Graph Theory 4901 - All

If a graph G has a Hamiltonian circuit, then G has a subgraph H with the following properties:

(1) H contains every edge of G (2) H is connected (3) H has the same number of edges as vertices (4)Every vertex of H has degree 2

Prove that a map ϕ is an isomorphism of G onto H

(1) the two graphs have the same number of vertices and edges, (2) aϕ is a vertex of H whenever a is a vertex of G, and (3) aϕ and bϕ are adjacent vertices in H whenever a and b are adjacent in G.

All Hamiltonian graphs are

2-connected

Theorem 8.6 - If a connected planar graph has v vertices, e edges, and f faces, and no component has fewer than three vertices, then

3f≤2e

Havel & Hakimi

A collection S={d_0,d_1,...,d_(v-1)} of v vertices with d_0≥ d_1≥ ...≥ d_(v-1), where d_0≥1 and v≥2, is graphical iff the collection S={d_1-1,...,d_(d_0)-1, d_(d_0+1), ...,d_(v-1)} is graphical.

graphically valid

A collection of v nonnegative integers that represent degrees of a connected graph.

Bipartite/Odd Cycle - Koenig

A graph is bipartite iff it contains no cycle of odd length

If G is n-colorable, then

G is an n-partite graph with n=χ(G)

Theorem 8.3 - G is planar iff

G is homeomorphic to a graph containing no subgraph homeomorphic to K_5 or K_(3,3)

Theorem 7.1 - χ(G)=2 iff

G is not null and G contains no cycles of odd length

Weighted Distance Properties

If G is a connected graph, then W(x,y) has the following properties: (1) W(x,y)=0 iff x=y; (2) W(x,y)=W(y,x); (3) W(x,y)+W(y,z) ≥ W(x,z) for all vertices x,y,z in G.

Pigeonhole Principle

If a set consisting of more than kn objects is partitioned into n classes, then some class receives more than k objects

Theorem 2.1 - Walk/ Path

If there is a walk from vertex y to vertex z in the graph G, where y is not equal to z, then there is a path in G with first vertex y and last vertex z.

Two Circuit Lemmas

If vertices v and w are part of a circuit in G and one edge is removed from the circuit, then there exists a trail from v to w in G. If G is connected and G contains a circuit, then an edge of the circuit can be removed without disconnecting G.

Hand Shaking Lemma Corollary

In any graph or multigraph, the number of vertices of odd degree is even. In particular, a regular graph of odd degree has an even number of vertices.

Hand Shaking Lemma

In any graph or multigraph, the sum of the degrees of the vertices equals twice the number of edges; ∑ _(x ϵ V)〖d(x)〗=2e

Sorted Edges Method

List all edges in ascending order. Choose the cheapest edge in order as long as adding the edge does not result in a vertex of degree 3, and the collection of edges contains no cycle of length less than v.

Nearest Neighbor Method

Start at one vertex x, and choose the edge incident with x whose cost is least, say that edge is xy. Then choose the edge incident with y that is not xy and of least cost. Different solutions may come from different choices of initial vertex x.

induced subgraph

Subgraph created by deleting vertices (and therefore some edges); a subgraph consisting of U and all the edges of G that join two vertices of U, which is a subset of vertices of G

complete bipartite graph

a bipartite graph in which every vertex in one partition is adjacent to all of the vertices in the other partition and vice versa

Euler circuit

a circuit that contains every edge in a multigraph

maximal clique

a clique in which no vertex outside of the subgraph is adjacent to all vertices of the subgraph; the clique of the largest possible size

cycle

a closed walk of length n, n≥3, in which the vertices x_0, x_1,..., x_(n-1) are all different; C_n

clique

a complete subgraph; a subgraph in which every vertex is adjacent to every other vertex in the subgraph.

tree

a connected acyclic graph

tree

a connected graph that contains no cycles; a connected acyclic graph

Hamilton cycle

a cycle that passes through every vertex in a graph

simple graph

a graph having no loops or multiple edges

cubic

a graph in which all of the vertices have degree 3

regular

a graph in which all of the vertices have the same degree;

connected graph

a graph in which every pair of nonadjacent vertices is joined by a walk

clique structure

a graph of the cliques of a graph; the vertices represent the maximal cliques of G and the edges show when a clique has a common vertex with another clique

