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