Discrete Mathematics for BSAI

01 · Opening Challenge: Inspect Every Walkway Once

A campus robot must inspect each walkway exactly once. It may pass a building more than once. Could a route cover all the walkways without repeating one, and must the robot finish where it started?

Predict informally: Imagine tracing the network with a pencil without lifting it or drawing an edge twice. An arrival at a middle building usually needs a different walkway to leave. We will turn that observation into a precise test and then try an actual route.

02 · Learning Objectives and Three-Hour Guide

By the end of Week 13, you should be able to draw a small graph from a route description; identify vertices and edges; distinguish a walk, trail, path and cycle; read degree and adjacency representations; apply the Euler trail/circuit degree test to a connected graph; and justify a campus inspection route or explain why none exists.

Classroom essentials Sections 01–11 and Practice 1–6. Optional self-study Colab and Practice 7. The campus case continues in Week 14 with a different route question.

03 · From a Map to a Graph

A graph describes objects and their connections. In our campus model, a vertex is a building and an edge is a two-way walkway directly connecting two buildings. Drawing positions do not affect which buildings are connected.

For a first tiny example, let V={A,B,C} and E={{A,B},{B,C},{A,C}}. Each pair in E is one undirected edge; {A,B} is the same edge as {B,A}. This triangle has three vertices and three edges.

We model one undirected walkway between a pair of buildings, with no walkway from a building to itself. These assumptions make the subsequent degree counts and adjacency table unambiguous.

04 · Walk, Trail, Path and Cycle

A walk is a sequence of vertices where each consecutive pair is joined by an edge. It may repeat vertices or edges. A trail is a walk with no repeated edge; a vertex may still repeat. A path is a walk with no repeated vertex. A cycle is a closed route returning to its starting vertex without repeating other vertices or edges.

For a path we do not call the closing repetition in a cycle a new distinct vertex; “cycle” names that special closed pattern. In our inspection task, the key restriction is on edges, so a trail is the useful type of route. A trail can revisit a building.

05 · Degree Counts Incident Walkways

The degree of a vertex is the number of edges meeting there. In the triangle A–B–C–A, each vertex has degree 2. Every edge contributes one to the degree of each of its two endpoints, so the sum of all degrees is twice the number of edges:

For the triangle: 2+2+2=6=2×3. This is a useful check on a small undirected graph. An odd-degree vertex has 1,3,5,… connections; an even-degree vertex has 0,2,4,… .

06 · Two Ways to Record Adjacency

Two vertices are adjacent if an edge joins them. For the triangle, a neighbor list says A: B,C; B: A,C; C: A,B. An adjacency matrix gives a 1 if a pair is directly connected and 0 otherwise.

The matrix is symmetric because our walkways are two-way. Its diagonal entries are zero because the model has no walkway from a building back to itself. A row's 1s count that vertex's degree. We will use a compact neighbor list for the larger campus graph.

07 · Use Every Edge Exactly Once

An Euler trail uses every edge in the graph exactly once; it may start and end at different vertices. An Euler circuit is an Euler trail that returns to its starting vertex. In both cases, every edge must be included. These names describe a route condition, not a method for computing the route.

A simple cycle is also an Euler circuit if it includes every edge in the graph. But an Euler circuit in a larger graph may revisit a vertex and therefore need not be a simple cycle.

08 · Why Odd Degrees Matter

At a building used in the middle of a trail, each arrival needs a different edge for departure. Those edges pair up, so the degree at that building is even. The starting point of an open trail can have one unmatched departure, and its ending point can have one unmatched arrival. These are its two odd-degree vertices.

The test is stated here for a connected finite undirected graph: every building can be reached through the network. If two groups of walkways are disconnected, one uninterrupted route cannot inspect edges in both groups, regardless of the degree counts.

09 · Major Case Study 4 Begins: Campus Inspection and Delivery Robot

Scenario. Five buildings are connected by six two-way walkways. The inspection robot must traverse each walkway exactly once. It may pass a building several times. We treat each named walkway as one edge, and assume all six are open and connected.

