Use the fact that every planar graph with fewer than 12 vertices
has a vertex of degree <= 4 to prove that every planar graph
with than 12 vertices can be 4-colored.
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.
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.
Assume that in addition to having capacities for edges we have
capacities for vertices, meaning that the incoming flow to a vertex
cannot be more that its capacity. Describe an efficient algorithm to
solve the max-flow problem in such a network.
Prove or disprove the following statements.
(a) There is a simple graph with 6 vertices with degree sequence
(3, 3, 5, 5, 5, 5)?
(b) There is a simple graph with 6 vertices with degree sequence
(2, 3, 3, 4, 5, 5)?
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?