Graph Coloring Calculator

Introduction

Graph coloring sounds visual, but the heart of the idea is really about conflict management. A graph has vertices and edges. If two vertices are joined by an edge, they represent items that cannot share the same resource. A proper coloring gives each vertex a label so that neighboring vertices always get different labels. The smallest number of labels that works is the graph's chromatic number. This calculator solves that exact problem for one common teaching size: a graph with five vertices entered as a 5×5 adjacency matrix.

Because the graph is small, the page can do something very useful: it does not guess. It uses an exact search, so the number it returns is not a heuristic or approximation. If the page says the chromatic number is 3, that means the graph cannot be colored with 1 or 2 colors, but it can be colored with 3. That direct connection between definition and result makes this calculator a good bridge between graph theory notation and the way the concept behaves in actual examples.

Formally, the graph is written as $G=\left(V,E\right)$, and the quantity we want is the chromatic number $\chi \left(G\right)$. Even though the page only handles five vertices, you can still model many important families: empty graphs, complete graphs, paths, stars, odd cycles, and small dense graphs containing triangles.

How to Use

The workflow is simple, but it helps to read the matrix carefully. Each box in the editor corresponds to a pair of vertices. A value of 1 means there is an edge between those two vertices, and a value of 0 means there is not. After entering the matrix, click Color Graph. The result area will show the chromatic number first, followed by a table telling you which color label was assigned to each vertex.

1. Decide which graph you want to test, using vertices numbered 0 through 4.
2. Enter the graph as a 5×5 adjacency matrix with 0s and 1s.
3. Keep the diagonal at 0 and mirror entries across the diagonal for an undirected graph.
4. Click Color Graph to see the minimum number of colors and one valid assignment.

The result labels are just category numbers such as 1, 2, and 3. They are not literal paint colors. In an application, you might reinterpret them as time slots, radio channels, seating groups, or register numbers. The important rule is always the same: adjacent vertices must have different labels.

How to enter the graph (adjacency matrix)

The calculator reads a matrix $A$ where each entry $a_{ij}$ is 1 if there is an edge between vertex i and vertex j, and 0 otherwise. For a simple undirected graph, three habits keep the input meaningful and easy to interpret.

• Diagonal entries should be 0, because a simple graph has no self-loops: a00 = a11 = a22 = a33 = a44 = 0.
• Symmetry should hold, so aij = aji. If you enter a13 = 1, then a31 should also be 1.
• Only 0 and 1 are intended. Empty fields are interpreted as 0 by the script, but clean 0/1 input is best.

If your matrix is not symmetric, the code still runs. In that case, the safety test effectively treats the row being checked as a directed constraint description for that vertex. For most graph-coloring exercises, that is not what you want, so it is worth pausing for a quick symmetry check before you interpret the answer.

Formula and backtracking method

The defining rule of a proper coloring can be written in a compact way. A coloring is a function $c:V\to \left(1,,,2,,,\dots ,,,k\right)$ that assigns one of $k$ color labels to each vertex. The assignment is valid only if every edge $(u,v)$ connects two vertices with different labels, meaning $c_u\ne c_v$. The chromatic number is the minimum $k$ for which such a coloring exists.

That is the formula-level definition. The page turns it into a concrete search. It first tries to color the graph with one color. If that fails, it tries two colors, then three, and so on up to five. For each candidate value of $k$, it assigns colors to vertices in order from 0 to 4. At each step, it checks whether the proposed color would clash with any already colored adjacent vertex. If the choice is safe, it moves forward. If the choice leads to a dead end later, the algorithm undoes it and tries the next color. That cycle of trying, recursing, and undoing is backtracking.

Algorithm overview (backtracking search)

In the implementation, the helper isSafe(v, color, adj, colors) checks whether assigning a candidate color to vertex v would duplicate the color of any adjacent vertex already placed in the partial solution. The recursive routine assign(v, maxColors, adj, colors) then tries each color label from 1 through maxColors. If a label works, the function recurses on the next vertex. If the deeper call fails, the choice is erased and the next label is tested. Once all five vertices are assigned successfully, the search stops and the current color vector is a valid coloring.

