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Combinatorics and Graph Theory

A graph G consists of a vertex set V and a collection E of 2-element subsets of V. Elements of E are called edges.

Combinatorics and Graph Theory

A graph \(G\) consists of a vertex set \(V\) and a collection \(E\) of \(2\)-element subsets of \(V\). Elements of \(E\) are called edges. In our course, we will (almost always) use the convention that \(V=\{1,2,3,\dots,n\}\) for some positive integer \(n\). With this convention, graphs can be described precisely with a text file:

  1. The first line of the file contains a single integer \(n\), the number of vertices in the graph.

  2. Each of the remaining lines of the file contains a pair of distinct integers and specifies an edge of the graph.

We illustrate this convention in with a text file and the diagram for the graph\(G\) it defines.

Much of the notation and terminology for graphs is quite natural. See if you can make sense out of the following statements which apply to the graph \(G\) defined above:

  1. \(G\) has \(9\) vertices and \(10\) edges.

  2. \(\{2,6\}\) is an edge.

  3. Vertices \(5\) and \(9\) are adjacent.

  4. \(\{5,4\}\) is not an edge.

  5. Vertices \(3\) and \(7\) are not adjacent.

  6. \(P = (4, 3,1, 7,9,5)\) is a path of length\(5\) from vertex \(4\) to vertex\(5\).

  7. \(C=(5,9,7,1)\) is cycle of length\(4\).

  8. \(G\) is disconnected and has two components. One of the components has vertex set \(\{2,6,8\}\).

  9. \(\{1,5,7\}\) is a triangle.

  10. \(\{1,7,5,9\}\) is a clique of size\(4\).

  11. \(\{4,2,8,5\}\) is an independent set of size\(4\).

Equipped only with this little bit of background material, we are already able to pose a number of interesting and challenging problems.

Example

Consider the graph \(G\)

  1. What is the largest \(k\) for which \(G\) has a path of length\(k\)?

  2. What is the largest \(k\) for which \(G\) has a cycle of length\(k\)?

  3. What is the largest \(k\) for which \(G\) has a clique of size\(k\)?

  4. What is the largest \(k\) for which \(G\) has an independent set of size\(k\)?

  5. What is the shortest path from vertex\(7\) to vertex\(6\)?

Suppose we gave the class a text data file for a graph on \(1500\) vertices and asked whether the graph contains a cycle of length at least \(500\). Raoul says yes and Carla says no. How do we decide who is right?

Suppose instead we asked whether the graph has a clique of size\(500\). Helene says that she doesn't think so, but isn't certain. Is it reasonable that her classmates insist that she make up her mind, one way or the other? Is determining whether this graph has a clique of size\(500\) harder, easier or more or less the same as determining whether it has a cycle of size\(500\).

We will frequently study problems in which graphs arise in a very natural manner. Here's an example.

Condensed: the full section is in Keller & Trotter, Applied Combinatorics.

Haddaad Ma jiro xisaabis oo ku nool midkan, laakiin qaybaha ka mid ah waa la xisaabi karaa. Ku raaxayso mid ka hooseeya, ama ku qor adiga kuu gaar ah.

Ku hay shaqadaada

A xisaab bilaash ah oo ku daray qoraalada waxbarasho kasta, diiwaanka waxa aad dhamaatay, dhibaatooyinka aad xal u tahay meel kaliya, iyo macallinka aad weydiisan kartaa oo ku saabsan bogga. Xisaabta oo keliya waa u furan yahay qof kasta, la diiwaan geliyey ama aan.

Ku soo biir Ku soo gal

Calaamadaha la isticmaalay halkan

Taab calaamad kasta oo loogu talagalay qeexidda buuxda, sawir, iyo waxa xarfaha oo dhan ka mid ah ka dhigan tahay.

Su'aalaha dadku waydiiyaan

Permutation or combination?

Ask whether order matters. A lock code is a permutation (order matters); a hand of cards is a combination (it does not).

What is a graph in this sense?

Dots (vertices) joined by lines (edges), not a plot. Road maps, social networks and molecules are graphs; questions like "is there a route" and "how few colours" are graph theory.

Qaybaha ee boggaan waxaa la beddelay ka Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Condensed iyo dib-oo halkan lagu sharaxay; khaladaadka waa inoo.

In ka badan Combinatorics & Graph Theory

The counting principlesPigeonhole principle and inclusion, exclusionBinomial coefficients and Pascal's triangleRecurrences and generating functionsGraphs: vertices, edges, degreesPaths, cycles, trees, Euler and HamiltonColouring and planar graphs