uni/year2/semester1/logseq-stuff/pages/Colouring Graphs; Eulerian & Hamiltonian Graphs.md

• #MA284 - Discrete Mathematics
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• Vertex Colouring

  • There are maps that can be coloured with a single colour, two colours, three colours, or four colours.
  • For all maps, no matter how complicated, at most four colours is sufficient.

• Colouring Graphs

  • If we think of a map as a way of showing which regions share borders, then we can represent it as a graph, where:
    • A vertex in the graph corresponds to a region in the map.
    • There is an edge between two vertices in the graph if the corresponding regions share a border.
  • Colouring regions of a map corresponds to colouring vertices of the graph. Since neighbouring regions in the map must have different colours, so too must adjacent vertices.
  • More precisely:
    • Vertex Colouring: An assignment of colours to the vertices of a graph.
    • Proper Colouring: If the vertex colouring has the property that adjacent vertices are coloured differently, then the colouring is called proper.
    • Minimal Colouring: A proper colouring that is done with the fewest possible number of colours.
  • Lots of different proper colourings are possible. If the graph has v vertices, then clearly at most v colours are needed. However, usually, we need far fewer.

  • Chromatic Numbers

    • What is the chromatic number of a graph? #card card-last-interval:: 2.8 card-repeats:: 1 card-ease-factor:: 2.6 card-next-schedule:: 2022-11-17T10:50:36.778Z card-last-reviewed:: 2022-11-14T15:50:36.779Z card-last-score:: 5
      • The chromatic number of the graph, written \chi(G) is the smallest number of colours needed to get a proper vertex colouring of a graph G.

    • Example

      • Determine the chromatic number of the graphs C_2, C_3, C_4, & C_5. background-color:: green
        • \chi(C_2) = 2
        • \chi(C_3) = 3
        • \chi(C_4) = 2
        • \chi(C_3) = 5
      • Determine the chromatic number of the K_n & K_{p,q} for any n, p, q. background-color:: green
        • \chi(K_4) = 4
        • \chi(K_n) = n
        • \chi(K_{3,3}) = 2
        • \chi(K_{p,q}) = 2
    • In general, calculating \chi(G) is not easy, but there are some ideas that can help. For example, it is clearly true that if a graph has v vertices, then
      • 1 \leq \chi(G) \leq v

    • Cliques

      • If the graph happens to be complete, then \chi(G) = v. If it is not complete, then we can look at cliques in the graph.
        • What is a clique of a graph? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T16:14:53.505Z card-last-score:: 1
          • A clique is a subgraph of a graph, all of whose vertices are connected to each other.
        • (Clique numbers will not be on the exam).
        • What is the clique number of a graph? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-23T00:00:00.000Z card-last-reviewed:: 2022-11-22T13:37:29.890Z card-last-score:: 1
          • The clique number of a graph, G, is the number of vertices in the largest clique in G.
        • Lower Bound: The chromatic number of a graph is at least its clique number.
        • Upper Bound: \chi(G) \leq \Delta(G) + 1, where \Delta(G) denotes the largest degree of any vertex in the graph G.

• Algorithms for \chi(G)

  • In general, finding a proper colouring for a graph is hard. There are some algorithms that are efficient, but not optimal. We'll look at two:
    • The Greedy Algorithm.
    • The Welsh-Powell Algorithm.
  • The Greedy Algorithm is simple & efficient, but the result can depend on the ordering of the vertices.
  • The Welsh-Powell Algorithm is slightly more complicated, but can give better colourings.

  • Greedy Algorithm #card

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    • 1. Number all the vertices. Number your colours.
      2. Give a colour to vertex 1.
      3. Take the remaining vertices in order. Assign each one the lowest numbered colours that is different from the colours of its neighbours.

  • Welsh-Powell Algorithm #card

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    • 1. List all vertices in decreasing order of their degree (i.e., largest degree first). If two or more share the same degree, list them in any way you want.
      2. Colour the first listed vertex (with the first unused colour).
      3. Work down the list, giving that colour to all vertices not conencted to one previously coloured.
      4. Cross (verb.) coloured vertices of the list, and return to the start of the list.

• Eulerian Paths & Circuits

  • Recall that a path is a sequence of adjacent vertices in a graph.
  • What is a Eulerian Path? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T16:18:26.028Z card-last-score:: 1
    • A Eulerian Path (also called an Euler Path and an Eulerian trail) in a graph is path which uses every edge exactly once.
    • 
  • Recall that a circuit is a path that begins & ends at that same vertex, and no edge is repeated.
  • What is an Eulerian Circuit? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T16:21:29.522Z card-last-score:: 1
    • An Eulerian Circuit (also called an Eulerian Cycle) is an Eulerian path that that starts & finishes on at the same vertex.
    • If a graph has such a circuit, we say that it is Eulerian.
  • It is possible to come up with a condition that guarantees that a graph has an Eulerian Path, and, additionally, one that ensures that ensures that it has an Eulerian Circuit.
    • To begin with, we'll reason that the following graph could not have an Eulerian circuit, although it does have an Eulerian path.
      • 
    • Suppose, first, that we have a graph that ==does have an Eulerian circuit.== Then, for every edge in the circuit that "exits" a vertex, there is another that "enters" that vertex. So, every vertex must have even degree.
      • A graph has an Eulerian Circuit if and only if every vertex has even degree.
    • Next, suppose that a graph==does not have an Eulerian circuit==, but does have an Eulerian path. Then, the degree at the "start" & "end" verticwes must be odd, and every other vertex has even degree.
      • A graph has an Eulerian Path if and only if it has either zero or two vertices with odd degree.

• Hamiltonian Paths & Cycles

  • What is a Hamiltonian Path? #card card-last-interval:: 2.8 card-repeats:: 2 card-ease-factor:: 2.6 card-next-schedule:: 2022-11-24T08:06:13.770Z card-last-reviewed:: 2022-11-21T13:06:13.770Z card-last-score:: 5
    • A Hamiltonian Path is a graph that visits every vertex exactly once.
  • What is a Hamiltonian Cycle? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T16:23:21.600Z card-last-score:: 1
    • A Hamiltonian Cycle is a cycle which visits the start / end vertex twice, and every other vertex exactly once.
    • A graph that has a Hamiltonian Cycle is called a Hamiltonian Graph.
      • Important examples of Hamiltonian Graphs include cycle graphs, complete graphs, & graphs of the platonic solids.
  • In general, the problem of finding a Hamiltonian path or cycle in a large graph is hard - it is known to be NP-complete. However, there are two relatively simple sufficient conditions to testing if a graph is Hamiltonian:

    • Ore's Theorem #card

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      • A graph with v vertices, where v \geq 3, is Hamiltonian if, for every pair of non-adjacent vertices, the sum of their degrees is \geq v.

    • Dirac's Theorem #card

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      • A simple graph with v vertices, where v \geq 3, is Hamiltonian if every vertex has degree \geq v / 2.