Eulerian graph

a graph that contains an Eulerian walk

acyclic graph

a graph that contains no cycles

subgraph

a graph that contains the edges and vertices of another graph; we say that G contains H or H is contained in G; V(H)⊆V(G), E(H)⊆E(G), H⊆G

planar graph

a graph that has a planar representation; or one in which v(G)=0

disconnected graph

a graph that has at least one pair of vertices not joined by a path; a graph whose vertex set can be partitioned into two subsets that have no common element or no edge with one endpoint in one subset and the other endpoint in the other subset

connected graph

a graph that is not disconnected; every partition of a graph's vertex set into two nonempty sets results in at least one edge with one endpoint in each set

multigraph

a graph that may have more than one edge corresponding to a pair of vertices

directed graph

a graph whose edges have direction; also called a digraph

bipartite graph

a graph whose vertex set can be partitioned into two subsets so that every edge has one end in one subset and the other end in the other subset

Hamiltonian graph

a graph with a Hamilton cycle

The deletion of a bridge from a connected graph yields

a graph with exactly two components

k-coloring

a labeling f: V(G)->{1,...,k}

component

a maximal connected subgraph

cutset

a minimal collection of edges whose deletion disconnects G

Eulerization of G

a multigraph with a closed Euler walk, that is formed from G by duplicating some edges.

isomorphism

a one-to-one map ϕ from V(G) onto V(H) with the property that a and b are adjacent vertices in G iff aϕ and bϕ are adjacent vertices in H

Kempe chains

a path on which the colors alternate between two specified colors

proper path

a path other than P_1

weight

a positive function associated with each edge xy; denoted w(x,y)

planar

a representation that contains no crossings

independent sets

a set of edges such that no two of its elements are adjacent; also called a stable set

Euler walk

a simple walk that contains every edge in a multigraph

spanning tree

a spanning subgraph that is a tree when considered as a graph on its own right

minimal spanning trees

a spanning tree such that the weight of the tree is minimal

spanning subgraph

a subgraph of G that has the same vertex set as G; it is obtained by deleting edges only.

proper subgraph

a subgraph that does not equal the graph

proper tree

a tree other than K_1

leaf

a vertex of degree 1 in a tree, together with the edge incident with it

pendant

a vertex which has exactly one edge as an endpoint; d(x)=1

isolated vertex

a vertex which has no edges; d(x)=0

cutpoint

a vertex x in G such that G-x contains more components than G does; in other words, if G is connected, G-x is disconnected; also known as vertex cut or separating set

path

a walk in which no vertex is repeated; P_n

circuit

a walk in which the first and last vertices, x_0 and x_n, are the same; also called a closed walk

forests

acyclic graphs

trail

also simple walk; a walk in which no edge is repeated

face

an area of the plane entirely surrounded by edges of the graph and containing no edge

bridge

an edge whose deletion increases the number of components

loop

an edge whose endpoints are equal

automorphism

an isomorphism from a graph G to itself; it reflects the symmetries of the graph

Corollary 8.8 - Every planar graph has

at least one vertex of degree smaller than 6

Corollary 4.3 - Every tree other than K_1 has

at least two leaves

Theorem 8.10 - Every planar graph can be

colored using at most four colors

greedy coloring

coloring vertices in the order v_1, ..., v_n of V(G), assigning to v_i the smallest-indexed color not already used

graph

consists of a finite set V of vertices together with a set E of edges and an incidence function that associates with each edge, a pair of vertices; denoted usually by G

spans

contains every vertex

eliding a vertex

deleting a vertex of degree 2 and joining the two vertices adjacent to it

representations

different drawings of the same graph

arcs

directed edges

parallel edges

edges that have the same pair of endpoints

Theorem 4.1 - A connected graph is a tree iff

every edge is a bridge

G is 2-connected iff

every two vertices of G lie on a cycle

Theorem 8.7 - If a connected planar graph has v vertices and e edges, where v≥3, then

e≤3v-6

Whitney - Connectivity & Degree

for any graph G, κ(G) ≤ κ'(G) ≤ δ(G)

complement of G in H

formed by deleting all edges of G from H; H-G

homeomorphic

graphs that can be obtained from each other by dividing an edge or eliding a vertex

class 1

graphs that satisfy χ'(G)=Δ(G)

class 2

graphs that satisfy χ'(G)=Δ(G)+1

Theorem 4.5 - Every connected graph G

has a spanning tree

G is Hamiltonian

if G is a graph with v vertices, such that v≥3, and either (1) d(x)+d(y)≥v whenever x and y are nonadjacent vertices of G or (2) every vertex has degree at least v/2

