Math 225 Midterm 3
How to look for isomorphism
Vertex degrees, relationship between vertices, number of edges, graph complements, look for shared subgraphs
What is complete induction?
We assume one thru k
Define Forest
if we have a graph with no cycles, and our graph has more than one component, we call the graph a forest
If G is a graph with n>=3 vertices, the graph has a Hamiltonian circuit if the degree of each vertex is at ______.
least n/2.
If we see Kn as a subgraph, we automatically know we need at least __ colors
n colors
Cycle
no edges are repeated but we can possibly repeat vertices
The degree of a region (in a plane drawing) is the...
number of edges in its border, where an edge is counted twice if it borders only one region
Any two plane drawings of the same connected graph G will have the same # of _____
regions
The chromatic number X(G) of a graph is...
the minimum number of different colors needed in a graph coloring of G
A graph has an Euler trail if it has ____ vertices of ___ degree
2, odd
If G has a Hamiltonian circuit, then for every subset S in V(G)...
(# components of G-S) <= |S|
connected components
(disjoint connected subgraphs)
A graph has an Euler cycle if it has ___ vertices of ___ degree
0, odd
To show graphs are not isomorphic show:
1) number of vertices are not the same 2) number of edges is not the same 3) degree sequence is different 4) complements are non-isomorphic 5) find a subgraph of one that does not exist in the other
How many colors are needed for bipartite graphs?
2
For any plane drawing of a graph G, the sum of the degrees of regions is _______. Why?
2 times the number of edges. Every edge has two sides.
Define Hamiltonian circuit
A Hamiltonian circuit in a graph is a circuit that visits every vertex in a graph G exactly once (and comes back to where it starts)
Define Hamiltonian path
A Hamiltonian path in a graph is a path that visits every vertex in a graph G exactly once (but since it is a path it does not need to come back to where it starts)
A tree
A connected graph with no cycles
Bipartite graph
A graph G=(V,E) is bipartite if we can partition V = (X U Y) so that every edge has one end pt in X and the other in Y. AKA: No xs touch other xs and no ys touch other ys
Multigraph
A graph in which loops and multiple edges are allowed
A connected graph
A graph is connected if, for any vertices x, y there is a path in G from x to y
Define leaf
A leaf of a tree is a vertex of degree 1
A plane drawing
A plane drawing of a graph G is a drawing of G in the plane, with no edges crossing
Complement of a graph
All of the edges that were not in the original graph
Define Euler trail
An Euler trail in a multigraph G is a trail that uses every edge of G and visits every vertex of G
Define a Euler's cycle
An Euler's cycle in a multigraph G is a cycle that uses every edge of G and visits every vertex of G. It must end where it starts.
Incident
An edge is incident to a vertex if that vertex is an endpoint of the edge
In order to have an Eulerian cycle...
G must be connected and every vertex of G must have an even degree.
How can we show a graph is bipartite?
Coloring the vertices. If the graph is bipartite, we can color the vertices with two colors so that no two adjacent vertices are the same color
Pruning lemma
If T is a tree and v is a leaf of T, then T^I = T-v is also a tree
In any graph, the number of vertices of ___ degrees is ____.
In any graph, the number of vertices of odd degrees is even.
K4 is a subgraph of_____
K5
What subgraphs can we use to prove something is not planar?
K5 or K3,3
In general, on n vertices, the maximum number of edges is
N choose 2
Real life example of Euler trails/cycles
Snowplow
Complete graph Kn=(V,E) has how many edges?
The complete graph Kn=(V,E) has |V|=n and E is the set of all possible edges, so |E|=(n choose 2)
Degree
The degree of a vertex is the # edges incident to it
The sum of the degrees is....
The sum of the degrees is 2x the # of edges
Real life example of Hamiltonian trails/cycles
Traveling salesman
True or false: Every circuit is a cycle but not every cycle is a circuit
True
Isomorphic graphs
Two graphs are isomorphic if they are "the same". This means there must exist a one-to-one and onto relationship for the vertices
Adjacent
Two vertices are adjacent if they are joined by an edge
What is the value of the pruning lemma?
When proving results about trees, we often prove by induction on |V| or |E| by removing a leaf.
Is K4 planar?
Yes, K4 is planar
To show graphs are isomorphic, we must exhibit
a one-to-one onto relationship for the vertices and equivalence with the edges
Define Bridge
an edge that when removed would create two islands
In a wheel, if the number of vertices on the rim are odd, what type of coloring will we have?
an even # coloring
A coloring of a graph is an....
assignment of a color to each vertex s.t. adjacent vertices get different colors
A tree is a graph that is ______ and has ______
connected, no cycles
If G is a graph with n>=3 vertices and if _________for all non adjacent vertices, then G ____ a Hamiltonian circuit.
deg(u) + deg(v) >= n........has
Euler's Polyhedral Formula
for any plane drawing of a connected graph with |V| vertices, |E| edges and |R| regions, |V|-|E|+|R|=2
Every complete graph has a _____ circuit
hamiltonian circuit
Define trail
has all distinct edges but can repeat vertices
Define path
has all distinct vertices
Circuit
has no repeated edges and no repeated vertices (except as we close)
How many edges in a tree?
|E| = |V| - 1