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Study Guides > Mathematics for the Liberal Arts

Euler and Hamiltonian Paths and Circuits

In the next lesson, we will investigate specific kinds of paths through a graph called Euler paths and circuits. Euler paths are an optimal path through a graph. They are named after him because it was Euler who first defined them. By counting the number of vertices of a graph, and their degree we can determine whether a graph has an Euler path or circuit. We will also learn another algorithm that will allow us to find an Euler circuit once we determine that a graph has one.

Learning Objectives

  • Determine whether a graph has an Euler path and/ or circuit
  • Use Fleury's algorithm to find an Euler circuit
  • Add edges to a graph to create an Euler circuit if one doesn't exist
  • Identify whether a graph has a Hamiltonian circuit or path
  • Find the optimal Hamiltonian circuit for a graph using the brute force algorithm, the nearest neighbor algorithm, and the sorted edges algorithm
  • Identify a connected graph that is a spanning tree
  • Use Kruskal's algorithm to form a spanning tree, and a minimum cost spanning tree
 

Euler Circuits

In the first section, we created a graph of the Königsberg bridges and asked whether it was possible to walk across every bridge once. Because Euler first studied this question, these types of paths are named after him.

Euler Path

An Euler path is a path that uses every edge in a graph with no repeats. Being a path, it does not have to return to the starting vertex.

Example

In the graph shown below, there are several Euler paths. One such path is CABDCB. The path is shown in arrows to the right, with the order of edges numbered. Fig2_5_16

Euler Circuit

An Euler circuit is a circuit that uses every edge in a graph with no repeats. Being a circuit, it must start and end at the same vertex.

Example

The graph below has several possible Euler circuits. Here’s a couple, starting and ending at vertex A: ADEACEFCBA and AECABCFEDA. The second is shown in arrows. Fig2_5_17
Look back at the example used for Euler paths—does that graph have an Euler circuit? A few tries will tell you no; that graph does not have an Euler circuit. When we were working with shortest paths, we were interested in the optimal path. With Euler paths and circuits, we’re primarily interested in whether an Euler path or circuit exists. Why do we care if an Euler circuit exists? Think back to our housing development lawn inspector from the beginning of the chapter. The lawn inspector is interested in walking as little as possible. The ideal situation would be a circuit that covers every street with no repeats. That’s an Euler circuit! Luckily, Euler solved the question of whether or not an Euler path or circuit will exist.

Euler’s Path and Circuit Theorems

A graph will contain an Euler path if it contains at most two vertices of odd degree. A graph will contain an Euler circuit if all vertices have even degree

Example

In the graph below, vertices A and C have degree 4, since there are 4 edges leading into each vertex. B is degree 2, D is degree 3, and E is degree 1. This graph contains two vertices with odd degree (D and E) and three vertices with even degree (A, B, and C), so Euler’s theorems tell us this graph has an Euler path, but not an Euler circuit. Fig2_5_18

Example

Is there an Euler circuit on the housing development lawn inspector graph we created earlier in the chapter? All the highlighted vertices have odd degree. Since there are more than two vertices with odd degree, there are no Euler paths or Euler circuits on this graph. Unfortunately our lawn inspector will need to do some backtracking. Fig2_5_19

Example

When it snows in the same housing development, the snowplow has to plow both sides of every street. For simplicity, we’ll assume the plow is out early enough that it can ignore traffic laws and drive down either side of the street in either direction. This can be visualized in the graph by drawing two edges for each street, representing the two sides of the street. Fig2_5_20
Notice that every vertex in this graph has even degree, so this graph does have an Euler circuit. The following video gives more examples of how to determine an Euler path, and an Euler Circuit for a graph. https://youtu.be/5M-m62qTR-s

Fleury's Algorithm

Now we know how to determine if a graph has an Euler circuit, but if it does, how do we find one? While it usually is possible to find an Euler circuit just by pulling out your pencil and trying to find one, the more formal method is Fleury’s algorithm.

Fleury’s Algorithm

1. Start at any vertex if finding an Euler circuit. If finding an Euler path, start at one of the two vertices with odd degree.
2. Choose any edge leaving your current vertex, provided deleting that edge will not separate the graph into two disconnected sets of edges.
3. Add that edge to your circuit, and delete it from the graph.
4. Continue until you’re done.

