Graph coloring is a fundamental problem in graph theory with farāreaching applications. Given a graph , the objective is to assign a color to each vertex such that no two adjacent vertices share the same color. The minimum number of colors needed for such an assignment is the chromatic number . Our calculator accepts a fiveāvertex graph in the form of an adjacency matrix and employs a backtracking search to discover a coloring using the least number of colors. The small graph size keeps the algorithm tractable while still demonstrating the complexity of the general problem, which is NPācomplete for arbitrary graphs.
The adjacency matrix of a simple graph is a square matrix where each entry equals if an edge connects vertex and vertex , and otherwise. Because the graph is undirected, is symmetric and diagonal entries are zero. The calculator reads the matrix, stores it in a twoādimensional array, and then tries to color the vertices using one color, then two colors, and so forth, until a valid coloring is found. For each attempted number of colors , we attempt to assign a color to every vertex using a recursive backtracking routine.
The core constraint that defines a proper coloring is expressed with the
logical condition
that no edge
has endpoints of the same color:
. Programmatically we check this rule in a helper function
isSafe. If a conflict is detected, the algorithm tries the
next available color. When all vertices have been assigned without
conflicts, the recursion unwinds and we report both the chromatic number
and the color of each vertex.
Consider a simple example. Suppose we enter the matrix shown below.
| 0 | 1 | 1 | 0 | 0 |
| 1 | 0 | 1 | 1 | 0 |
| 1 | 1 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 1 |
| 0 | 0 | 1 | 1 | 0 |
This structure is the adjacency matrix of a fiveāvertex line graph with one additional diagonal connection. The algorithm first tries one color and fails because adjacent vertices must differ. With two colors, backtracking explores several partial assignments but ultimately fails again because a triangle subgraph requires at least three colors. Finally, using three colors it finds the assignment , demonstrating that the chromatic number is three.
Backtracking is exhaustive, so its performance suffers as graphs grow. The algorithm runs in time in the worst case, where is the number of vertices and the number of colors. This exponential growth explains why graph coloring has inspired many heuristics: greedy strategies, DSATUR (degree of saturation), simulated annealing, and evolutionary approaches. For small graphs, backtracking is fine and guarantees an optimal solution, making it perfect for educational purposes.
Graph coloring is not just an abstract puzzle. It models scheduling problems where tasks that share a resource must be assigned different time slots, register allocation in compilers where variables that interfere cannot share a register, and frequency assignment where nearby transmitters must avoid using the same channel. Map coloring, which inspired the famous Four Color Theorem, is a planar case of graph coloring where regions of a map correspond to vertices and shared borders correspond to edges. Our calculator can illustrate such scenarios by encoding adjacency information.
From a theoretical standpoint, the chromatic number is linked with many other graph properties. Brooksā theorem provides a bound for most graphs, where is the maximum degree. The chromatic polynomial counts the number of distinct ācolorings and is itself a rich object that satisfies deletionācontraction recurrences.
Because we limit our calculator to five vertices, its interface is straightforward. Each matrix entry can be set to 1 or 0. The algorithm assumes the matrix is symmetric; asymmetries are treated as directed edges but the coloring condition still checks both directions. For convenience, diagonal entries default to zero. The output color labels are numeric, but you can interpret them as actual colors when applying the result to a real problem. If no edges are present, the chromatic number is one, because all vertices can share the same color.
The backtracking routine implemented in the script follows a standard template: choose a vertex, attempt colors sequentially, and recursively proceed. If a contradiction arises later, the algorithm backtracksāhence the nameāand tries the next color. The small size allows the algorithm to be implemented in a few lines of JavaScript while still capturing the essence of general constraint satisfaction problems. This technique is akin to solving Sudoku puzzles, where backtracking explores possible digits for each empty cell.
For a broader perspective, graph coloring intersects with linear algebra and combinatorics. For example, the adjacency matrix can be used to study eigenvalues that bound the chromatic number through Hoffmanās inequality. Researchers investigate random graphs to understand expected chromatic numbers, and probabilistic methods reveal surprising results. There are also variants like list coloring, where each vertex has its own allowable colors, and edge coloring, where edges rather than vertices must differ. Our calculator focuses on the classical vertex coloring problem but the underlying framework could be adapted.
To practice, try encoding a cycle graph by placing ones on the superā and subādiagonals and wrapping around. You will find that it requires three colors, as bipartite graphs have chromatic number two while odd cycles break bipartiteness. Experiments like these help develop intuition about graph structure. Although the calculator targets small graphs, understanding the process builds foundations for tackling larger problems with specialized software or optimization packages.
In summary, this tool offers a handsāon way to explore graph coloring. It emphasizes the definition via adjacency matrices, the search for the minimal number of colors, and the algorithmic challenge underlying the problem. By experimenting with different graphs, observing the recursive search, and reflecting on the theory described here, students and practitioners gain deeper appreciation for the richness of graph theory and its many realāworld applications.
After computing a coloring, consider saving the adjacency matrix and output in a notebook. Building a library of example graphs and their chromatic numbers can aid study sessions or serve as benchmarks when testing more advanced algorithms.
