edge exactly once. For a…

Q: (10) A graph is connected if, and only if, (Choose one answer from below) A) It consists of a few…

A: A connected graph is graph that is connected in the sense of a topological space, i.e., there is a…

Q: True or false: If a connected graph G can be decomposed into exactly k paths then exactly 2k…

A: To Prove: A connected graph can be decomposed into exactly k paths then exactly 2k vertices of G…

Q: All cycle graphs are also K-regular graphs. True or False

A:

Q: 2 3

A: For a vertex x, the pair a,b under consideration are such that  i) a≠x,b≠x and  ii) there is a…

Q: Show that if G is a connected graph and e is an edge in G that is part of a cycle, then the graph G'…

A: