Hamiltonian Circuits

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    Hamilton CircuitTopics in Contemporary MathematicsMA 103

    Summer II, 2013

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    In Euler paths and Euler circuits, thegame was to find paths or circuitsthat include every edge of the graph once (andonly once).

    In Hamilton paths and Hamilton circuits, the game is to find pathsand circuits that include every vertex of the graph once and only once.

    Hamilton Paths and Hamilton Circuits2

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    ! A Hamilton pathin a graph is a path

    that includes each vertex of the graph

    once and only once.! A Hamilton circuit is a circuit that

    includes each vertex of the graph

    onceand only once. (At the end, ofcourse, the circuit must return to the

    startingvertex.)

    HAMILTON PATHS & CIRCUITS

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    The figure shows a graph that

    (1)has Euler circuits (the vertices are

    all even) and

    (2)

    has Hamilton circuits.

    Example 1 Hamilton versus Euler

    One suchHamilton circuit is A,

    F, B, C, G, D, E, A there are

    plenty more.

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    Note that if a graph has aHamilton circuit, then it

    automatically has a Hamilton path(the Hamilton circuitcan always be truncated into a Hamilton path bydropping the last vertex of the circuit.)

    For example, the Hamilton circuit A, F, B, C, G, D, E, Acan be truncated into the Hamilton path A, F, B, C, G, D, E.

    Contrast this with the mutually exclusive relationship

    between Euler circuits and paths:If a graph has anEuler circuit it cannot have an Euler path and vice versa.

    Example 1 Hamilton versus Euler5

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    This figure shows a graph that

    (1)has no Euler circuits but does have Euler paths (for

    example C, D, E, B, A, D) and

    (2)has no Hamilton circuits (sooner or later you have to go to

    C, and then you are stuck) but does have Hamilton paths (forexample, A, B, E, D, C).

    Example 1 Hamilton versus Euler

    This illustrates that a graph can

    have aHamilton path but no

    Hamilton circuit!

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    This figure shows a graph that

    (1)has neither Euler circuits nor paths (it has four odd

    vertices) and

    (2) has Hamilton circuits (for example A, B, C, D, E, A

    there are plenty more)

    Example 1 Hamilton versus Euler

    and consequently has

    Hamilton paths (for example, A,

    B, C, D, E).

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    This figure shows a graph that(1) has no Euler circuits but hasEuler paths (F and G are the twoodd vertices) and

    (2) has neither Hamilton circuitsnor Hamilton paths.

    Example 1 Hamilton versus Euler8

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    This figure shows a graphthat

    (1) has neither Euler circuits nor

    Euler paths (too many oddvertices) and

    (2) has neither Hamilton circuits

    nor Hamilton paths.

    Example 1 Hamilton versus Euler9

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    The lesson of Example 1 is that the existence of anEuler path or circuit in a graph tells us nothing aboutthe existence of a Hamilton path or circuit in thatgraph.

    Euler versus Hamilton10

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    Unlike Euler circuit and path, there exist noHamiltoncircuit and path theoremsfor determining if a graph has

    a Hamilton circuit, a Hamilton path, or neither.

    Determining when a given graph does or does nothave a Hamilton circuit or path can be very easy, butit also can be very hardit all depends on the graph.

    Euler versus Hamilton11

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    Sometimes the question,Does the graph have a

    Hamilton circuit? has an obviousyesanswer, and the

    more relevant question turns out to be,How many

    different Hamilton circuits does it have?

    In this section we will answer this question for an

    important family of graphs called complete graphs.

    How Many Hamilton Circuits?12

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    Complete Graph

    A graph with Nvertices in which every pair of

    distinct vertices is joined by an edge is called

    a complete graph on N vertices and is denoted by

    the symbolKN

    .

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    One of the key properties of KNis that every vertex has

    degree N 1.

    This implies that the sum of the degrees of all the

    vertices isN(N 1), and it follows from Eulers sum

    of degrees theorem that the number of edges inKNis

    N(N 1)/2.

    For a graph withNvertices and no multiple edges orloops,N(N 1)/2 is the maximum number of edges

    possible, and this maximum can only occur when the

    graph isKN.

    How Many Hamilton Circuits?14

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    ! KNhas N(N 1)/2edges.

    ! Of all graphs with Nvertices and no

    multiple edges or loops, KNhas the

    most edges.

    NUMBER OF EDGES IN KN

    Because KNhas a complete set of edges (every

    vertex is connected to everyother vertex), italso has a complete set of Hamilton circuits

    you can travel thevertices in any sequence you

    choose and you will not get stuck.

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    There are two additional sequences that describe this

    same Hamilton circuit: D, B, C, A, D (with reference pointD) and B, C, A, D, B (with reference point B). Taking allthis into account, there are six different Hamilton

    Example 2 Hamilton Circuits in K4

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    19

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    Example 2 Hamilton Circuits in K420

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    Lets try to list all the Hamilton circuits in K5.

    For simplicity, we will write eachcircuit just once, using a commonreference point say A. (As long as we are consistent, it doesnt reallymatter which reference point wepick.)

    Each of the Hamilton circuits will be described bya sequence that startsand ends with A, with the letters B,C, D, and E sandwiched in between

    in some order.

    Thereare 4 !3 !2 !1 = 24 different ways to shuffle the letters B, C,D, and E, each producing a different Hamiltoncircuit.

    Example 2 Hamilton Circuits in K521

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