Graph theory
Chapter 1: Introduction
Definitions:
Order
Size
Induced subgraph
Number of vertices
Number of edges
A subgraph F of a graph G whenever u and v are vertices of F and uv is an
edge in G, then it is an edge in F.
Trail
Walk with no edge traversed more than once.
Path
Walk with no vertice traversed more than once.
Circuit
Closed trail with length 3 or more.
Geodesic
A path from u to v with length d(u,v).
Diameter
The greatest distance between any two vertices in a graph.
Complete
A graph is complete if every two distinct vertices of G are adjacent.
Complement of graph The complement has every edge graph G does not have
Bipartite graph
A graph of 2 components, each component is a partite set
Complete bipartite
If every vertice of a component is connected to the vertices of the other
component
k-partite graph
A graph of k components
Join
The union between a graph G en H where all vertices are connected to the
vertices of the other graph
Cartesian product
2 graphs multiplied
Theorems:
Theorem 1.6:
Theorem 1.7:
Theorem 1.8:
Theorem 1.9:
Theorem 1.10:
Theorem 1.11:
Theorem 1.12:
If a graph G contains a u-v walk of length l, then G contains a u-v path of length
at most l
Let R be the relation defined on the vertex set of a praph G by u R v, where u,v є
V(G) such that u R v and u R w. Hence G contains a u-v path P’ and a v-w path P’’.
As we have seen earlier, following P’ by P’’ produces a u-w walk W. By Theorem
1.6, G contains a u-w path and so u R w.
Let G be a graph of order 3 or more. If G contains two distinct vertices u and v
such G-u and G-v are connected, then G itself is connected
If G is a connected graph of order 3 or more, then G contains two distinct
vertices u and v such that G-u and G-v are connected.
Let G be a graph of order 3 or more. Then G is connected if and only if G
contains two distinct vertices u and v such that G-u and G-v are connected
If G is a disconnected graph, then the complement of G is connected
A non-trivial graph G is a bipartite graph if and only if G contains no odd cycles
Chapter 2: Degrees
Definitions:
Degree of a vertex
Neighbors
Neighborhood
Even (odd) vertex
Sharpness
r-regular graph
Petersen graph
Number of edges incident with an edge
Two adjacent vertices
All neighbors
Vertex with even(odd) degree
The sharpness of a theory means that some
statement could is not the bound of the statement,
for example bla ≤ n-1 is true, but is bla≤ n-2 also true.
A graph where every vertex has the degree r
A 3-regular graph of degree 10(see picture)
Harary graphs
Degree sequence
Graphical
Adjacency matrix
r-regular graphs Hr,n with order n and the properties of theorem 2.6
All degrees of the vertices of a graph sorted in a sequence.
A degree sequence is graphical if a graph can be constructed from it
Matrix with n columns and n rows with where
Aij =
Incidence matrix
n x m matrix where
Bij =
Equal walks
Two walks are equal if they are equal term to term
Theorems:
Theorem 2.1
If G is a graph of size m, then
Corollary 2.3
Theorem 2.4
Every graph has an even number of odd vertices
Let G be a graph of order n. If
Corollary 2.5
Theorem 2.6
Theorem 2.7
Theorem 2.10
Theorem 2.13
for every two non-adjacent vertices u and v of G, then G is connected and
diam(G) ≤ 2.
If G is a graph of order n with δ(G) ≥ (n-1)/2, then G is connected
Let r and n be integers with 0 ≤ r ≤ n-1. There exists an r-regular graph of
order n if and only if at least one of r and n are even.
For every graph G and every integer r ≥ Δ(G), there exists an r-regular graph H
containing G as an induced subgraph.
A non-increasing sequence s : d1, d2,…., dn (n≥2) of non-negative integers,
where d1≥1, is graphical if and only if the sequence
s1 : d2 – 1, d3 – 1, dd1+1 -1, dd1 + 2,…dn
is graphical.
Let G be a graph with vertex set V(G) = {v1, v2, … , vn} and adjacency matrix A
= [aij]. Then the entry aij(k) in row i and column j of Ak is the number of distinct
vi – vj walks of length k in G.
Chapter 4: Trees
Definitions:
Bridge
An edge uv is a brige when G – uv is disconnected, while G is connected
Tree
A tree is an acyclic graph
Caterpillar
A tree of order 3 or more
Spine
A caterpillar without end-vertices
Forest
A cyclic graph where each components is a tree
Isomorphic Graphs
Graphs that are structurally equivalent
Spanning subgraph
Subgraph H is a spanning subgraph of G if it contains every vertex of G.
