An m × n grid graph has m rows of n vertices with vertices
closest to each other connected by an edge. Find the greatest
length of any path in such a graph, and provide a brief explanation
as to why it is maximum. You can assume m, n ≥ 2. Please provide an
explanation without using Hamilton Graph Theory.
Given a connected graph G with n vertices. We say an edge of G
is a bridge if the graph becomes a disconnected graph after
removing the edge. Give an O(m + n) time algorithm that finds all
the bridges. (Partial credits will be given for a polynomial time
algorithm.) (Hint: Use DFS)
The vertices of a triangle determine a circle, called the
circumcircle of the triangle. Show that if P is
any point on the circumcircle of a triangle, and X,
Y, and Z are the feet of the perpendiculars from
P to the sides of the triangle, then X,
Y and Z are collinear.
Let G be a simple undirected graph with n vertices where n is an
even number. Prove that G contains a triangle if it has at least
(n^2 / 4) + 1 edges using mathematical induction.
let
G be a simple graph. show that the relation R on the set of
vertices of G such that URV if and only if there is an edge
associated with (u,v) is a symmetric irreflexive relation on
G
Consider an undirected graph G that has n distinct
vertices. Assume n≥3.
How many distinct edges will there be in any circuit for G that
contains all the vertices in G?
What is the maximum degree that any vertex in G can have?
What is the maximum number of distinct edges G can have?
What is the maximum number of distinct edges that G can have if
G is disconnected?
Let Q:= {1,2,...,q}. Let G be a graph with the elements of Q^n
as vertices and an edge between (a1,a2,...,an) and (b1,b2,...bn) if
and only if ai is not equal to bi for exactly one value of i. Show
that G is Hamiltonian.