Example

Find an Euler Circuit on this graph using Fleury’s algorithm, starting at vertex A. Step 1: Original Graph.Choosing edge AD. Circuit so far: AD. Step 2: AD deleted. D is current. Can’t choose DC since that would disconnect graph. Choosing DE.Circuit so far: ADE. Step 3: E is current. From here, there is only one option, so the rest of the circuit is determined. Circuit: ADEBDCA.  

Try It Now

Does the graph below have an Euler Circuit? If so, find one.
The following video presents more examples of using Fleury's algorithm to find an Euler Circuit. https://youtu.be/vvP4Fg4r-Ns

Eulerization and the Chinese Postman Problem

Not every graph has an Euler path or circuit, yet our lawn inspector still needs to do her inspections. Her goal is to minimize the amount of walking she has to do. In order to do that, she will have to duplicate some edges in the graph until an Euler circuit exists.

Eulerization

Eulerization is the process of adding edges to a graph to create an Euler circuit on a graph. To eulerize a graph, edges are duplicated to connect pairs of vertices with odd degree. Connecting two odd degree vertices increases the degree of each, giving them both even degree. When two odd degree vertices are not directly connected, we can duplicate all edges in a path connecting the two.
Note that we can only duplicate edges, not create edges where there wasn’t one before. Duplicating edges would mean walking or driving down a road twice, while creating an edge where there wasn’t one before is akin to installing a new road!

Example

For the rectangular graph shown, three possible eulerizations are shown. Notice in each of these cases the vertices that started with odd degrees have even degrees after eulerization, allowing for an Euler circuit.
2 by 4 grid of rectangles. Each intersection has an open dot. 2 by 3 grid with dots at intersections. curved lines connecting upper dots on cell 2:1, right-hand dots on cell 1:1, upper dots on cell 1:3, right-hand dots on cell 1:3, and lower dots on cell 2:2
In the example above, you’ll notice that the last eulerization required duplicating seven edges, while the first two only required duplicating five edges. If we were eulerizing the graph to find a walking path, we would want the eulerization with minimal duplications. If the edges had weights representing distances or costs, then we would want to select the eulerization with the minimal total added weight.

Try It now

Eulerize the graph shown, then find an Euler circuit on the eulerized graph. Equilateral triangle with dots at vertices labeled A, B, C. There is a smaller triangle which shares the C,D edge with the larger triangle. The smaller triangle has vertices labeled B, C, D.
 

Example

Looking again at the graph for our lawn inspector from Examples 1 and 8, the vertices with odd degree are shown highlighted. With eight vertices, we will always have to duplicate at least four edges. In this case, we need to duplicate five edges since two odd degree vertices are not directly connected. Without weights we can’t be certain this is the eulerization that minimizes walking distance, but it looks pretty good. Graph with 25 edges and 20 vertices.
The problem of finding the optimal eulerization is called the Chinese Postman Problem, a name given by an American in honor of the Chinese mathematician Mei-Ko Kwan who first studied the problem in 1962 while trying to find optimal delivery routes for postal carriers.  This problem is important in determining efficient routes for garbage trucks, school buses, parking meter checkers, street sweepers, and more. Unfortunately, algorithms to solve this problem are fairly complex. Some simpler cases are considered in the exercises The following video shows another view of finding an Eulerization of the lawn inspector problem. https://youtu.be/lUqCtywkskU  

Hamiltonian Circuits

The Traveling Salesman Problem

In the last section, we considered optimizing a walking route for a postal carrier. How is this different than the requirements of a package delivery driver? While the postal carrier needed to walk down every street (edge) to deliver the mail, the package delivery driver instead needs to visit every one of a set of delivery locations. Instead of looking for a circuit that covers every edge once, the package deliverer is interested in a circuit that visits every vertex once.

Hamiltonian Circuits and Paths

A Hamiltonian circuit is a circuit that visits every vertex once with no repeats. Being a circuit, it must start and end at the same vertex. A Hamiltonian path also visits every vertex once with no repeats, but does not have to start and end at the same vertex.
Hamiltonian circuits are named for William Rowan Hamilton who studied them in the 1800’s.

