Let G be a simple undirected graph with n vertices where n is an
even number....
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.
Given an undirected graph G = (V,E),
consisting of n vertices and m edges, with each
edge labeled from the set {0,1}.
Describe and analyze the worst-case time complexity of an
efficient algorithm to find any cycle consisting of edges whose
labels alternate 0,1.
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 Kn denote the simple graph on n vertices.
(a) Let v be some vertex of Kn and consider K n −
v, the graph obtained by deleting
v. Prove that K n − v is isomorphic to K n−1 .
(b) Use mathematical induction on n to prove the following
statement:
K n , the complete graph on n vertices, has
n(n-1)/2
edges
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.
Let G be a bipartite graph with 107 left vertices and 20 right
vertices. Two vertices u, v are called twins if the set of
neighbors of u equals the set of neighbors of v (triplets,
quadruplets etc are defined similarly).
Show that G has twins.
Bonus: Show that G has triplets. What about quadruplets, etc.?
(a) What is the maximum degree of a vertex in a simple graph
with n vertices?
(b) What is the maximum number of edges in a simple graph of
n vertices?
(c) Given a natural number n, does there exist a simple
graph with n vertices and the maximum number of edges?
Let G be a graph whose vertices are the integers 1
through 8, and let the adjacent vertices of each vertex be given by
the table below:
vertex
adjacent vertices
1
(2, 3, 4)
2
(1, 3, 4)
3
(1, 2, 4)
4
(1, 2, 3, 6)
5
(6, 7, 8)
6
(4, 5, 7)
7
(5, 6, 8)
8
(5, 7)
Assume that, in a traversal of G, the adjacent vertices
of a given vertex are returned in the...
You are given a directed graph G(V,E) with n vertices and m
edges. Let S be the subset of vertices in G that are able to reach
some cycle in G. Design an O(n + m) time algorithm to compute the
set S. You can assume that G is given to you in the adjacency-list
representation.