1 Graph Theory Overview
1.1 Development History
The first stage:
1736: Euler published the first article on graph theory to study the Seven Bridges of Konigsberg, known as the father of graph theory
1750: first a theorem is proposed topology, polyhedron Euler's formula: V-E + F = 2
The second stage (19 to 20 century):
1852: Francis Guthrie made a four-color problem
1856: Thomas P. Kirkman & William R.Hamilton study the hamiltonian
1878: Alfred Kempe given given the four-color theorem prover
1890: Heawood (Heawood) to overthrow the existing four-color theorem prover
1891: Peterson (Petersen Denmark) gives first papers on theoretical knowledge of graph theory
1936: Geni Ge (Dénes Kőnig Hungarian), wrote the first book on graph theory monograph "theory of finite and infinite graphs chart" graph theory became an independent discipline
The third stage (Modern graph theory):
1941: FP Ramsey create Extremal graph theory
1959: Erd˝os and Rényi FIG introduce random theory (probability of existence of edges p)
1976: Kenneth Appel & Wolfgang Haken use a computer the ultimate proof that the four-color problem
1.2 Reference materials
Graph Theory with Application - J.A. Bondy and U.S.R. Murty, Elsevier, 1976
"Graph Theory and Its Applications" classic textbook, Wu Wang name translation, electronic version
Graph theory - J.A. Bondy and U.S.R. Murty, Springer, 2008
"Graph theory" GTM244, it can be considered "Graph Theory with Application" second edition, recommended textbooks
Graph Theory, 5th - Reinhard Diestel, Springer, 2017
"Graph theory" GTM173, electronic version
Introduction to Graph Theory, 2nd- Douglas B. West, 2017
Introductory textbooks
2, the initial knowledge graph
(Note: general considerations simple graph (no graph loops or multiple edges), and the order of 2 or greater)
Figure 2.1 Rule
Definition:
FIG irregular irregular: the same for all vertices
FIG irregular almost almost irregular: the same only one pair of vertices
Theorem:
1) Irregular FIG absent
2) there is exactly the same order of nearly two irregular FIG and FIG complement each other (the same vertices, edges together are complete graph)
3) For any positive integer n is the maximum value of the set, the presence of order n + 1 in FIG, it is exactly equal to the integer number of vertices
(Conclusion The above does not apply and multiple FIGS weighted graph)
2.2 Regular Graphs
Definition:
Regular Graphs r-regular: for all vertices is r
(0-regular simply connected to a single point, 1-regular communication is a diagram illustrating a single-side, 2-regular communication is a single ring, 3-regular cube FIG single communication is referred to)
Theorem:
FIG r regular n order exists, as long as r, n is not an odd number and r <= n-1
Common Regular Graphs:
Kn: n order complete graph, r = n-1
Cn: n (n> = 3) order circle, r = 2
Qn: 2 ^ n hypercube of order (n-cube), r = n
Kr, r: 2r order bipartite
2.3 bipartite FIG.
Definition：
Bipartite graph (bipartite graph): vertices are divided into two sets, all edges connected only between two sets.
Theorem：
G is a bipartite graph G no odd cycle <=>
2.4 subgraph
FIG G, subgraph (subgraph) H
subgraph ---> spanning subgraph
---> induced subgraph ---> vertex-delete subgraph
spanning subgraph: subgraph, G and H in the same vertices
induced subgraph: FIG elicitor, H = G [S] (by removing one or more of the vertices from the graph)
vertex-delete subgraph: subgraph to a vertex, a vertex is removed from FIG.
Theorem：
Arbitrary graph can be expressed as subgraph a regular graph
FIG isomorphism and reconstruction
Go to all the vertices of a given subgraph of a graph G, is able to reconstruct the unique graph G (only the isomorphism)? The problem unsolved
2.5 distance
Definition
FIG communication (connected), do not communicate FIG (disconnected): composed of a plurality of communication branches (component).
Cut points (cut-vertex): Gv ratio G has more connected components
Bridge (bridge): Ge ratio G has a plurality of communication with a branch
Theorem:
FIG communication G, e is e bridge <=> not exist any vertex u of a circle <=> G, v, so that the path through any path uv e
FIG communication G, w is a cut point is present <=> vertex u, v, so that the path through any path uv w
Definition
The distance between the vertices xy (Distance): xy minimum length of all paths.