Example

One Hamiltonian circuit is shown on the graph below. There are several other Hamiltonian circuits possible on this graph. Notice that the circuit only has to visit every vertex once; it does not need to use every edge. This circuit could be notated by the sequence of vertices visited, starting and ending at the same vertex: ABFGCDHMLKJEA. Notice that the same circuit could be written in reverse order, or starting and ending at a different vertex. Rectangular graph with 12 vertices labeled a through M (without I)
Unlike with Euler circuits, there is no nice theorem that allows us to instantly determine whether or not a Hamiltonian circuit exists for all graphs.[1]

Example

Does a Hamiltonian path or circuit exist on the graph below? Arrow shaped graph with 5 vertices labeled A- E, Edge from C to E is not part of the arrow shape. A and C are connected by two edges. We can see that once we travel to vertex E there is no way to leave without returning to C, so there is no possibility of a Hamiltonian circuit. If we start at vertex E we can find several Hamiltonian paths, such as ECDAB and ECABD

try it now

  The Brute force algorithm is optimal; it will always produce the Hamiltonian circuit with minimum weight. Is it efficient? To answer that question, we need to consider how many Hamiltonian circuits a graph could have. For simplicity, let’s look at the worst-case possibility, where every vertex is connected to every other vertex. This is called a complete graph. Suppose we had a complete graph with five vertices like the air travel graph above. From Seattle there are four cities we can visit first. From each of those, there are three choices. From each of those cities, there are two possible cities to visit next. There is then only one choice for the last city before returning home. This can be shown visually:    
Counting the number of routes, we can see thereare [latex]4\cdot{3}\cdot{2}\cdot{1}[/latex] routes. For six cities there would be [latex]5\cdot{4}\cdot{3}\cdot{2}\cdot{1}[/latex] routes.

Number of Possible Circuits

For N vertices in a complete graph, there will be [latex](n-1)!=(n-1)(n-2)(n-3)\dots{3}\cdot{2}\cdot{1}[/latex] routes. Half of these are duplicates in reverse order, so there are [latex]\frac{(n-1)!}{2}[/latex] unique circuits. The exclamation symbol, !, is read “factorial” and is shorthand for the product shown.

Example

How many circuits would a complete graph with 8 vertices have? A complete graph with 8 vertices would have = 5040 possible Hamiltonian circuits. Half of the circuits are duplicates of other circuits but in reverse order, leaving 2520 unique routes. While this is a lot, it doesn’t seem unreasonably huge. But consider what happens as the number of cities increase:
Cities Unique Hamiltonian Circuits
9 8!/2 = 20,160
10 9!/2 = 181,440
11 10!/2 = 1,814,400
15 14!/2 = 43,589,145,600
20 19!/2 = 60,822,550,204,416,000
Watch these examples worked again in the following video. https://youtu.be/DwZw4t0qxuQ   As you can see the number of circuits is growing extremely quickly. If a computer looked at one billion circuits a second, it would still take almost two years to examine all the possible circuits with only 20 cities! Certainly Brute Force is not an efficient algorithm.

Nearest Neighbor Algorithm (NNA)

1.     Select a starting point. 2.     Move to the nearest unvisited vertex (the edge with smallest weight). 3.     Repeat until the circuit is complete.
Unfortunately, no one has yet found an efficient and optimal algorithm to solve the TSP, and it is very unlikely anyone ever will. Since it is not practical to use brute force to solve the problem, we turn instead to heuristic algorithms; efficient algorithms that give approximate solutions. In other words, heuristic algorithms are fast, but may or may not produce the optimal circuit.

Example

Consider our earlier graph, shown to the right. Starting at vertex A, the nearest neighbor is vertex D with a weight of 1. From D, the nearest neighbor is C, with a weight of 8. From C, our only option is to move to vertex B, the only unvisited vertex, with a cost of 13. From B we return to A with a weight of 4. triangular graph with 4 vertices and 6 edges. There is one vertex in the center of the triangle. The resulting circuit is ADCBA with a total weight of [latex]1+8+13+4 = 26[/latex].
Watch the example worked out in the following video. https://youtu.be/SqOP5n9bNX4 We ended up finding the worst circuit in the graph! What happened? Unfortunately, while it is very easy to implement, the NNA is a greedy algorithm, meaning it only looks at the immediate decision without considering the consequences in the future. In this case, following the edge AD forced us to use the very expensive edge BC later.