Spanning tree
If H is a tree it is called a spanning tree
Cost/weight
Edge can have certain costs or weights
Weight of a graph
The weight of a graph is all weights summed up
Minimum spanning tree: Spanning tree with minimum weight
Algorithms for mst-problem:
- Kruskal’s algorithm:
For a connected weighted graph G, a spanning tree T of G constructed follows: For the first
edge e1 of T, we select any edge of G of minimum weight and for the second edge e2 of T, we
select any remaining edge of minimum weight. For the third edge e3 of T, we choose any
-
remaining edge of G of minimum weight that does not produce a cycle with the previously
selected edges. We continue in this manner until a spanning tree is produced.
Prim’s algorithm:
For a connected weighted graph G, a spanning tree T of is constructed as follows: For an
arbitrary vertex u for G, an edge minimum weight incident with u is selected as first edge e1
of T. For subsequent edges e2, e3,..,en-1 , we select an edge of minimum weight among those
edges having exactly one of its vertices incident with an edge already selected.
Theorems:
Theorem 4.1
Theorem 4.2
Theorem 4.3
Theorem 4.4
Corollary 4.6
Theorem 4.7
Theorem 4.8
Theorem 4.9
Theorem 4.10
Theorem 4.11
An edge e of a graph G is a bridge if and only if e lies on no cycle of G
A graph G is a tree if and only if every two vertices of G are connected by a
unique path
Every nontrivial tree has at least two end-vertices
Every tree of order n has size n-1
Every forest of order n with k components has size n – k
The size of every connected graph of order n is at least n-1
Let G be a graph of order n and size m. If G satisfies any two properties:
1. G is connected,
2. G is acyclic,
3. M = n-1,
then G is a tree.
Let T be a tree of order k. If G is a graph with δ(G) ≥ k -1, then T is isomorphic
to some subgraph of G
Every connected graph contains a spanning tree
Kruskal’s algorithm produces a minimum spanning tree in a connected
weighted graph.
Chapter 6 Traversability
Definitions:
Eulerian circuit
Eulerian graph
Eulerian trail
Hamiltonian cycles
Hamiltonian graph
Hamiltonian path
Closure; C(g)
Cut-vertex
Theorems:
Theorem 6.1
Corollary 6.2
Theorem 6.3
Theorem 6.4
A circuit C in a graph G which contains every edge exactly once
A graph with an eulerian circuit
A trail that contains all edges in a graph
A cycle that contains all vertices in a graph
A graph that contains a Hamiltonian cycle
A path that contains all vertices in a graph
A graph G with order n, the closure of G is the graph obtained by recursively
joining pairs of non-adjacent, whose degree sum is at least n, until no such
pair remains.
A vertex in a connected graph G if G – v is disconnected
A nontrivial connected graph G is Eulerian if and only if every vertex has an
even degree.
A connected graph G contains an Eulerian trail if and only if exactly two
vertices of G have odd degrees. Furthermore, each Eulerian trail of G begins
or ends at one of these odd vertices and ends at the other
Let G and H be nontrivial connected graphs. Then G x H is Eulerian if and only
if both G and H are Eulerian or every vertex of G and H is odd.
The Petersen graph is not Hamiltonian
Theorem 6.5
Theorem 6.6
Corollary 6.7
Theorem 6.8
Theorem 6.9
Corollary 6.10
Theorem 6.11
If G is a Hamiltonian graph, then for every nonempty proper set S of vertices
of G,
k(G – S) ≤ |S|. (k(G) is the number of components in G)
Let G be a graph of order n ≥ 3. If
Deg u + deg v ≥ n
for each pair u, v of nonadjacent vertices of G, then G is Hamiltonian.
Let G be a graph of order n ≥ 3. If deg v ≥ n/2 for each vertex v of G, the G is
Hamiltonian.
Let u and v be nonadjacent vertices in a graph G of order n such that deg u +
deg v ≥ n. Then G + uv is Hamiltonian if and only if G is Hamiltonian.
A graph is Hamiltonian if and only if its closure is Hamiltonian
If G is a graph of order at least 3 such that C(G) is complete, then G is
Hamiltonian
Let G be a graph of order n ≥ 3. If for every integer j with 1 ≤ j ≤ , the
number of vertices of G with degree at most j is less than j, then G is
Hamiltonian.
Chapter 8 Matchings and factorization
Definitions:
Independent
Matching
Neighborly
Cover
Edge cover
Theorems:
Theorem 8.3
Theorem 8.4
Theorem 8.5
A set of edges(vertices) is independent if no two edges(vertices) are adjacent
Let G be a bipartite graph with partite sets U and W, where r = |U| ≤ |W|. A
matching in G is therefore a set M = {e1,e2,…,ek} of edges, where ei = uiwi for 1
≤ i ≤ k such that u1, u2, … uk are k distinct vertices of U and w1, w2, … , wk are k
distinct vertices of W.
