This document describes a college timetable scheduling system that aims to optimally allocate resources by preventing overlaps. It uses graph coloring algorithms to model the scheduling problem. Sets, relations and constraints are defined mathematically. Cards representing timetable events are converted to a graph, with collisions as edges. A graph coloring algorithm is applied to assign timeslots, coloring adjacent vertices differently. The purpose is to simulate real-world scheduling as a mathematical problem to find optimal solutions, as manual scheduling of large timetables is tedious. References on discrete mathematics, graph coloring applications are also provided.

1. College Timetable Scheduling
System
Shardul Kulkarni
shardulind@gmail.com

2. Introduction

3. Problem Statement
To schedule timetable for a institution with considering proper constraints so as to generate a
timetable with optimal distribution of resources.
Objective
College timetable scheduling system should give output in terms of timeslot, where no
resources must get overlapped on a single timeslot.

4. Usecase Diagram

5. Class Diagram

6. Object Diagram

7. State Diagram

8. Deployment Diagram
Keyboard
Mouse
Monitor

9. Mathematical Modeling
SETS
1) F denotes Set of all Faculty. f 𝟄 F , where f is a Faculty
2) C denotes Set of all Courses offered by Department D
c 𝟄 C, where c is a Course and d 𝟄 D, where d is a Department
3) R denotes Set of all Classrooms r 𝟄 R, where r is a Classroom
4) B denotes Set of all Blocks. b 𝟄 B , where b is a Block
Relations:
1) R1 = {(f,d) | f 𝟄 F , d 𝟄 D , where f is faculty of department d}
2) R2 = {(c,d) | c 𝟄 C, d 𝟄 D, where c is course offered by department d}
3) R3 = {(c,f) | c 𝟄 C, f 𝟄 F, where c is course taught by faculty f}
4) R4 = {(b,d) | b 𝟄 B, d 𝟄 D, where block b belongs to department d}

10. Constraints:
1) Multiple Cards must not have students from same block in same timeslot.
2) Multiple Cards must not have same lecturer in same timeslot.
3) Multiple Cards must not have arranged lectures in same classroom on same timeslot.

11. The data about the cards is converted into Graph.
A graph G = (V,E) is undirected graph , where v ε V and represents a Card.
e ε E and represents the collision occurring between two Cards as per above mentioned constraints.
Adjacency Matrix ( 2D array can be used to represent the Matrix)
Algorithm:
1) Creating the Graph G based on stored data about Cards.
2) Give Color A to the vertex with having highest degree.
3) Give Color A to all the Non-adjacent and Uncoloured vertices.
4) Give another color (say Color B) to the next vertex with following highest degree.
5) Give that color to all Non-adjacent and Uncoloured vertices of newly selected vertex.
6) Repeat 4 and 5 till all vertices are coloured.
7) End

12. Sr.
No
Cards Conflicting Cards
1 A K,C,F
2 B K,D,E
3 C B,A,J,I,K,F
4 D B,J,K,F
5 E B,K
6 F D,C,K,J,A,G
7 G F,K,J,I
8 H K
9 I G,K,C
10 J C,D,F,G
11 K A,B,C,D,E,F,G,H,I
Example

14. CONCLUSION
With the help of Graph Coloring problem, a local algorithm is designed which considers few
constraints and suggests the time-slots for generating the Time Table. The sole purpose of
Discrete Structure and Graph theory is to transcend the real life in the mathematical notions
so as to simulate the real life situation and find the optimal solution. Time table scheduling
is tedious job if done manually on large scale, the Graph coloring problem simplifies the
solution and helps to provide optimal solution. Using the graph coloring a simple set of time
slots can be defined.

15. REFERENCES
Discrete Mathematics and its application by Kenneth Rosen
https://pdfs.semanticscholar.org/5df9/1356015f0f29234c1934e449820c06128b7c.pdf
https://www.geeksforgeeks.org/graph-coloring-applications/