Vertex v eccentricity (eccentricity): v farthest distance from the vertex u v, u is called a vertex v centrifuged (eccentric vertix)
Central apex: G in heart rate from the minimum vertices
2.6 Tree
Definition：
Tree: FIG communication ring does not contain G
Theorem：
G is a tree <=> any two vertices of G has one and only one communication path
tree of order n has the n-1 sides
Add any one side in the G, it will form a loop.
Removing any of its sides, is no longer in communication.
Any two vertices of G can be unique within the communication path.
Definition：
Leaf (leaf): Moderate tree node 1
Theorem：
Tree has at least two nodes
Each node of the tree can be used as a root
Definition：
Spanning Tree (spanning graph): generating a subgraph of G T is the tree is called a spanning tree T of G
Minimum Spanning Tree: weighted sum of the minimum spanning tree edge
Crooks may be generated using the minimum spanning tree algorithm
Crooks algorithm: Select the right edge of the smallest, but does not guarantee the formation lap
3, FIG traversal
Definition：
For a sequence of vertices (v1, v2, ..., vk)
Passage (walk) ---> path (path) ---> ring (Circle)
| |
v v
---> trace (trail) ---> loop (circuit)
Closed meanings: v1 = vk
walk: All side v_i-v_i + 1 is a graph
path: all vertices not repeated, closed path for the circle
trail: not all edges of a reclosable trail circuit.
注：path -> trail, circle -> circuit
3.1 Euler problem (traversing all edges can not be repeated, Eulerian path)
Eulerian cycle (Eulerian circuit): G contains all edges of the circuit
Euler trace (Eulerian trail or Eulerian tour): G contains all edges of trace
G is communicating Euler or Euler: G contains an Euler
Theorem：
G is a connected graph of Euler <=> each vertex is an even number, (note: FIG multiple G may be)
Communication graph G comprising the Euler <=> trace graph G is exactly two odd vertices (Note: FIG. G may be multiple, the Euler two odd track from the start and end of vertices)
3.2 Chinese postman problem (through all the edges can be repeated)
Definition：
Euler path: through the shortest of all edges of the close path G
Euler method to find the path of: adding an odd multiple edges of the vertices, so that an even number of each vertex
Theorem：
Euler path must bridge two through G in (Note: that is to say multiple stresses replication Euler FIG bridge)
NOTE: weighted graph similar idea can be solved
3.3, Hamilton problem (through all the points can not be repeated)
Definition：
Hamiltonian cycle: After all the vertices of the ring, the ring comprising become Hamilton Hamilton FIG.
No theory can accurately determine whether a graph is hamiltonian, but there are the following conclusions
Theorem:
Let G be a graph of order n, of each vertex> = n / 2, then G is a Hamiltonian graph;
(Orr theorem) graph G of order n if any nonadjacent vertices of the pair of degrees and greater than equal to n, then G is Hamiltonian FIG;
If G is a Hamilton FIG, then removing any k vertices of G, FIG remaining connected component containing at most k;
3.4 Traveling Salesman Problem (through all the points can be repeated)
TSP (Traveling Salesman Problem), has been optimized algorithm
4, and FIG matching decomposition
4.1 match
(Study bipartite matching problem set two elements in the drawing)
Definition:
Consisting of a collection of sides adjacent to each other, called matching (matching) (matching concept can be used in all of the figures)
Distinct Representatives: n non-empty, each pick a set of elements, which may be different from n elements. Thus, there is a set of lines representative of different lines, equivalent to the corresponding bipartite graph (set of lines and a set of vertex elements bipartite graph) containing size n of match.
Theorem：
(Hall Theorem)
n non-empty lines independently represents the presence <=> distinct set of k and contains less than k elements (k = 1, ..., n).
Definition:
Maximum matching: M G is contained in the most matching edge
Perfect Match: G match covers all the vertices (a perfect match of order n of FIG size of n / 2)
Theorem:
(Tate theorem) <=> G has a perfect match for any subset of the vertex S, are k0 (GS) <= | S |, wherein kO (G) represents the number of communicating branches in the odd order G
1-regular graph has a perfect match
2-regular graph with a perfect match for each communication <=> G is an even-order branch ring
3-regular graph (FIG cube) (Peterson theorem) each comprising a bridgeless cube FIG perfect match
4.2 Regular Graphs 1- factorization
Definition：
1-n generate the subgraph is referred to as G 1 -factor FIG.