This method matters pedagogically because it mirrors the definition of chromatic number. The page does not merely produce a coloring; it proves minimality by checking smaller color counts first. That is why the displayed chromatic number is trustworthy for the graph you entered.

Worked example (triangle forces 3 colors)

A good first test is a graph that contains a triangle. Any triangle already forces at least three colors, because each of the three vertices touches the other two. Enter the following matrix and click Color Graph. The page will report a chromatic number of 3.

One valid output coloring for that graph is $\left(1,2,3,1,2\right)$. That means vertex 0 has color 1, vertex 1 has color 2, vertex 2 has color 3, vertex 3 has color 1, and vertex 4 has color 2. A textbook or another solver may show a different labeling, and that is fine. Colorings are usually not unique; what matters is that adjacent vertices differ and the number of colors is minimal.

What the result means

Once the calculation finishes, read the output in two layers. First, look at the chromatic number. That tells you the minimum number of groups or resources needed. Second, look at the per-vertex table. That table shows one specific way to realize the minimum. If you are checking a hand-drawn graph, this second part is especially useful because it lets you compare the algorithm's structure with your own attempted coloring.

More intuition: common graph families you can test

Trying a few standard families is the fastest way to build intuition. The five-vertex limit is not a weakness here; it is a teaching advantage, because you can switch between examples quickly and see how one extra edge changes the answer.

• No edges (empty graph): every off-diagonal entry is 0, so all vertices can share one color and χ(G) = 1.
• Complete graph K5: every off-diagonal entry is 1, so each vertex conflicts with all the others and χ(G) = 5.
• Path graph P5: connect 0–1–2–3–4 and keep symmetry. Paths are bipartite, so χ(G) = 2.
• Cycle graph C5: connect 0–1–2–3–4–0. Because the cycle length is odd, two colors fail and χ(G) = 3.
• Complete bipartite K2,3: split the vertices into {0,1} and {2,3,4}, then connect every cross pair. Bipartite graphs need only two colors, so χ(G) = 2.

Assumptions, limitations, and interpretation

• Fixed size: this interface is built specifically for 5 vertices. The script reads a00 through a44.
• Input values: use 0 or 1 for clean interpretation. Empty cells are treated as 0.
• Undirected graphs: standard vertex coloring assumes a symmetric adjacency matrix.
• Color labels: the output labels are abstract categories, not literal hues.
• Optimality: because the algorithm tries smaller values first, the first successful color count is the chromatic number for the entered graph.

Why graph coloring matters

The reason graph coloring appears in so many courses is that it converts real-world exclusion rules into a single clean model. In scheduling, two exams that share students should not happen at the same time. In wireless networking, nearby transmitters should not use the same frequency. In compiler design, variables that are simultaneously live should not share a register. Each situation can be represented by a graph, and a coloring assigns limited resources so that conflicts are avoided.

This page uses a deliberately small instance size, but the underlying problem is famous precisely because it becomes difficult on larger graphs. That contrast is helpful. With five vertices, you can see the structure clearly and verify exact answers instantly. Once that is comfortable, it becomes easier to understand why larger problems often rely on greedy strategies, DSATUR-style heuristics, integer programming, or specialized solvers instead of plain exhaustive search.

Tips for getting reliable results

If a result surprises you, check the matrix before assuming the algorithm is wrong. A single missing mirrored entry can change an undirected graph into an asymmetric input. A mistaken diagonal 1 can introduce a self-loop that falls outside the usual simple-graph assumptions discussed above. It also helps to sketch the graph from the matrix and compare each edge one by one. That small habit catches most data-entry errors quickly.

Also remember that the output is only one valid coloring. A solution key may use a different numbering scheme while still describing the same partition into compatible groups. Swapping labels 1 and 2 does not change the mathematical quality of the coloring. The real checks are adjacency and minimum color count.