Assignment (AMA)

Q1: Two opposite corners are removed from an 8-by-8 checkerboard. Prove that it is impossible to cover the remaining 65 squares with 31 dominoes, such that each domino covers two adjacent squares?

Q2: Draw a 5-vertex connected graph G that has no cut-vertices, and then verify that G satisfies each of the following properties.

a. Given any two vertices, there exists a cycle containing both.
b. For any vertex v and any edge e of G, there exists a cycle containing v and e.
c. Given any two vertices x and y, and any edge e, there exists a path from x to y that contains e.
d. Given any two edges, there exists a cycle containing both.
e. Given any three distinct vertices u, v, and w, there exists a u-v path that contains w.
f. Given any three distinct vertices u, v, and w, there exists a u-v path that does not contain w.

Q3: Through A Write an algorithm to construct the indicated graph operation, using only the primary graph operations of additions and deletions of vertices and edges. Test your algorithm on the pair ( P4, W5) and on the pair ( K4- K2, C4 ):

A- Cartesian product of two graphs (psedocode)?

Q4: Through A,Beither draw the required graph or explain why no such graph exists:

A- An 8-vertex, 2-component, simple graph with exactly 10 edges and three cycles?
B- An 11-vertex, simple, connected graph with exactly 14 edges that contains five edge-disjoint cycles?

Q5: Prove or disprove: If a simple graph G has no cut-edge, then every vertex pf G has even degree?

Q6: Prove that if a graph has exactly two vertices of odd degree, then there must be a path between them?

Q7: Show that any nontrivial simple graph contains at least two vertices that are not cut-vertices?

Q8: Through A Draw the specified tree(s) or explain why on such a tree(s) can exist?

A- A 14-vertex binary tree of height 3.

Q9: Prove that a directed tree that has more than one vertex with in degree 0 cannot be a rooted tree?