FIG edge set G may be represented as divided into a plurality 1- factor graph, known as the G 1 -factor exploded view
By definition, each factorization 1- FIG even order must be a regular r- FIG.
Note: not vice versa, e.g. Peters diagram (FIG. 10 the order 3-n) instead of l- factorization FIG.
Theorem: even order completely exploded view of FIG factor is 1;
Conjecture: If G is a regular graph of order n r- and r> = n / 2, then G is 1 factorization FIG. R can be decomposed into a 1 -factor FIG.
4.3 canonical factorization FIG 2- (ring decomposition)
Theorem：
(Peterson given proof) G is 2- <=> factor G is an exploded view of a regular r-, r is an even number;
Complete graph decomposition theorem (2012 prove, formerly Al Shiba guess)
(Bryant - Horsley - Peterson Theorem)
1, n is an odd number (n> = 3), m1 + m2 + ... + mt = n (n-1) / 2, Kn can be decomposed into Cm1, Cm2, ..., Cmt;
2, n is an even number (n> = 4), m1 + m2 + ... + mt = n (n-2) / 2, Kn-M can be decomposed into Cm1, Cm2, ..., Cmt, where M is a perfect Kn of match;
Corollary: n = an odd number, and m is divisible by n (n-1) / 2, then Kn is decomposable FIG Cm-;
Corollary then: n = odd number, Kn is an exploded view factor Hamilton (Hamilton each circle is the circle)
application:
Steiner ternary (Steiner triple systems) issues
n elements, each element of a group of three (called triples), each of the elements of a triplet falls exactly, this packet is called a ternary Steiner
Is equivalent to Kn is decomposed into a plurality of C3
5, FIG painting (Graphs, crossing points)
Definition: if G can be drawn in the plane, and the two edges do not intersect any, to be called G plan view (planar graph), the picture shows a plan view of the plane can be fitted on the plane shown in FIG obtained (planar embedding) ;
Three Houses and Three Utilities Problem can not be equivalent to a plan view K3,3
Polyhedron Euler equation: V-E + F = 2
Plan view of the polyhedron can be projected onto a plane formed polyhedron
Euler equation: G is a plan embedded FIG communication, with n vertices, m edges, R & lt regions, the n-m + r = 2
Theorem: plan G, then m <= 3n-6, where n is the order, m is the number of edges
Corollary: Each vertex contains a plan view of no more than 5 degrees;
(The following two theorems equivalent)
Kuratowski Theorem: G is a plan view of FIG. <=> G does not contain a cross-sectional drawing of a molecule as its K5 and K3,3 subgraph
NOTE: insertion of arbitrary edge graph G groups (which may be 0) of the vertex 2 of FIG resulting molecule is referred sectional G of FIG.
Wagner theorem: G is a plan view of FIG. <=> K5 and K3,3 are not the thumbnail G
NOTE: The graph G of an edge shrinkage of a vertex, and deleting the overlapping vertices and edges, thumbnail picture shows the formation (FIG split, then H is the thumbnail G)
Thumbnail theorem: For any set containing an infinite number of FIG graph, there must be a thumbnail is another diagram of FIG.
Definition
FIG drawn on the G plane, the minimum number of cross points generated by crossing number cr referred graph G (G) (crossing number)
The graph G by a straight line drawn on a plane, the minimum number of cross points generated by crossing straight segments called number cr_ graph G (G) (rectilinear crossing number)
Theorem：cr(G) >= m-3n+6
cr (Kn) <= ¼ floor (n / 2) floor ((n-1) / 2) floor ((n-2) / 2) floor ((n-3) / 2), wherein, when n <= 12, the equality holds in the above formula
(Method in Theorem) plan may be no intersection point with a line drawing, i.e. cr_ (G) = 0
6, the colored pattern (colored vertex, edge coloring)
Four color theorem: a plan view of each vertex can be colored to four or fewer colors, different color and each two adjacent vertices