Graphs
connected
for all u,v in V, there exists a path u to v (d(u,v) < infinity)
Card(E) <= 3Card(V)-6 if G has no cycles of length 3, then Card(E) <= 2Card(V)-4
Let G=(V,E) be connected planar with more than 2 vertices. Give non-planarity conditions
length
the number of edges in a path
connected component
All v reachable from u (there exists a path from u to v)
G is connected and is drawn in a plane with no edges crossing and r regions
Card(V) - Card(E) + r = 2
G has exactly two odd-degree vertices (which are the start and end vertices)
G has a Eulerian path (but not cycle) iff
all vertices have even degrees
G is a Eulerian cycle iff
Card(E) = Card(V)-1 and G is connected For all u,w in V, there exists a unique path from u to w G is connected but removing any edge disconnects G G has no simple cycles, but adding an edge creates a simple cycle
G is a tree iff... (4 equivalent statements)
X(G) = 2
G is bipartite iff X(G) = ...
G has no odd-length cycles
G is bipartite iff...
G does not contain a subgraph obtained from K5, K3,3, by a sequence of subdivisions
G is planar iff
bipartite
G=(V,E) is _________ if there is a partition V=(V1)U(V2) such that all edges have one endpoint in V1, other in V2
colors
G=(V,E) is an arrangement of "______" (labels) to the vertices, such that adjacent vertices do not have the same _____
simple finite graph
G=(V,E) such that V is a set of finite vertices and E is a subset of (v1, v2) in V (not equal to each other) called edges
C(n,2)
How many edges are in a complete graph?
C(n,2)
How many edges does a complete graph have?
then G is non-planar
If G has a non-planar subgraph, then...
X(G) <= 6
If G is planar, then X(G) =
non planar
K5, K3,3 are ___-______
Card(E) = mn
Km,n has Card(E) =
complete bipartite graph
Km,n is a graph with V=(V1)U(V2), Card(V1)=m and Card(V2)=n, and for all u in V1, v in V2, there exists an edge (u,v) in E (no other edge)
complete graph
Kn is the ________ graph with n vertices where all pairs of vertices are connected by an edge
2Card(E)
The sum of all degrees in a simple finite graph is equal to:
dg + 1 (Card(degrees of G) + 1)
The upperbound of colors you can use in G is:
chromatic number
X(G) is the least number of colors that a graph G can be colored with
tree
a connected graph with no simple cycles
Hamiltonian cycle/path
a cycle (or path) which visits every vertex exactly once
planar embedding
a drawing of graph G in the plane (defines regions)
n-dimensional Hamming cube
a graph Q(n) where V={0,1}^n = binary strings of length n; (s1,s2) is an edge in E iff s1, s2 differ in exactly one bit
planar
a graph is ______ if you can draw it in the plane without the edges crossing
subgraph
a graph obtained by erasing some vertices and edges
simple
a path is ______ if no vertex is visited more than once a cycle is ______ if the only repeated vertex is v1=vn and the length is at least 3
trail
a path that doesn't repeat an edge
Eulerian path/trail
a path that uses every edge exactly once
cycle
a path where v1=vn (returns to starting vertex)
path
a sequence of vertices from u to v
two leaves
a tree with at least one edge has how many leaves?
leaf
a vertex of degree 1
all bipartite
all trees are _________
neighbors
all u are _________ of v if (v,u) is in E
<= 5
every planar graph has a vertex of degree
n-regular (you can change exactly one out of each n bits, giving (n neighbors)-(deg(v))=n for all vertices)
hypercube Qn is _-_______
Euler characteristic
if G is drawn with r regions, the _____ ______________ is Card(V)-Card(E)+r and is always 2
odd-length simple cycle
if a graph has an odd-length cycle, then it has...
incident
if e=(v1,v2) is an edge, its endpoints are v1, v2 and it is said to be ________ on v1, v2
k-regular
if for all v in V, deg(v)=k
there is also a simple path from u to v
if there is a path from u to v, then...
even
in a simple, finite graph, the number of odd-degree vertices is ____
subdivision
obtained by removing edge e, adding a new vertex w, and two new edges (u,w), (w,v)
distance
the length of the shortest path from u to v (d(u,v)) (equal to infinity if it dne)
degree
the number of edges incident on a vertex
adjacent
u, v are adjacent if (u,v) is in E