The six walkways are LH, HC, CG, GL, LS, SH. In this undirected graph, LH and HL name the same walkway. Use the drawing to understand adjacency, then rely on the edge list when checking a route.

10 · Represent, Reason and Interpret

Represent the same graph in a neighbor list

Check: 3+3+2+2+2=12=2×6, consistent with six edges. The graph is connected. Exactly two vertices, L and H, have odd degree. Therefore an open Euler trail exists starting at L and ending at H, or vice versa. A closed Euler circuit does not exist.

Test one route edge by edge

The consecutive walkways are LH, HC, CG, GL, LS, SH: all six appear once. Vertex L occurs twice, which is allowed in a trail. The route starts at L and ends at H, matching the two odd-degree vertices.

Interpret the decision

The robot can inspect every walkway once if it may finish at a different building. If the task additionally requires returning to L without reusing an inspected walkway, the requirement cannot be met on this graph. The model does not include travel time, walkway length, direction restrictions, or temporary closures.

11 · Route Inspector

Predict → Test → Explain Choose a start, then select one walkway at a time. Only a walkway touching your current building can be used next, and each walkway can be used once. Try L first; then try an even-degree building and explain what changes.

12 · Optional Colab: Read Neighbors and Degrees

Self-study · mathematics first Draw the campus graph and fill its neighbor table before running Python. A dictionary pairs a key with a value. Here a building code is a key, and its value is a list of directly adjacent building codes. The colon means “maps to”; commas separate entries.

neighbors = {
    "L": ["H", "G", "S"],
    "H": ["L", "C", "S"],
    "C": ["H", "G"],
    "G": ["C", "L"],
    "S": ["L", "H"]
}
print(neighbors["L"])

Predict ["H", "G", "S"]. Notice that if L lists H, H also lists L: the walkway is two-way. The list contains neighbors, not a full inspection route.

Check degrees after the hand count

The familiar for loop visits each key. len(...) counts the entries in that neighbor list, so it reports the degree in this simple graph.

for building in neighbors:
    print(building, len(neighbors[building]))

Predict L 3, H 3, C 2, G 2, S 2. Explain why degree data alone is used with the connectedness condition in the Euler test; the code here verifies the table but does not construct a route.

13 · Practice and Case Record

Use a small drawing or edge list for every route question. Selected solutions appear at the end.

1. The graph has V={A,B,C,D} and E={{A,B},{B,C},{C,D}}. Find each degree and decide whether an Euler trail or circuit exists.
2. For the triangle with edges AB, BC, CA, classify A–B–A, A–B–C, and A–B–C–A as walk/trail/path/cycle where applicable.
3. The campus graph has edges LH, HC, CG, GL, LS, SH. Give the neighbors and degree of S. Check the sum of all degrees.
4. Find an Euler trail on the campus graph different from L–H–C–G–L–S–H. List its six edges and justify why none repeats.
5. Four leaves A, B, C, D each connect only to center O. Count odd-degree vertices. Could one trail use all four edges?
6. A connected graph has exactly two odd-degree vertices P and Q. Where must an Euler trail begin and end? Can it be a circuit?
7. Draw two disconnected triangles. All six vertices have degree 2. Explain why one uninterrupted route still cannot inspect every edge.

14 · Common Mistakes

15 · Summary

• A graph consists of vertices and edges. For a two-way campus walkway, one undirected edge joins its two endpoint buildings.
• A trail cannot repeat an edge; a path cannot repeat a vertex. An Euler trail uses every edge once, and an Euler circuit also returns to its start.
• Degree counts incident edges; the sum of degrees equals twice the number of edges in a finite undirected graph.
• For a connected graph, all even degrees allow an Euler circuit; exactly two odd degrees allow an open Euler trail; more than two odd degrees do not allow one.
• The campus graph has odd endpoints L and H and admits an L-to-H inspection trail, conditional on the stated walkway assumptions.

16 · Visual Concept Map

17 · Looking Ahead

Week 14 returns to the campus graph and asks about visiting every building once, which is different from inspecting every walkway once. It also introduces crossing-free drawings and coloring a separate conflict graph. The Euler degree test from this week answers the edge question only.