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 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
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.
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?
Null graph,Nn, n=1,2,3,4...,the graph with n vertices and no
edges. (N4=4 vertices with no edges)
4 a) find a graph with 8 vertices with no 3-cycles and no
induced sub graph isomorphic to N4
b)prove that every simple graph with 9 vertices with no 3-cycles
has an induced sub graph isomorphic to N4