A bipartite graph with vertex sets of size m and n can contain a Hamilton path

if m and n differ by at most 1

A bipartite graph with vertex sets of size m and n can contain a Hamilton cycle

if m=n

walk

in a graph G, a finite sequence of vertices x_0, x_1, ..., x_n and edges a_1, a_2, ..., a_n of G: x_0, a_1, x_1, a_2, ..., a_n, x_n, where the endpoints of a_i are x_(i-1) and x_i for each i.

dividing an edge

inserting a new vertex of degree 2 into the middle of an edge

independence number

is the number of elements in the largest independent set

chromatic number

is the smallest integer n such that G has an n-coloring; denoted by χ(G)

An edge xy in a connected graph is a bridge iff

it belongs to no cycle in the graph.

Theorem 4.2 - A finite connected graph G with v vertices is a tree iff

it has exactly v-1 edges

There is one and only one path

joining any two vertices of a tree

If G is a cycle of length n, then

n spanning trees can be constructed by deleting one edge and τ(G)=n

colorings

partitions of the vertex set, or edge set, of a graph, into subsets such that no two elements of the same subset are adjacent; labelings

Hamilton path

path that includes each vertex only once

complete graph

the graph formed by joining each pair of vertices in S, denoted K_s

Δ(G)

the largest of all degrees of vertices of G

distance

the length of the shortest path joining x to y; D(x,y)=0

weighted distance

the minimum among the weights of all of the paths between two connected vertices, x and y; denoted W(x,y)

crossing number of G

the minimum number of crossings in any representation of G; v(G)

edge connectivity

the minimum number of edges whose removal disconnects G; the smallest cutset in G

Eulerization number

the minimum number of new edges of an Eulerization; eu(G)

incidence matrix

the n x m matrix, where a_ve is the number of times that vertex v and edge e are incident

adjacency matrix

the n x n matrix where a_uv is the number of edges joining vertices u and v

crossing number of a representation

the number of different pairs of edges that cross

length

the number of edges in a walk

degree

the number of edges that have a specific vertex as an endpoint

τ(G)

the number of spanning trees of a graph

exterior face

the plane outside the representation

Corollary 8.5 - All plane representations of the same connected planar graphs have

the same number of faces

complement of G

the set of all edges of a complete graph of G that are not edges of G

neighborhood

the set of all vertices that are adjacent to a particular vertex

clique number

the size of the largest clique; denoted by ω(G)

edge chromatic number

the smallest number k such that G is k-edge colorable; also called the chromatic index of G

connectivity

the smallest number of vertices whose removal from G results in either a disconnected graph or a single isolated vertex; κ(G)

δ(G)

the smallest of all degrees of vertices of G

colors

the subsets of a coloring; labels

color class

the vertices receiving a particular label or a color i

If a multigraph has no odd vertices,

then it has an Euler walk starting from any given point and finishing at that point

If a multigraph has exactly two odd vertices,

then it has an Euler walk whose start and finish are the odd vertices

If a multigraph has more than two odd vertices,

then it has no Euler walk

If any vertex of degree 2 or any edge is deleted from a tree

then the resulting graph is not connected

Theorem 8.4 - Suppose that a plane representation of the connected planar graph G has v vertices, e edges, and f faces, then

v-e+f=2

neighbors

vertices that are adjacent

map

what is usually meant by a map of a continent or country with no disconnected parts

incident

when a vertex is an endpoint of an edge; describes the relationship between an edge and a vertex

adjacent

when two vertices are the endpoints of one edge in a graph; when two edges share a common endpoint

Triangle Inequality

|a| + |b| ≥ |a+b|; d(x,y) + d(y,z) ≥ d(x,z)

Theorem 7.13 - For any graph G,

Δ(G)≤χ'(G)≤Δ(G)+1

k-connected

κ(G)≥k where k is a positive integer; a graph that has no vertex cut with fewer than k vertices

Theorem 7.11 - If G is a bipartite graph, then

χ'(G)=Δ(G)

It requires d(x) colors to color the edges at x, so

χ'(G)≥Δ(G)

Theorem 7.5 - If G is a connected graph other than a complete graph or an odd cycle, then

χ(G)≤Δ(G)

Theorem 7.2 - If Δ(G) is the maximum degree in G, then

χ(G)≤Δ(G)+1