Given neighborhood N(X) a set U is neighborly if |N(X)|≥ |X| where for every
nonempty subset X of U and N(X) the union of all neighbors x є X. If the
neighborhood of a subset of X of U is bigger or equal to the subset X for every
subset of X of U, then U is neighborly.
A vertex and an incident edge cover each other
A graph without isolated vertices is a set of edges of G that covers all vertices
of G
α(G)
Vertex covering number
β (G)
Vertex independence number
α1(G)
Edge covering number
β 1(G)
Edge independence number
Minimum number of vertices that
cover all edges of G
Maximum number of vertices, no
two which are adjacent
Minimum number of edges that
cover all vertices of G
Maximum number of edges, no
two which are adjacent
Let G be a bipartite graph with partite sets U and W such that r = |U| ≤ |W|.
Then G contains a matching of cardinality r if and only if U is neighborly.
A collection {S1, S2, … , Sn} of a nonempty finite sets has a system of distinct
representatives if and only if for each integer k with 1 ≤ k ≤ n, the union of
any k of these sets contains at least k elements.
In a collection of r women and s men, where 1 ≤ r ≤ s, a total of r marriages
(Marriage Theorem)
Theorem 8.6
Theorem 8.7
Theorem 8.8
8.14
between acquainted couples is possible if and only if for each integer k with
1 ≤ k ≤ r, every subset of k women is collectively acquainted with at least k
men.
Every r-regular bipartite graph (r ≥ 1) has a perfect matching
For every graph G of order n containing no isolated vertices,
α1(G) + β 1(G) = n.
For every graph G of order n containing no isolated vertices,
α(G) + β (G) = n.
A graph G without isolated vertices contains a has a perfect a perfect
matching if and only if α1(G) = β1(G).
For 8.2 see handout Hungarian method
Chapter 9 Planarity
Definitions:
Planar graph
Plane graph
Regions
Exterior region
Boundary of a region
Maximal planar
Subdivision
Theorems:
Theorem 9.1
Theorem 9.2
Corollary 9.3
Corollary 9.4
Theorem 9.5
Theorem 9.7
A graph that can we drown without two edges crossing each other
A graph is drawn in a plane without no two edges of G cross
The connected pieces of a plane graph
The unbounded part of every plane graph
The subgraph of all vertices and edges that are
incident with the region
G is planar, but the addition of an edge between
two nonadjacent vertices of G makes it nonplanar
One or more vertices of degree 2 are inserted
into one or more edges of G(see picture)
If G is a connected plane graph of order n, size m, and having r regions, then
n–m+r=2
If G is a planar graph of order n ≥ 3 and size m, then
m ≤ 3n – 6.
Every planar graph contains a vertex of
degree 5 or less
The complete graph K5 is nonplanar(see
picture)
The graph K3,3 is nonplanar (see picture)
A graph G is planar if and only if G does not contain K5, K3,3 or a subdivision of
K5 or K3,3 as a subgraph.
Chapter 10 Coloring
Definitions:
Dual
Coloring
Chromatic number
k-coloring
k-coloring
If you represent a map by a graph it is called a dual
Assigning colors to regions, such that every adjacent vertex has a separate
color
Smallest number of colors needed for coloring denoted as χ(G)
If it is possible to color G from a set of k colors, then G is said to be kcolorable
A coloring that uses k colors
Color classes
Clique
Clique number
Shadow graph
Theorems:
Theorem 10.1
Theorem 10.2
Theorem 10.5
Theorem 10.7
Theorem 10.8
Theorem 10.9
Theorem 10.10
if G is k-chromatic then it is possible to divide G in k independent sets, these
are called color classes
A complete subgraph of graph G.
The order of the largest clique, denoted as ω(G)
Obtained from graph G by adding, for each vertex v of G, a new vertex v’,
called the shadow vertex of v, and joining v’ to the neighbors of v in G.
The chromatic number of every planar graph is at most 4(The Four Color
Theorem)
A graph G has chromatic number 2 if and only if G is a nonempty bipartite
graph
For every graph G of order n: χ(G) ≥ ω(G) and χ(G) ≥
For every graph:
χ(G) ≤ 1 + Δ(G)
For every connected graph G that is not an odd cycle or a complete graph
χ(G) ≤ Δ(G)
For every graph G,
χ(G) ≤ 1 + max{δ(H)},
where the maximum is taken over all induced subgraphs H of G.
For every integer k ≥ 3, there exists a triangle-free graph with chromatic
number k.