Example

Consider again our salesman. Starting in Seattle, the nearest neighbor (cheapest flight) is to LA, at a cost of $70. From there: LA to Chicago: $100 Chicago to Atlanta: $75 Atlanta to Dallas: $85 Dallas to Seattle: $120 Total cost: $450   In this case, nearest neighbor did find the optimal circuit.
Watch this example worked out again in this video. https://youtu.be/3Eq36iqjGKI   Going back to our first example, how could we improve the outcome? One option would be to redo the nearest neighbor algorithm with a different starting point to see if the result changed. Since nearest neighbor is so fast, doing it several times isn’t a big deal.

Example

We will revisit the graph from Example 17. triangular graph with 4 vertices and 6 edges. There is one vertex in the center of the triangle. Starting at vertex A resulted in a circuit with weight 26. Starting at vertex B, the nearest neighbor circuit is BADCB with a weight of 4+1+8+13 = 26. This is the same circuit we found starting at vertex A. No better. Starting at vertex C, the nearest neighbor circuit is CADBC with a weight of 2+1+9+13 = 25. Better! Starting at vertex D, the nearest neighbor circuit is DACBA. Notice that this is actually the same circuit we found starting at C, just written with a different starting vertex. The RNNA was able to produce a slightly better circuit with a weight of 25, but still not the optimal circuit in this case. Notice that even though we found the circuit by starting at vertex C, we could still write the circuit starting at A: ADBCA or ACBDA.
 

try it now

The table below shows the time, in milliseconds, it takes to send a packet of data between computers on a network. If data needed to be sent in sequence to each computer, then notification needed to come back to the original computer, we would be solving the TSP. The computers are labeled A-F for convenience.
A B C D E F
A -- 44 34 12 40 41
B 44 -- 31 43 24 50
C 34 31 -- 20 39 27
D 12 43 20 -- 11 17
E 40 24 39 11 -- 42
F 41 50 27 17 42 --
a.     Find the circuit generated by the NNA starting at vertex B. b.     Find the circuit generated by the RNNA.
While certainly better than the basic NNA, unfortunately, the RNNA is still greedy and will produce very bad results for some graphs. As an alternative, our next approach will step back and look at the “big picture” – it will select first the edges that are shortest, and then fill in the gaps.

Example

Using the four vertex graph from earlier, we can use the Sorted Edges algorithm. triangular graph with 4 vertices and 6 edges. There is one vertex in the center of the triangle. The cheapest edge is AD, with a cost of 1. We highlight that edge to mark it selected.
The next shortest edge is AC, with a weight of 2, so we highlight that edge. triangular graph with 4 vertices and 6 edges. There is one vertex in the center of the triangle. The edge between A and D is highlighted triangular graph with 4 vertices and 6 edges. There is one vertex in the center of the triangle. Edges between A and D and A and C are highlighted. For the third edge, we’d like to add AB, but that would give vertex A degree 3, which is not allowed in a Hamiltonian circuit. The next shortest edge is CD, but that edge would create a circuit ACDA that does not include vertex B, so we reject that edge. The next shortest edge is BD, so we add that edge to the graph.
triangular graph with 4 vertices and 6 edges. There is one vertex in the center of the triangle. Edges between A and D and A and C are highlighted. Edge between A and B is highlighted. BAD
triangular graph with 4 vertices and 6 edges. There is one vertex in the center of the triangle. Edges between A and D and A and C are highlighted. BAD
triangular graph with 4 vertices and 6 edges. There is one vertex in the center of the triangle. Edges between A and D and A and C are highlighted. OK
We then add the last edge to complete the circuit: ACBDA with weight 25.
Notice that the algorithm did not produce the optimal circuit in this case; the optimal circuit is ACDBA with weight 23. Notice that the algorithm did not produce the optimal circuit in this case; the optimal circuit is ACDBA with weight 23. While the Sorted Edge algorithm overcomes some of the shortcomings of NNA, it is still only a heuristic algorithm, and does not guarantee the optimal circuit.

Example

Your teacher’s band, Derivative Work, is doing a bar tour in Oregon. The driving distances are shown below. Plan an efficient route for your teacher to visit all the cities and return to the starting location. Use NNA starting at Portland, and then use Sorted Edges.
 Ashland Astoria  Bend  Corvallis  Crater Lake  Eugene  Newport  Portland  Salem  Seaside
Ashland - 374 200 223 108 178 252 285 240 356
Astoria 374 - 255 166 433 199 135 95 136 17
Bend 200 255 - 128 277 128 180 160 131 247
Corvallis 223 166 128 - 430 47 52 84 40 155
Crater Lake 108 433 277 430 - 453 478 344 389 423
Eugene 178 199 128 47 453 - 91 110 64 181
Newport 252 135 180 52 478 91 - 114 83 117
Portland 285 95 160 84 344 110 114 - 47 78
Salem 240 136 131 40 389 64 83 47 - 118
Seaside 356 17 247 155 423 181 117 78 118 -

To see the entire table, scroll to the right

Using NNA with a large number of cities, you might find it helpful to mark off the cities as they’re visited to keep from accidently visiting them again. Looking in the row for Portland, the smallest distance is 47, to Salem. Following that idea, our circuit will be:

Portland to Salem                    47 Salem to Corvallis                   40 Corvallis to Eugene                 47 Eugene to Newport                 91 Newport to Seaside                117 Seaside to Astoria                   17 Astoria to Bend                      255 Bend to Ashland                     200 Ashland to Crater Lake           108 Crater Lake to Portland          344 Total trip length:                     1266 miles

Using Sorted Edges, you might find it helpful to draw an empty graph, perhaps by drawing vertices in a circular pattern. Adding edges to the graph as you select them will help you visualize any circuits or vertices with degree 3.

We start adding the shortest edges:

Seaside to Astoria       17 miles

Corvallis to Salem       40 miles

Portland to Salem        47 miles

Corvallis to Eugene     47 miles

ring of dots with city names in problem as labels. edges between Seaside and Astoria, Eugene and Corvallis, Salem and Corvallis, and Salem and Portland.

The graph after adding these edges is shown to the right.   The next shortest edge is from Corvallis to Newport at 52 miles, but adding that edge would give Corvallis degree 3.   Continuing on, we can skip over any edge pair that contains Salem or Corvallis, since they both already have degree 2. Portland to Seaside                 78 miles Eugene to Newport                 91 miles Portland to Astoria                 (reject – closes circuit) Ashland to Crater Lk  108 miles

The graph after adding these edges is shown to the right. At this point, we can skip over any edge pair that contains Salem, Seaside, Eugene, Portland, or Corvallis since they already have degree 2.  

Newport to Astoria                (reject – closes circuit)

Newport to Bend                    180 miles

Bend to Ashland                     200 miles

 

At this point the only way to complete the circuit is to add:

Crater Lk to Astoria   433 miles.  The final circuit, written to start at Portland, is:

Portland, Salem, Corvallis, Eugene, Newport, Bend, Ashland, Crater Lake, Astoria, Seaside, Portland.  Total trip length: 1241 miles.

While better than the NNA route, neither algorithm produced the optimal route. The following route can make the tour in 1069 miles:

Portland, Astoria, Seaside, Newport, Corvallis, Eugene, Ashland, Crater Lake, Bend, Salem, Portland

Watch the example of nearest neighbor algorithm for traveling from city to city using a table worked out in the video below. https://youtu.be/GFp-046PQx0 In the next video we use the same table, but use sorted edges to plan the trip. https://youtu.be/_gXyujMsrmw

try it now

Find the circuit produced by the Sorted Edges algorithm using the graph below. graph with 6 vertices and 14 edges. between B and E is 13, between E and G is 45, between G and F is 19, between F and C is 37, between c and A is 33 between A and B is 11. Between B and C is 25, between B and F is 23, between E and A is 14, between E and F is 15. Between G and B is 13, and between G and C is 36. Between G and A is 27.
    [1] There are some theorems that can be used in specific circumstances, such as Dirac’s theorem, which says that a Hamiltonian circuit must exist on a graph with n vertices if each vertex has degree n/2 or greater.

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